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\title[Polynomial growth and subgroups of $\Out(F_{N})$]{Polynomial growth and subgroups of $\Out(F_{N})$}
\alttitle{Croissance polynomiale et sous-groupes de $\Out(F_N)$}


\subjclass{20E05, 20E08, 20E36, 20F65}
\keywords{Nonabelian free groups, outer automorphism groups, space of currents, group actions on trees}


\author[\initial{Y.} \lastname{Guerch }]{\firstname{Yassine} \lastname{Guerch}}
\address{Laboratoire de mathématique\\
d'Orsay UMR 8628\\
CNRS Université Paris-Saclay\\
91405 ORSAY Cedex, France}
\email{yassine.guerch@ens-lyon.fr}



\begin{abstract}
This paper, which is the last of a series of three papers, studies dynamical properties of elements of $\Out(F_{N})$, the outer automorphism group of a nonabelian free group $F_{N}$. We prove that, for every subgroup $H$ of $\Out(F_{N})$, there exists an element $\phi \in H$ such that, for every element $g$ of $F_{N}$, the conjugacy class $[g]$ has polynomial growth under iteration of $\phi$ if and only if $[g]$ has polynomial growth under iteration of every element of $H$.
\end{abstract}

\begin{altabstract}
Dans cet article, nous étudions des propriétés dynamiques des éléments de $\Out(F_N)$, le groupe des automorphismes extérieurs d'un groupe non abélien libre $F_N$ de rang $N \geq 2$. Nous montrons que, pour tout sous-groupe $H$ de $\Out(F_N)$, il existe un élément $\phi \in H$, appelé \emph{dynamiquement générique}, qui capture la croissance polynomiale de $H$ au sens suivant. La classe de conjugaison d'un élément $g \in F_N$ est à croissance polynomiale sous itération de tous les éléments de $H$ si, et seulement si, la classe de conjugaison de $g$ est à croissance polynomiale sous itération de $\phi$.
\end{altabstract}

\datereceived{2022-04-19}
\daterevised{2023-01-06}
\dateaccepted{2023-03-17}

\editors{X. Caruso and V. Guirardel}
\begin{DefTralics}
\newcommand{\Out}{\mathrm{Out}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\dateposted{2023-10-02}
\begin{document}

\maketitle

\section{Introduction}\label{Section Introduction}

Let $N \geq 2$. This paper, which is the last of a series of three papers~\cite{Guerch2021currents,Guerch2021NorthSouth}, studies the exponential growth of elements in $\Out(F_{N})$. An outer automorphism $\phi \in \Out(F_{N})$ is \emph{exponentially growing} if there exist a conjugacy class $[g] \subseteq F_{N}$, a free basis $\frkB$ of $F_{N}$ and a constant $K>0$ such that, for every $m \in \NN^*$, we have
\begin{equation}\label{Equation intro}
\ell_{\frkB}(\phi^m([g])) \geq e^{Km},
\end{equation}
where $\ell_{\frkB}(\phi^m([g]))$ denotes the length of a cyclically reduced representative of $\phi^m([g])$ in the basis $\frkB$.

If $g \in F_{N}$ satisfies Equation~\eqref{Equation intro}, then $g$ is said to be \emph{exponentially growing under iteration of $\phi$}. Otherwise, one can show, using for instance the technology of relative train tracks introduced by Bestvina and Handel~\cite{BesHan92}, that $g$ has \emph{polynomial growth under iteration of $\phi$}, replacing $\geq e^{Km}$ by $\leq (m+1)^K$ in Equation~\eqref{Equation intro} (see also~\cite{Levitt09} for a complete description of all growth types that can occur under iteration of an outer automorphism $\phi$).

We denote by $\Poly(\phi)$ the set of elements of $F_{N}$ which have polynomial growth under iteration of $\phi$. If $H$ is a subgroup of $F_{N}$, we set $\Poly(H)=\bigcap_{\phi\,\in\,H} \Poly(\phi)$. Note that $\Poly(\phi)$ and $\Poly(H)$ are invariant under conjugation. In this article, we prove the following theorem.

\begin{theo}\label{Theo intro 1}
Let ${N} \geq 2$ and let $H$ be a subgroup of $\Out(F_{N})$. There exists $\phi \in H$ such that $\Poly(\phi)=\Poly(H)$.
\end{theo}

In other words, there exists an element of $H$ which encaptures all the exponential growth of $H$: there exists $\phi \in H$ such that if $g \in F_{N}$ has exponential growth for some element of $H$, then $g$ has exponential growth for $\phi$.

Theorem~\ref{Theo intro 1} has analogues in other contexts. For instance, one has a similar result in the context of the mapping class group of a closed, connected, orientable surface $S$ equipped with a hyperbolic structure. Indeed, a consequence of the Nielsen--Thurston classification (see for instance~\cite[Theorem~13.2]{FarMar12}) and the work of Thurston~\cite[Proposition~9.21]{FatLauPoe79} is that the growth of the length of the geodesic representative of the homotopy class of an essential closed curve under iteration of an element of $\Mod(S)$ is either exponential or linear. Moreover, linear growth comes from twists about essential curves while exponential growth comes from pseudo-Anosov homeomorphisms of subsurfaces of $S$.

In~\cite{Ivanov92} (see also the work of McCarthy~\cite{McCarthy85}), Ivanov proved that, for every subgroup $H$ of $\Mod(S)$, up to taking a finite index subgroup of $H$, there exist finitely many homotopy classes of pairwise disjoint essential closed curves $C_1,\ldots,C_k$ elementwise fixed by $H$ and such that, for every connected component $S'$ of $S-\bigcup_{i=1}^k C_i$, the restriction \mbox{$H|_{S'} \subseteq \Mod(S')$} is either the trivial group or contains a pseudo-Anosov element. One can then construct an element $f \in H$ such that the element $f|_{S'} \in \Mod(S')$ is a pseudo-Anosov whenever $H|_{S'} \subseteq \Mod(S')$ contains a pseudo-Anosov element.

In the context of $\Out(F_{N})$, Clay and Uyanik~\cite{clay2019atoroidal} proved Theorem~\ref{Theo intro 1} when $H$ is a subgroup of $\Out(F_{N})$ such that $\Poly(H)=\{1\}$. Indeed, by a result of Levitt~\cite[Proposition~1.4, Lemma~1.5]{Levitt09}, if $\phi \in \Out(F_N)$ and if $\Poly(\phi) \neq\{1\}$, there exist a nontrivial element $g \in F_{N}$ and $k \in \NN^*$ such that $\phi^k([g])=[g]$. In this context, Clay and Uyanik proved that, if $H$ does not virtually preserve the conjugacy class of a nontrivial element of $F_{N}$, there exists an element $\phi \in H$ which is \emph{atoroidal}: no power of $\phi$ fixes the conjugacy class of a nontrivial element of $F_{N}$.

\begin{proof}
We now sketch the proof of Theorem~\ref{Theo intro 1}. It is inspired by the proof of~\cite[Theorem~A]{clay2019atoroidal}. However, technical difficulties emerge due to the presence of elements of $F_{N}$ with polynomial growth under iteration of elements of the considered subgroup of $\Out(F_{N})$. The main difficulties are dealt with in the second article of the series~\cite{Guerch2021NorthSouth}. Let $H$ be a subgroup of $\Out(F_{N})$. We first consider $H$-invariant \emph{free factor systems} $\clF$ of $F_{N}$, that is, $\clF=\{[A_1],\,\ldots,\,[A_k]\}$, where, for every $i \in \{1\ldots,\,k\}$, $[A_i]$ is the conjugacy class of a subgroup $A_i$ of $F_{N}$ and there exists a subgroup $B$ of $F_{N}$ such that $F_{N}=A_1 \ast \ldots \ast A_k \ast B$. There exists a partial order on the set of free factor systems of $F_{N}$, where $\clF_1 \leq \clF_2$ if for every free factor $A_1$ of $F_{N}$ such that $[A_1] \in \clF_1$, there exists a free factor $A_2$ of $F_{N}$ such that $[A_2] \in \clF_2$ and $A_1$ is a subgroup of $A_2$. Hence we may consider a maximal $H$-invariant sequence of free factor systems
\[
\varnothing=\clF_0 \leq \clF_1 \leq\ldots \leq \clF_k=\{[F_{N}]\}.
\]

The proof is now by induction on $i \in \{1,\,\ldots,\,k\}$: for every $i \in \{0\ldots,\,k\}$, we construct an element $\phi_i \in H$ such that $\Poly(\phi_i|_{\clF_i})=\Poly(H|_{\clF_i})$ (we define the meaning of the restrictions in Section~\ref{Section currents associated automorphism}). Let $i \in \{1,\,\ldots,\,k\}$ and suppose that we have constructed $\phi_{i-1}$. There are two cases to consider. If the extension $\clF_{i-1} \leq \clF_i$ is \emph{nonsporadic} (see the definition in~Section~\ref{Subsection malnormal}) then the construction of $\phi_i$ from $\phi_{i-1}$ follows from the works of Handel--Mosher~\cite{HandelMosher20}, Guirardel--Horbez~\cite{Guirardelhorbez19} and Clay--Uyanik~\cite{ClayUya2018}.

If the extension $\clF_{i-1} \leq \clF_i$ is \emph{sporadic}, the construction of $\phi_i$ relies on the action of $H$ on some natural (compact, metrizable) space that we introduced in~\cite{Guerch2021currents}. This space is called the \emph{space of currents relative to $\Poly(H|_{\clF_{i-1}})$} and it is denoted by $\PCurr(F_{N},\Poly(H|_{\clF_{i-1}}))$. It is defined as a subspace of the space of Radon measures on a natural space $\partial^2(F_{N},\Poly(H|_{\clF_{i-1}}))$, the double boundary of $F_{N}$ relative to $\Poly(H|_{\clF_{i-1}})$ (see Section~\ref{Section relative currents} for precise definitions).

In~\cite{Guerch2021NorthSouth}, we proved that the element $\phi_{i-1}$ that we have constructed acts with a \emph{North-South dynamics} on the space of relative currents $\PCurr(F_{N},\Poly(H|_{\clF_{i-1}}))$: there exist two proper disjoint closed subsets of $\PCurr(F_{N},\Poly(H|_{\clF_{i-1}}))$ such that~every point of $\PCurr(F_{N},\Poly(H|_{\clF_{i-1}}))$ which is not contained in these subsets converges to one of the two subsets under positive or negative iteration of $\phi_{i-1}$. This North-South dynamics result allows us, applying classical ping-pong arguments similar to the one of Tits~\cite{Tits72}, to construct the element $\phi_i \in H$ such that $\Poly(\phi_i|_{\clF_i})=\Poly(H|_{\clF_i})$, which concludes the proof.
\end{proof}

The element constructed in Theorem~\ref{Theo intro 1} is in general not unique. Indeed, when the subgroup $H$ of $\Out(F_{N})$ is such that $\Poly(H)=\{1\}$, Clay and Uyanik~\cite[Theorem~B]{clay2019atoroidal} give necessary and sufficient conditions for $H$ to contain a nonabelian free subgroup consisting in atoroidal elements.

We now outline some consequences of Theorem~\ref{Theo intro 1}. The first one is a result concerning the periodic subset of a subgroup of $\Out(F_{N})$. From Clay and Uyanik's theorem cited above, one can ask the following question. Let $H$ be a subgroup of $\Out(F_{N})$. If $H$ is a subgroup of $\Out(F_{N})$ such that $H$ virtually fixes the conjugacy class of a nontrivial subgroup $A$ of $F_{N}$, is it true that either $H$ virtually fixes the conjugacy class of a nontrivial element $g \in F_{N}$ such that $g$ is not contained in a conjugate of $A$, or there exists $\phi \in H$ such that the only conjugacy classes of elements of $F_{N}$ virtually fixed by $\phi$ are contained in a conjugate of $A$?

Unfortunately, such a result is not true. Indeed, let $F_3=\left\langle a,b,c \right\rangle$ be a nonabelian free group of rank $3$. Let $\phi_a$ (resp. $\phi_b$) be the automorphism of $F_3$ which fixes $a$ and $b$ and which sends $c$ to $ca$ (resp. $c$ to $cb$), and let $H=\left\langle [\phi_a],[\phi_b] \right\rangle \subseteq \Out(F_3)$. Then every element $\phi \in H$ has a representative which fixes $\left\langle a,b\right\rangle$ and sends $c$ to $cg_{\phi}$ with $g_{\phi} \in \left\langle a,b \right\rangle$. Thus, $\phi$ fixes the conjugacy class of $g_{\phi}cg_{\phi}c^{-1}$. However, there always exist $\phi' \in H$, such that $\phi'$ does not preserve the conjugacy class of $g_{\phi}cg_{\phi}c^{-1}$.

We denote by $\Per(H)$ the set of conjugacy classes of $F_{N}$ fixed by a power of every element of $H$. In the above example, we constructed a subgroup $H$ of $\Out(F_{N})$ such that $\Per(H)$ contains the conjugacy class of a nonabelian subgroup of rank $2$. This is in fact the lowest possible rank where a generalization of the theorem of Clay and Uyanik using $\Per(H)$ instead of $\Poly(H)$ cannot work, as shown by the following result, which is a consequence of Corollary~\ref{Coro alternative single elements} and Theorem~\ref{Theo intro 1}.

\begin{theo}\label{Theo intro 2}
Let $N \geq 3$ and let $g_1,\,\ldots,\,g_k$ be nontrivial root-free elements of $F_{N}$. Let $H$ be subgroup of $\Out(F_{N})$ such that, for every $i \in \{1,\,\ldots,\,k\}$, every element of $H$ has a power which fixes the conjugacy class of $g_i$. Then one of the following (mutually exclusive) statements holds.
\begin{enumerate}
\item\label{theo1.2.1} There exists $g_{k+1} \in F_{N}$ such that $[\langle g_{k+1}\rangle] \notin \{[\langle g_1\rangle],\,\ldots,\,[\langle g_k\rangle]\}$ and whose conjugacy class is fixed by a power of every element of $H$.

\item\label{theo1.2.2} There exists $\phi \in H$ such that $\Per(\phi)=\{[\langle g_1 \rangle],\,\ldots,\,[\langle g_k \rangle]\}$.
\end{enumerate}
\end{theo}

As proved by Ivanov~\cite{Ivanov92}, Case~\eqref{theo1.2.2} of Theorem~\ref{Theo intro 2} naturally occurs when we are working with a subgroup of a mapping class group of a compact, connected surface $S$ whose fundamental group is identified with $F_{N}$. Finally, in Corollary~\ref{Coro pseudo Anosov subgroup Mod}, we prove a characterization of subgroups of the mapping class group of such a surface $S$ using periodic conjugacy classes.



\section{Preliminaries}


\subsection{Malnormal subgroup systems of \texorpdfstring{$F_{N}$}{FN}}\label{Subsection malnormal}

Let ${N}$ be an integer greater than $1$ and let $F_{N}$ be a free group of rank ${N}$. A \emph{subgroup system of $F_{N}$} is a finite (possibly empty) set $\clA$ whose elements are conjugacy classes of nontrivial (that is distinct from $\{1\}$) finite rank subgroups of $F_{N}$. Note that a subgroup system $\clA$ is completely determined by the set of subgroups $A$ of $F_{N}$ such that $[A] \in \clA$.

There exists a partial order on the set of subgroup systems of $F_{N}$, where $\clA_1 \leq \clA_2$ if for every subgroup $A_1$ of $F_{N}$ such that $[A_1] \in \clA_1$, there exists a subgroup $A_2$ of $F_{N}$ such that $[A_2] \in \clA_2$ and $A_1$ is a subgroup of $A_2$. In this case we say that $\clA_2$ is an \emph{extension} of $\clA_1$.

The \emph{stabilizer in $\Out(F_{N})$ of a subgroup system} $\clA$, denoted by $\Out(F_{N},\clA)$, is the set of all elements $\phi \in \Out(F_{N})$ such that $\phi(\clA)=\clA$. An element of $\Out(F_{N},\clA)$ is called an \emph{outer automorphism relative to $\clA$} or a \emph{relative outer automorphism} if the context is clear. Note that $\phi$ might permute the conjugacy classes of subgroups of $F_N$ contained in $\clA$. If $\clA_1$ and $\clA_2$ are two subgroup systems, we set $\Out(F_{N},\clA_1,\clA_2)=\Out(F_{N},\clA_1) \cap \Out(F_{N},\clA_2)$.

If $\clA$ is a subgroup system of $F_{N}$, we denote by $\Out(F_{N},\clA^{(t)})$ the subgroup of $\Out(F_{N})$ consisting in every element $\phi \in \Out(F_{N})$ such that, for every subgroup $P$ of $F_N$ such that $[P] \in \clA$, there exists $\Phi \in \phi$ such that $\Phi(P)=P$ and $\Phi|_{P}=\id_P$.

Recall that a subgroup $A$ of $F_{N}$ is \emph{malnormal} if for every element $x \in F_{N}-A$, we have $xAx^{-1} \cap A=\{e\}$.

\begin{defi}[Malnormal subgroup system, nonperipheral element]\label{Defi malnormal subgroup system}
Let $\clA$ be a subgroup system of $F_N$.
\begin{enumerate}
\item\label{defi2.1.1} The subgroup system $\clA$ is \emph{malnormal} if every subgroup $A$ of $F_{N}$ such that $[A] \in \clA$ is malnormal and, for all subgroups $A_1,A_2$ of $F_{N}$ such that $[A_1],[A_2] \in \clA$, if $A_1 \cap A_2$ is nontrivial then $A_1=A_2$.

\item\label{defi2.1.2} An element $g \in F_{N}$ is \emph{$\clA$-peripheral} (or simply \emph{peripheral} if there is no ambiguity) if it is trivial or conjugate into one of the subgroups of $\clA$, and \emph{$\clA$-nonperipheral} otherwise.
\end{enumerate}
\end{defi}

An important class of examples of malnormal subgroup systems is given by the \emph{free factor systems}. A \emph{free factor system of $F_{N}$} is a (possibly empty) set $\clF$ of conjugacy classes $\{[A_1],\,\ldots,\,[A_r]\}$ of nontrivial subgroups $A_1,\,\ldots,\,A_r$ of $F_{N}$ such that there exists a subgroup $B$ of $F_{N}$ with $F_{N}=A_1 \ast \ldots \ast A_r \ast B$. An ascending sequence of free factor systems $\clF_1 \leq \ldots \leq \clF_i=\{[F_{N}]\}$ of $F_{N}$ is called a \emph{filtration of $F_{N}$}.

\begin{defi}[Sporadic extension]\label{Defi sporadic}\leavevmode
\begin{enumerate}
\item\label{defi2.2.1} An extension of free factor systems $\clF_1 \leq \clF_2=\{[A_1],\,\ldots,\,[A_k]\}$ of $F_N$ is \emph{sporadic} if there exists $\ell \in \{1,\,\ldots,\,k\}$ such that, for every $j \in \{1,\,\ldots,\,k\}-\{\ell\}$, we have $[A_j] \in \clF_1$ and if one of the following holds:
\begin{enumerate}\alphenumi
\item\label{defi2.2.1.a} there exist subgroups $B_1,B_2$ of $F_{N}$ such that $[B_1],[B_2] \in \clF_1$ and $A_{\ell}=B_1 \ast B_2$;

\item\label{defi2.2.1.b} there exists a subgroup $B$ of $F_{N}$ such that $[B] \in \clF_1$ and $A_{\ell}$ is an HNN extension of $B$ over the trivial group (thus $A_{\ell}$ is isomorphic to $B \ast \ZZ$);
\item\label{defi2.2.1.c} there exists $g \in F_{N}$ such that $\clF_2=\clF_1\cup\{[g]\}$ and $A_{\ell}=\left\langle g \right\rangle$.
\end{enumerate}
Otherwise, the extension $\clF_1 \leq \clF_2$ is \emph{nonsporadic}.

\item\label{defi2.2.2} A free factor system $\clF$ of $F_{N}$ is \emph{sporadic} (resp. \emph{nonsporadic}) if the extension $\clF \leq \{[F_{N}]\}$ is sporadic (resp. nonsporadic).
\end{enumerate}
\end{defi}

Given a free factor system $\clF$ of $F_{N}$, a \emph{free factor of $(F_{N},\clF)$} is a subgroup $A$ of $F_{N}$ such that there exists a free factor system $\clF'$ of $F_{N}$ with $[A] \in \clF'$ and $\clF \leq \clF'$. A free factor of $(F_{N},\clF)$ is \emph{proper} if it is nontrivial, not equal to $F_{N}$ and if its conjugacy class does not belong to $\clF$.

In general, we will work in a finite index subgroup of $\Out(F_{N})$ defined as follows. Let
\[
\IA_{N}(\ZZ/3\ZZ)=\ker\big(\Out(F_{N}) \to \Aut(H_1(F_{N},\ZZ/3\ZZ))\big).
\]
For every $ \phi \in \IA_{N}(\ZZ/3\ZZ)$, we have the following properties:
\begin{enumerate}
\item any $\phi$-periodic conjugacy class of free factor of $F_{N}$ is fixed by $\phi$~\cite[Theorem~II.3.1]{HandelMosher20};

\item any $\phi$-periodic conjugacy class of elements of $F_{N}$ is fixed by $\phi$~\cite[Theorem~II.4.1]{HandelMosher20}.
\end{enumerate}

Another class of examples of malnormal subgroup systems is the following one. Let $g \in F_{N}$ and let $\frkB$ be a free basis of $F_{N}$. The length of the conjugacy class of $g$ with respect to $\frkB$ is
\[
\ell_{\frkB}([g])=\min_{h\,\in\,[g]} \ell_{\frkB}(h),
\]
where $\ell_{\frkB}(h)$ is the word length of $h$ with respect to the basis $\frkB$. An outer automorphism $\phi \in \Out(F_{N})$ is \emph{exponentially growing} if there exists $g \in F_{N}$ such that the length of the conjugacy class $[g]$ of $g$ in $F_{N}$ with respect to some basis of $F_{N}$ grows exponentially fast under positive iteration of $\phi$. One can show that if $g$ is exponentially growing with respect to some free basis of $F_{N}$, then it is exponentially growing for every free basis of $F_{N}$.

If $\phi \in \Out(F_{N})$ is not exponentially growing, one can show, using for instance the technology of train tracks due to Bestvina and Handel~\cite{BesHan92}, that for every $g \in F_{N}$, the conjugacy class $[g]$ has polynomial growth under positive iteration of~$\phi$. In this case, we say that $\phi$ is \emph{polynomially growing}.  
For an automorphism $\alpha \in \Aut(F_{N})$, we say that $\alpha$ is \emph{exponentially growing} if there exists $g \in F_{N}$ such that the word length of $[g]$ grows exponentially fast under iteration of $[\alpha] \in \Out(F_N)$. Otherwise, $\alpha$ is polynomially growing. The polynomial subgroup of $\alpha$ is the subgroup of $F_N$ consisting in all elements $g \in F_N$ whose word length grows polynomially fast under iteration of $\alpha$.

Let $\phi \in \Out(F_{N})$ be exponentially growing. A subgroup $P$ of $F_{N}$ is a \emph{polynomial subgroup} of $\phi$ if there exist $k \in \NN^*$ and a representative $\alpha$ of $\phi^k$ such that $\alpha(P)=P$ and $\alpha|_P$ is polynomially growing. By~\cite[Proposition~1.4]{Levitt09}, there exist finitely many conjugacy classes $[H_1],\,\ldots,\,[H_k]$ of maximal polynomial subgroups of $\phi$. Moreover, the proof of~\cite[Proposition~1.4]{Levitt09} implies that the set $\clH=\{[H_1],\,\ldots,\,[H_k]\}$ is a malnormal subgroup system (see~\cite[Section~2.1]{Guerch2021NorthSouth}). We denote this malnormal subgroup system by $\clA(\phi)$.

Note that, if $H$ is a subgroup of $F_{N}$ such that $[H] \in \clA(\phi)$, there exist $p \in \NN^*$ and $\Phi^{-1} \in \phi^{-1}$ such that $\Phi^{-p}(H)=H$. By for instance~\cite[Theorem~1.1]{BesFeiHan05}, up to taking a larger $p$, the image of $\phi^p$ in $\Out(H)$ preserves a sequence $\clS$ of free factor systems of $H$ such that every extension of the sequence is sporadic. Hence the image of $\phi^{-p}$ in $\Out(H)$ preserves $\clS$. This implies that $H$ is a polynomially growing subgroup of $\phi^{-1}$. Hence we have $\clA(\phi) \leq \clA(\phi^{-1})$. By symmetry, we have
\begin{equation}\label{Equation p 6}
\clA(\phi)=\clA\left(\phi^{-1}\right).
\end{equation}
Moreover, for every element $\psi \in \Out(F_{N})$, we have
\[
\clA\left(\psi\phi\psi^{-1}\right)=\psi(\clA(\phi)).
\]

In order to distinguish between the set of elements of $F_{N}$ which have polynomial growth under positive iteration of $\phi$ and the associated malnormal subgroup system, we will denote by $\Poly(\phi)$ the former. We have $\Poly(\phi)=\Poly(\phi^{-1})$ by Equation~\eqref{Equation p 6}. If $H$ is a subgroup of $\Out(F_{N})$, we set $\Poly(H)=\bigcap_{\phi\,\in\,H} \Poly(\phi)$.

\begin{defi}[Atoroidal, expanding outer automorphism]
Let $\clA$ be a malnormal subgroup system of $F_N$ and let $\phi \in \Out(F_{N},\clA)$ be a relative outer automorphism.
\begin{enumerate}
\item\label{defi2.3.1} The outer automorphism $\phi$ is \emph{atoroidal relative to $\clA$} if, for every $k \in \NN^*$, the element $\phi^k$ does not preserve the conjugacy class of any $\clA$-nonperipheral element.

\item\label{defi2.3.2} The outer automorphism $\phi$ is \emph{expanding relative to $\clA$} if $\clA(\phi) \leq \clA$.
\end{enumerate}
\end{defi}

Note that an expanding outer automorphism relative to $\clA$ is in particular atoroidal relative to $\clA$. When $\clA=\varnothing$, the outer automorphism $\phi$ is expanding relative to $\clA$ if and only if for every nontrivial element $g \in F_{N}$, the length of the conjugacy class $[g]$ of $g$ in $F_{N}$ with respect to some basis of $F_{N}$ grows exponentially fast under iteration of $\phi$. Therefore, using for instance a result of Levitt~\cite[Corollary~1.6]{Levitt09}, the outer automorphism $\phi$ is expanding relative to $\clA=\varnothing$ if and only if $\phi$ is atoroidal relative to $\clA=\varnothing$.


Let $\clA=\{[A_1],\,\ldots,\,[A_r]\}$ be a malnormal subgroup system and let $\clF$ be a free factor system. Let $i \in \{1,\,\ldots,\,r\}$. By for instance~\cite[Theorem~3.14]{ScoWal79} for the action of $A_i$ on one of its Cayley graphs, there exist finitely many subgroups $A_i^{(1)},\ldots,A_i^{(k_i)}$ of $A_i$ such that:

\begin{enumerate}
\item for every $j \in \{1,\,\ldots,\,k_i\}$, there exists a subgroup $B$ of $F_{N}$ such that $[B] \in \clF$ and $A_i^{(j)}=B \cap A_i$;

%\medskip

\item for every subgroup $B$ of $F_{N}$ such that $[B] \in \clF$ and $B \cap A_i \neq \{e\}$, there exists $j \in \{1,\,\ldots,\,k_i\}$ such that $A_i^{(j)}=B \cap A_i$;

%\medskip

\item the subgroup $A_i^{(1)} \ast \ldots \ast A_i^{(k_i)}$ is a free factor of $A_i$.
\end{enumerate}

Thus, one can define a new subgroup system as
\[
\clF \wedge \clA=\bigcup_{i=1}^r\left\{\left[A_i^{(1)}\right],\,\ldots,\,\left[A_i^{(k_i)}\right]\right\}.
\]
Since $\clA$ is malnormal, and since, for every $i \in \{1,\,\ldots,\,r\}$, the group $A_i^{(1)} \ast \ldots \ast A_i^{(k_i)}$ is a free factor of $A_i$, it follows that the subgroup system $\clF \wedge \clA$ is a malnormal subgroup system of $F_{N}$. We call it the \emph{meet of $\clF$ and $\clA$}. If $\phi \in \Out(F_{N},\clF,\clA)$ then $\phi \in \Out(F_{N},\clF\wedge\clA)$.

\subsection{Relative currents}\label{Section relative currents}

In this section, we define the notion of \emph{currents of $F_{N}$ relative to a malnormal subgroup system $\clA$}. The section follows~\cite{Guerch2021currents,Guerch2021NorthSouth} (see the work of Gupta~\cite{gupta2017relative} for the particular case of free factor systems and Guirardel and Horbez~\cite{Guirardelhorbez19laminations} in the context of free products of groups). It can be thought of as a functional space in which densely live the $\clA$-nonperipheral elements of $F_{N}$.

Let $\partial_{\infty}F_{N}$ be the Gromov boundary of $F_{N}$. The \emph{double boundary of $F_{N}$} is the metrisable locally compact, totally disconnected quotient topological space
\[
\partial^2F_{N}=\left(\partial_{\infty} F_{N} \times \partial_{\infty} F_{N} \setminus \Delta \right)/\sim,
\]
where $\sim$ is the equivalence relation generated by the flip relation $(x,y)\sim(y,x)$ and $\Delta$ is the diagonal, endowed with the diagonal action of $F_{N}$. We denote by $\{x,y\}$ the equivalence class of $(x,y)$.

Let $T$ be the Cayley graph of $F_{N}$ with respect to a free basis $\frkB$. The boundary of $T$ is naturally homeomorphic to $\partial_{\infty}F_{N}$ and the set $\partial^2F_{N}$ is then identified with the set of unoriented bi-infinite geodesics in $T$. Let $\gamma$ be a finite geodesic path in~$T$. The path $\gamma$ determines a subset in $\partial^2F_{N}$ called the \emph{cylinder set of $\gamma$}, denoted by $C(\gamma)$, which consists in all unoriented bi-infinite geodesics in $T$ that contain $\gamma$. Such cylinder sets form a basis for the topology on $\partial^2 F_{N}$, and in this topology, the cylinder sets are both open and compact, hence closed (see for instance~\cite[Section~5.4]{Martin95}). The action of $F_{N}$ on $\partial^2F_{N}$ has a dense orbit.

Let $A$ be a nontrivial subgroup of $F_{N}$ of finite rank. The induced $A$-equivariant inclusion $\partial_{\infty} A \hookrightarrow \partial_{\infty} F_{N}$ induces an inclusion $\partial^2 A \hookrightarrow \partial^2 F_{N}$. Let $\clA=\{[A_1],\,\ldots,\,[A_r]\}$ be a malnormal subgroup system. Let
\[
\partial^2\clA= \bigcup_{i=1}^r \bigcup_{g\,\in\, F_{N}} \partial^2 \left(gA_ig^{-1}\right).
\]

\begin{defi}[Relative double boundary]
Let $\clA$ be a malnormal subgroup system. The \emph{double boundary of $F_{N}$ relative to $\clA$} is
\[
\partial^2(F_{N},\clA)=\partial^2F_{N} -\partial^2\clA.
\]
\end{defi}

The double boundary of $F_N$ relative to a malnormal subgroup system is a subset of $\partial^2 F_N$ which is invariant under the action of $F_{N}$ on $\partial^2F_{N}$ and inherits the subspace topology of $\partial^2F_{N}$.


\begin{lemm}[{\cite[Lemmas~2.5, 2.6, 2.7]{Guerch2021currents}}]\label{Lem Properties of relative boundary}
Let ${N} \geq 3$ and let $\clA$ be a malnormal subgroup system of $F_{N}$. The space $\partial^2(F_{N},\clA)$ is an open subspace of $\partial^2 F_{N}$, hence is locally compact, and the action of $F_{N}$ on $\partial^2(F_{N},\clA)$ has a dense orbit.
\end{lemm}

We can now define a \emph{relative current}.

\begin{defi}[relative current]
Let $\clA$ be a malnormal subgroup system of $F_{N}$. A \emph{relative current on $(F_{N},\clA)$} is a (possibly zero) $F_{N}$-invariant nonnegative Radon measure $\mu$ on $\partial^2(F_{N},\clA)$.
\end{defi}

The set $\Curr(F_{N},\clA)$ of all relative currents on $(F_{N},\clA)$ is equipped with the weak-$\ast$ topology: a sequence $(\mu_n)_{n\,\in\,\NN}$ in $\Curr(F_{N},\clA)^{\NN}$ converges to a current $\mu \in \Curr(F_{N},\clA)$ if and only if for every Borel subset $B \subseteq \partial^2(F_{N},\clA)$ such that $\mu(\partial B)=0$ (where $\partial B$ is the topological boundary of $B$), the sequence $(\mu_n(B))_{n\,\in\,\NN}$ converges to~$\mu(B)$.

The group $\Out(F_{N},\clA)$ acts on $\Curr(F_{N},\clA)$ as follows. Let $\phi \in \Out(F_{N},\clA)$ and let $\Phi$ be a representative of $\phi$. The automorphism $\Phi$ acts diagonally by homeomorphisms on $\partial^2F_{N}$. If $\Phi' \in \phi$, then the action of $\Phi'$ on $\partial^2F_{N}$ differs from the action of $\Phi$ by a translation by an element of $F_{N}$. Let $\mu \in \Curr(F_{N},\clA)$ and let $C$ be a Borel subset of $\partial^2(F_{N},\clA)$. Then, since $\phi$ preserves $\clA$, we see that $\Phi^{-1}(C) \in \partial^2(F_{N},\clA)$. Then we~set
\[
\phi(\mu)(C)=\mu\left(\Phi^{-1}(C)\right),
\]
which is well-defined since $\mu$ is $F_{N}$-invariant.

%\bigskip

Every conjugacy class of nonperipheral element $g \in F_{N}$ determines a relative current $\eta_{[g]}$ as follows. Suppose first that $g$ is \emph{root-free}, that is there do not exist $k \geq 2$ and $h \in F_N$ such that $g=h^k$. Let $\gamma$ be a finite geodesic path in the Cayley graph $T$. Then $\eta_{[g]}(C(\gamma))$ is the number of axes in $T$ of conjugates of $g$ that contain the path $\gamma$. By~\cite[Lemma~3.2]{Guerch2021currents}, $\eta_{[g]}$ extends uniquely to a current in $\Curr(F_{N},\clA)$ which we still denote by $\eta_{[g]}$. If $g=h^k$ with $k \geq 2$ and $h$ root-free, we set $\eta_{[g]}=k \;\eta_{[h]}$. Such currents are called \emph{rational currents}.

%\bigskip

Let $\mu \in \Curr(F_{N},\clA)$. The \emph{support of $\mu$}, denoted by $\Supp(\mu)$, is the support of the Borel measure $\mu$ on $\partial^2(F_{N},\clA)$. We recall that $\Supp(\mu)$ is a \emph{lamination of $\partial^2(F_{N},\clA)$}, that is, a closed $F_{N}$-invariant subset of $\partial^2(F_{N},\clA)$.

In the rest of the article, rather than considering the space of relative currents itself, we will consider the set of \emph{projectivized relative currents}, denoted by
\[
\bbP\Curr(F_{N},\clA)=\left(\Curr(F_{N},\clA)-\{0\}\right)/\sim,
\]
where $\mu \sim \nu$ if there exists $\lambda \in \RR_+^*$ such that $\mu=\lambda \nu$. The projective class of a current $\mu \in \Curr(F_{N},\clA)$ will be denoted by $[\mu]$. For every $\phi \in \Out(F_{N},\clA)$, the action $\phi\colon \mu \mapsto \phi(\mu)$ is positively linear. Therefore, the action of $\Out(F_{N},\clA)$ on $\Curr(F_{N},\clA)$ induces an action on $\PCurr(F_{N},\clA)$. We have the following properties.

\begin{lemm}\cite[Lemma~3.3]{Guerch2021currents}\label{Lem PCurr compact}
Let ${N} \geq 3$ and let $\clA$ be a malnormal subgroup system of $F_{N}$. The space $\PCurr(F_{N},\clA)$ is compact.
\end{lemm}


\begin{prop}
[{\cite[Theorem~1.2]{Guerch2021currents}}]\label{Prop density rational currents}
Let ${N} \geq 3$ and let $\clA$ be a malnormal subgroup system of $F_{N}$. The set of projectivised rational currents associated with nonperipheral elements of $F_{N}$ is dense in $\PCurr(F_{N},\clA)$.
\end{prop}

\subsection{Currents associated with an almost atoroidal outer automorphism of \texorpdfstring{$F_{N}$}{FN}}\label{Section currents associated automorphism}

Let ${N} \geq 3$ and let $\clF=\{[A_1],\ldots,[A_k]\}$ be a free factor system of $F_{N}$. If $\phi \in \IA_{N}(\ZZ/3\ZZ)$ preserves $\clF$, we denote by
\begin{equation}\label{Equation defi phi i}
\phi|_{\clF}=\left([\Phi_1|_{A_1}],\,\ldots,\,[\Phi_k|_{A_k}]\right) \in \prod_{i=1}^k \Out(A_i)
\end{equation}
where, for every $i \in \{1,\,\ldots,\,k\}$, the element $\Phi_i$ is a representative of $\phi$ such that $\Phi_i(A_i)=A_i$. Note that the outer class of $\Phi_i|_{A_i}$ in $\Out(A_i)$ does not depend on the choice of $\Phi_i$ since $A_i$ is a malnormal subgroup of $F_{N}$. Hence, for every $i \in \{1,\,\ldots,\,k\}$, we can naturally associate to $\phi$ the outer automorphism $[\Phi_i|_{A_i}] \in \Out(A_i)$ as in Equation~\eqref{Equation defi phi i}, and this notation will be used from now on.

Note that, for every $i \in \{1,\,\ldots,\,k\}$, the element $[\Phi_i|_{A_i}]$ is expanding relative to the free factor system $\clF \wedge \{[A_i]\}=\{[A_i]\}$, without additional assumption on $\phi$. We will say that \emph{$\phi|_{\clF}$ is expanding relative to $\clF$}.

Let
\[
\Poly(\phi|_{\clF})=\bigcup_{i=1}^k\bigcup_{g\,\in\,F_{N}} g\Poly([\Phi_i|_{A_i}])g^{-1} \subseteq F_{N}.
\]
If $H$ is a subgroup of $\IA_{N}(\ZZ/3\ZZ)$ which preserves $\clF$, we set
\[
\Poly(H|_\clF)=\bigcap_{\phi\,\in\,H} \Poly(\phi|_{\clF}).
\]

We now define a class of outer automorphisms of $F_{N}$ which we will study in the rest of the article.


\begin{defi}[Almost atoroidal]\label{Defi almost atoroidal outer automorphism}
Let ${N} \geq 3$ and let $\clF$ be a free factor system of $F_{N}$. Let $\phi \in \IA_{N}(\ZZ/3\ZZ)$ be an outer automorphism preserving $\clF$. The outer automorphism $\phi$ is \emph{almost atoroidal relative to} $\clF$ if $\Poly(\phi) \neq \{[F_{N}]\}$ and if $\phi$ is an atoroidal outer automorphism relative to $\clF$ whenever the extension $\clF \leq \{[F_{N}]\}$ is nonsporadic.
\end{defi}

Note that, if $\clF$ is a sporadic free factor system, then $\phi \in \IA_{N}(\ZZ/3\ZZ) \cap \Out(F_N,\clF)$ is almost atoroidal relative to $\clF$ if and only if $\Poly(\phi) \neq \{[F_{N}]\}$. Definition~\ref{Defi almost atoroidal outer automorphism} is a subcase of a larger definition of almost atoroidality studied in~\cite[Definition~4.3]{Guerch2021NorthSouth}.

Let $\clF \leq \clF_1=\{[A_1],\ldots,[A_k]\}$ be two free factor systems of $F_{N}$. Let $\phi$ be an element of $\IA_{N}(\ZZ/3\ZZ) \cap \Out(F_N,\clF,\clF_1)$. We say that $\phi|_{\clF_1}$ is \emph{almost atoroidal relative to $\clF$} if, for every $i \in \{1,\,\ldots,\,k\}$, the outer automorphism $[\Phi_i|_{A_i}]$ defined in Equation~\eqref{Equation defi phi i} is almost atoroidal relative to $\clF \wedge \{[A_i]\}$.


Let $\phi \in \IA_{N}(\ZZ/3\ZZ)$ be an almost atoroidal outer automorphism relative to $\clF$. We now recall from~\cite{Guerch2021NorthSouth} the definition and some properties of some subsets of the space $\PCurr(F_{N},\clF \wedge \clA(\phi))$ associated with $\phi$.


\begin{defi}[Polynomially growing currents]
Let ${N} \geq 3$ and let $\clF$ be a free factor system of $F_{N}$. Let $\phi \in \IA_{N}(\ZZ/3\ZZ) \cap \Out(F_N,\clF)$ be an almost atoroidal outer automorphism relative to $\clF$. The \emph{space of polynomially growing currents associated with $\phi$}, denoted by $K_{PG}(\phi)$, is the subspace of all currents in $\PCurr(F_{N},\clF \wedge \clA(\phi))$ whose support is contained in $\partial^2\clA(\phi) \cap \partial^2(F_{N},\clF \wedge \clA(\phi))$.
\end{defi}


We will need the following result which gives the existence and properties of an approximation of the length function of the conjugacy class of an element of $F_{N}$ in the context of the space of currents.

\begin{prop}[{\cite[Lemma~3.27, Lemma~3.28$\MK$(3)]{Guerch2021NorthSouth}}]\label{Prop moving exponential1}
Let ${N} \geq 3$ and let $\clF$ be a sporadic free factor system of $F_{N}$. Let $\phi \in \Out(F_{N},\clF)$ be an almost atoroidal outer automorphism relative to $\clF$. There exists a continuous, positively linear function
\[
\lVert. \rVert_{\clF} \colon \Curr(F_N,\clF \wedge \clA(\phi)) \to \RR_+
\]
such that the following holds.

\begin{enumerate}
\item\label{prop2.11.1} There exist a basis $\frkB$ of $F_{N}$ and a constant $C\geq 1$ such that, for every $\clF \wedge \clA(\phi)$-nonperipheral element $g \in F_{N}$, we have $\lVert \eta_{[g]} \rVert_{\clF} \in \NN^*$ and
\[
\ell_{\frkB}([g]) \geq C\;\left\lVert \eta_{[g]} \right\rVert_{\clF}.
\]

%\medskip

\item\label{prop2.11.2} For every $\eta \in \Curr(F_{N},\clF\wedge\clA(\phi))$, if $\lVert \eta \rVert_{\clF}=0$, then $\eta=0$.
\end{enumerate}
\end{prop}

\begin{prop}[{\cite[Propositions~4.4, 4.12, 5.24]{Guerch2021NorthSouth}}]\label{Prop Existence and properties of Delta}
Let ${N} \geq 3$ and let $\clF$ be a sporadic free factor system of $F_{N}$ ($\clF$ might be equal to $\{[F_N]\}$). Let $\phi \in \IA_{N}(\ZZ/3\ZZ)$ be an almost atoroidal outer automorphism relative to $\clF$. There exist two unique proper compact $\phi$-invariant subsets $\Delta_{\pm}(\phi)$ of $\PCurr(F_{N},\clF \wedge \clA(\phi))$ such that the following assertions hold.

\begin{enumerate}
\item\label{prop2.12.1} For every $[\mu] \in \Delta_+(\phi) \cup \Delta_-(\phi)$, the support of $\mu$ is contained in $\partial^2\clF$.

\item\label{prop2.12.2} Let $U_+$ be a neighborhood of $\Delta_+(\phi)$, let $U_-$ be a neighborhood of $\Delta_-(\phi)$, let $V$ be a neighborhood of $K_{PG}(\phi)$. There exists $N \in \NN^*$ such that for every $n \geq 1$ and every ($\clF \wedge \clA(\phi)$)-nonperipheral $w \in F_{N}$ such that $\eta_{[w]} \notin V$, one of the following holds
\[
\phi^{Nn}(\eta_{[w]}) \in U_+ \quad\text{or}\quad \phi^{-Nn}(\eta_{[w]}) \in U_-.
\]
\end{enumerate}
\end{prop}

The subsets $\Delta_+(\phi)$ and $\Delta_-(\phi)$ are called the \emph{simplices of attraction and repulsion of $\phi$}.

Let $\clF \leq \clF_1=\{[A_1],\,\ldots,\,[A_k]\}$ be a sporadic extension of two free factor systems of $F_{N}$. Let $\phi$ be an element of $\IA_{N}(\ZZ/3\ZZ) \cap \Out(F_N,\clF,\clF_1)$. Let $i \in \{1,\,\ldots,\,k\}$. If $\phi|_{\clF_1}$ is almost atoroidal relative to $\clF$, we denote by $\Delta_{\pm}([A_i],\phi) \subseteq \PCurr(A_i,\clF \wedge \{[A_i]\} \wedge \clA([\Phi_i|_{A_i}]))$ the convexes of attraction and repulsion of $[\Phi_i|_{A_i}]$. If $\psi \in \IA_{N}(\ZZ/3\ZZ)$ preserves the conjugacy class of $A_i$ and $\clF \wedge \{[A_i]\} \wedge \clA([\Phi_i|_{A_i}])$, then $\Delta_{\pm}([A_i],\psi\phi\psi^{-1})=\psi(\Delta_{\pm}([A_i],\phi))$.

%\bigskip

Let
\[
\widehat{\Delta}_{\pm}(\phi)=\big\{[t\mu +(1-t)\nu]\;\big|\; t \in [0,1], [\mu] \in \Delta_{\pm}(\phi), [\nu] \in K_{PG}(\phi), \lVert \mu \rVert_{\clF}=\lVert \nu \rVert_{\clF}=1\big\}
\]
be the \emph{convexes of attraction and repulsion of $\phi$}. We have the following results.

\begin{theo}
\cite[Theorem~6.4]{Guerch2021NorthSouth}\label{Theo North-South dynamics almost atoroidal}
Let ${N} \geq 3$ and let $\clF$ be a sporadic free factor system of $F_{N}$. Let $\phi \in \IA_{N}(\ZZ/3\ZZ)\cap \Out(F_{N},\clF)$ be an almost atoroidal outer automorphism relative to $\clF$. Let $\widehat{\Delta}_{\pm}(\phi)$ be the convexes of attraction and repulsion of $\phi$ and $\Delta_{\pm}(\phi)$ be the simplices of attraction and repulsion of $\phi$. Let $U_{\pm}$ be open neighborhoods of $\Delta_{\pm}(\phi)$ in $\PCurr(F_{N},\clF \wedge \clA(\phi))$ and $\widehat{V}_{\pm}$ be open neighborhoods of $\widehat{\Delta}_{\pm}(\phi)$ in $\PCurr(F_{N},\clF \wedge \clA(\phi))$. There exists $M \in \NN^*$ such that for every $n \geq M$, we have
\[
\phi^{\pm n}\left(\PCurr(F_{N},\clF \wedge \clA(\phi))-\widehat{V}_{\mp} \right) \subseteq U_{\pm}.
\]
\end{theo}


\begin{prop}[{\cite[Corollary~6.5]{Guerch2021NorthSouth}}]\label{Prop moving exponential}
Let ${N} \geq 3$ and let $\clF$ be a sporadic free factor system of $F_{N}$. Let $\phi \in \Out(F_{N},\clF)$ be an almost atoroidal outer automorphism relative to $\clF$. Let $\lVert. \rVert_{\clF} \colon \Curr(F_N,\clF \wedge \clA(\phi)) \to \RR_+$ be the function given by Proposition~\ref{Prop moving exponential1}.

For every open neighborhood $\widehat{V}_{-} \subseteq \PCurr(F_{N},\clF \wedge \clA(\phi))$ of $\widehat{\Delta}_-(\phi)$, there exists $M$ in $\NN^*$ and a constant $L_0>0$ such that, for every current $[\mu] \in \PCurr(F_{N},\clF \wedge \clA(\phi))-\widehat{V}_-$, and every $m \geq M$, we have
\[
\left\lVert\phi^m(\mu) \right\rVert_{\clF} \geq 3^{m-M}L_0 \lVert\mu \rVert_{\clF}.
\]
\end{prop}

\section{Nonsporadic extensions and fully irreducible outer automorphisms}

Let $N \geq 3$ and let $\clF$ and $\clF_1=\{[A_1],\,\ldots,\,[A_k]\}$ be two free factor systems of $F_{N}$ with $\clF \leq \clF_1$ such that the extension $\clF \leq \clF_1$ is nonsporadic. Let $H$ be a subgroup of $\IA_{N}(\ZZ/3\ZZ)$ which preserves $\clF$ and $\clF_1$. We suppose that $H$ is \emph{irreducible with respect to $\clF \leq \clF_1$}, that is, there does not exist a proper, nontrivial free factor system $\clF'$ of $F_{N}$ preserved by $H$ with $\clF < \clF' < \clF_1$.

Suppose that there exists $\phi \in H$ such that $\Poly(\phi|_{\clF})=\Poly(H|_{\clF})$. In this section, we show that there exists $\widehat{\phi} \in H$ such that $\Poly(\widehat{\phi}|_{\clF_1})=\Poly(H|_{\clF_1})$.

The key point is to construct \emph{fully irreducible outer automorphisms relative to $\clF$} in $H$ in the following sense. Let $\phi \in \Out(F_{N},\clF)$. We say that $\phi$ is \emph{fully irreducible relative to $\clF$} if no power of $\phi$ preserves a proper free factor system $\clF'$ of $F_{N}$ such that $\clF < \clF'$. If $\phi \in \Out(F_{N},\clF,\clF_1)$, we say that $\phi|_{\clF_1}$ is \emph{fully irreducible relative to $\clF$} (resp. \emph{atoroidal relative to $\clF$}) if, for every $i \in \{1,\,\ldots,\,k\}$, the outer automorphism $[\Phi_i|_{A_i}]$ defined in Equation~\eqref{Equation defi phi i} is fully irreducible relative to $\clF \wedge \{[A_i]\}$ (resp. atoroidal relative to $\clF \wedge \{[A_i]\}$).

If $H$ is a subgroup of $\Out(F_N,\clF,\clF_1)$, we say that $H|_{\clF_1}$ is \emph{atoroidal relative to $\clF$} if there does not exist a conjugacy class of $F_{N}$ which is $H$-invariant, $\clF$-nonperipheral and $\clF_1$-peripheral.

First, we recall some properties of fully irreducible outer automorphisms.

\begin{prop}\label{Prop properties fully irreducible}
Let ${N} \geq 3$ and let $\clF$ be a nonsporadic free factor system of $F_{N}$. Let $H$ be a subgroup of $\IA_{N}(\ZZ/3\ZZ)$ which preserves $\clF$ and such that $H$ is irreducible with respect to the extension $\clF \leq \{[F_{N}]\}$. Let $\phi \in H$ be a fully irreducible outer automorphism relative to $\clF$.

\begin{enumerate}
\item\label{prop3.1.1} \cite[Corollary~3.15]{Guerch2021NorthSouth} There exists at most one (up to taking inverse) conjugacy class $[g]$ of root-free $\clF$-nonperipheral element of $F_{N}$ which has polynomial growth under iteration of $\phi$. Moreover, the conjugacy class $[g]$ is fixed by $\phi$.

%\medskip

\item\label{prop3.1.2} \cite[Theorem~7.4]{Guirardelhorbez19} One of the following holds:
\begin{enumerate}
\item\label{prop3.1.2.a} there exists $\psi \in H$ such that $\psi$ is a fully irreducible, atoroidal outer automorphism relative to $\clF$;

%\medskip

\item\label{prop3.1.2.b} if $\phi$ fixes the conjugacy class of a root-free $\clF$-nonperipheral element $g$ of $F_{N}$, then $[g]$ is fixed by $H$.
\end{enumerate}
\end{enumerate}

Thus, there exists $\psi \in H$ such that $\psi$ is fully irreducible relative to $\clF$ and the conjugacy class of an $\clF$-nonperipheral element $g \in F_N$ has polynomial growth under iteration of $\psi$ if and only if it has polynomial growth under iteration of every element of $H$.
\end{prop}

Hence Proposition~\ref{Prop properties fully irreducible} suggests that, if $H$ is a subgroup of $F_{N}$ which satisfies the hypotheses in Proposition~\ref{Prop properties fully irreducible}, one step in order to prove Theorem~\ref{Theo intro 1} is to construct relative fully irreducible (atoroidal) outer automorphisms in $H$. This is contained in Theorem~\ref{Theo nonsporadic extension}. First we need the following lemma, whose statement is similar to an argument appearing in the proof of~\cite[Theorem~6.6]{ClayUya2018} (see also~\cite[Section~IV.2.1]{HandelMosher20}).

\begin{lem}\label{Lem uniqueness extension}
Let $N \geq 3$ and let $H$ be a subgroup of $\IA_{N}(\ZZ/3\ZZ)$. Let
\[
\varnothing=\clF_0 < \clF_1 <\ldots < \clF_k=\{[F_{N}]\}
\]
be a maximal $H$-invariant sequence of free factor systems. Let
\[
S=\left\{j\;\middle|\; \text{the extension } \clF_{j-1} \leq \clF_j \text{ is nonsporadic}\right\}
\]
and let $j \in S$. There exists a unique conjugacy class $[B_j]$ of a subgroup $B_j$ in $F_{N}$ such that $[B_j] \in \clF_j$ and $[B_j] \notin \clF_{j-1}$.
\end{lem}

\medskip

\begin{proof}
There exists at least one such conjugacy class since $\clF_{j-1}<\clF_j$. Suppose towards a contradiction that there exist two distinct subgroups $B_+$ and $B_-$ of $F_{N}$ such that $[B_+] \neq [B_-]$, $[B_+],[B_-] \in \clF_j$ and $[B_+],[B_-] \notin \clF_{j-1}$. Then
\[
\clF'([B_-])=\left(\clF_j-\{[B_+]\}\right) \cup \left(\clF_{j-1} \wedge \{[B_+]\}\right)
\]
is $H$-invariant and $\clF_{j-1} < \clF'([B_-])< \clF_j$, which contradicts the maximality hypothesis of the sequence of free factor systems.
\end{proof}
\begin{theo}\label{Theo nonsporadic extension}
Let $N \geq 3$ and let $H$ be a subgroup of $\IA_{N}(\ZZ/3\ZZ)$. Let
\[
\varnothing=\clF_0 < \clF_1 <\ldots < \clF_k=\{[F_{N}]\}
\]
be a maximal $H$-invariant sequence of free factor systems. There exists $\phi \in H$ such that for every $i \in \{1,\,\ldots,\,k\}$ such that the extension $\clF_{i-1} \leq \clF_i$ is nonsporadic, the element $\phi|_{\clF_i}$ is fully irreducible relative to $\clF_{i-1}$. Moreover, if $H|_{\clF_i}$ is atoroidal relative to $\clF_{i-1}$, one can choose $\phi$ so that $\phi|_{\clF_i}$ is atoroidal relative to $\clF_{i-1}$.
\end{theo}

\begin{proof}
The proof follows~\cite[Theorem~6.6]{ClayUya2018} (see also~\cite[Corollary~3.4]{clay2019atoroidal}). Let $S \subseteq \{0,\,\ldots,\,k\}$ be as in the statement of Lemma~\ref{Lem uniqueness extension} and let $j \in S$. Let $B_j$ be a subgroup of $F_{N}$ given by Lemma~\ref{Lem uniqueness extension}. Let $A_{j,1},\,\ldots,\,A_{j,s}$ be the subgroups of $B_j$ with pairwise disjoint conjugacy classes such that $\clA_{j-1}=\{[A_{j,1}],\,\ldots,\,[A_{j,s}]\} \subseteq \clF_{j-1}$ and $s$ is maximal for this property. Note that, for every $j \in S$, the free factor system $\clA_{j-1}$ is a nonsporadic free factor system of $B_j$ by Lemma~\ref{Lem uniqueness extension} and since the extension $\clF_{j-1} \leq \clF_j$ is nonsporadic.

By~\cite[Theorem~7.1]{Guirardelhorbez19} (see also~\cite[Theorem~D]{HandelMosher20} for the finitely generated case), for every $j \in S$, there exists an element $\phi \in H$ such that $[\Phi_j|_{B_j}] \in \Out(B_j,\clA_{j-1})$ is fully irreducible relative to $\clA_{j-1}$. By Proposition~\ref{Prop properties fully irreducible}$\MK$\eqref{prop3.1.2}, for every $j \in S$ such that $H|_{\clF_j}$ is atoroidal relative to $\clF_{j-1}$, there exists $\phi \in H$ such that $[\Phi_j|_{B_j}] \in \Out(B_j,\clA_{j-1})$ is fully irreducible and atoroidal relative to $\clA_{j-1}$.

Let $S_1$ be the subset of $S$ consisting in every $j \in S$ such that $H|_{\clF_j}$ is atoroidal relative to $\clF_{j-1}$, and let $S_2=S-S_1$. By~\cite[Theorems~4.1, 4.2]{Guirardelhorbez19} (see also~\cite{gupta18,Horbez17cyclic,Mann2014,Mann2014Thesis}), for every $j \in S_1$ (resp. $j \in S_2$) there exists a Gromov-hyperbolic space $X_j$ (the \emph{$\clZ$-factor complex of $B_j$ relative to $\clA_{j-1}$} when $j \in S_1$ and the \emph{free factor complex of $B_j$ relative to $\clA_{j-1}$} otherwise) on which $\Out(B_j,\clA_{j-1})$ acts by isometries and such that $\phi_0 \in \Out(B_j,\clA_{j-1})$ is a loxodromic element if and only if $\phi_0$ is fully irreducible atoroidal relative to $\clA_{j-1}$ (resp. fully irreducible relative to $\clA_{j-1}$). The conclusion then follows from~\cite[Theorem~5.1]{ClayUya2018}.
\end{proof}

\section{Sporadic extensions and polynomial growth}\label{Section sporadic extension}

Let $N \geq 3$ and let $\clF$ and $\clF_1=\{[A_1],\,\ldots,\,[A_k]\}$ be two free factor systems of $F_{N}$ with $\clF \leq \clF_1$. Suppose that the extension $\clF \leq \clF_1$ is sporadic. Let $H$ be a subgroup of $\IA_{N}(\ZZ/3\ZZ) \cap \Out(F_{N},\clF,\clF_1)$.

In order to prove Theorem~\ref{Theo intro 1}, we need to show that if $\Poly(\phi|_{\clF})=\Poly(H|_{\clF})$, there exists $\psi \in H$ such that $\Poly(\psi|_{\clF_1})=\Poly(H|_{\clF_1})$.

Let $\phi \in H$ be such that $\Poly(\phi|_{\clF})=\Poly(H|_{\clF})$. Note that, for every element $g$ of $\Poly(\phi|_{\clF})$, there exists a subgroup $A$ of $F_{N}$ such that $[A] \in \clF \wedge \clA(\phi)$ and $g \in A$. Conversely, for every subgroup $A$ of $F_{N}$ such that $[A] \in \clF \wedge \clA(\phi)$ and every element $g \in A$, we have $g \in \Poly(\phi|_{\clF})$.

Thus $\clF\wedge \clA(\phi)$ is the natural malnormal subgroup system associated with the set $\Poly(\phi|_{\clF})=\Poly(H|_{\clF})$. Thus, we see that $H$ preserves $\clF \wedge \clA(\phi)$ and hence $H$ acts by homeomorphisms on $\PCurr(F_N,\clF \wedge \clA(\phi))$.

We first need a general statement regarding the construction of an $\RR$-tree equipped with an action of $F_N$ stabilized by an exponentially growing outer automorphism.

\begin{lemm}\label{Lem construction r tree}
Let $\phi \in \Out(F_N)$ be an exponentially growing outer automorphism. Let $B_1,\,\ldots,\,B_n$ be subgroups of $F_N$ such that, for every $i \in \{1,\,\ldots,\,n\}$, we have $[B_i] \in \clA(\phi)$.
\begin{enumerate}
\item\label{lemm4.1.1} Suppose that there exist distinct $k,\ell \in \{1,\,\ldots,\,n\}$ with $B_k \neq B_{\ell}$. Then there exist:
\begin{enumerate}\romanenumi
\item\label{lemm4.1.1.i} a finitely generated subgroup $B$ of $F_N$ containing every $B_i$ with $i \in \{1,\,\ldots,\,n\}$;

\item\label{lemm4.1.1.ii} an $\RR$-tree $T$ equipped with a minimal, isometric action of $B$ with trivial arc stabilizers such that, for every $i \in \{1,\,\ldots,\,n\}$, the group $B_i$ is elliptic in $T$;

\item\label{lemm4.1.1.iii} distinct $i,j \in \{1,\,\ldots,\,n\}$ such that the point fixed by $B_i$ in $T$ is distinct from the point fixed by $B_j$.
\end{enumerate}
%\medskip

\item\label{lemm4.1.2} Suppose that there exist $g \in F_N$ and $k \in \{1,\,\ldots,\,n\}$ with $g \notin B_k$. Then there exist:
\begin{enumerate}\romanenumi
\item\label{lemm4.1.2.i} a finitely generated subgroup $B$ of $F_N$ containing $g$ and every $B_i$ with $i \in \{1,\,\ldots,\,n\}$;

\item\label{lemm4.1.2.ii} an $\RR$-tree $T$ equipped with a minimal, isometric action of $B$ with trivial arc stabilizers such that, for every $i \in \{1,\,\ldots,\,n\}$, the group $B_i$ is elliptic in $T$;

\item\label{lemm4.1.2.iii} $i \in \{1,\,\ldots,\,n \}$ such that the point fixed by $B_i$ is not fixed by $g$.
\end{enumerate}
\end{enumerate}
\end{lemm}

Note that, in the statement of Lemma~\ref{Lem construction r tree}$\MK$\eqref{lemm4.1.2}, the element $g$ is not necessarily contained in $\Poly(\phi)$. In particular, the action of $g$ on $T$ might be loxodromic.

%\medskip

\begin{proof}
We prove Assertion~\eqref{lemm4.1.1}. By~\cite[Lemma~1.2]{Levitt09}, there exists a nontrivial $\RR$-tree $T'$ equipped with a minimal, isometric action of $F_{N}$ with trivial arc stabilizers and such that every polynomial subgroup of $\phi$ fixes a point in $T'$.

If there exist distinct $i,j \in \{1,\,\ldots,\,n\}$ such that $B_i$ fixes a point in $T'$ distinct from the point fixed by $B_j$, then the tree $T=T'$ satisfies the assertion of Lemma~\ref{Lem construction r tree}$\MK$\eqref{lemm4.1.1}.

Suppose that there exists a point $x$ of $T'$ fixed by every $B_i$ with $i \in \{1,\,\ldots,\,n\}$. By~\cite{GabLev95}, there are only finitely many orbits of points in $T'$ with nontrivial stabilizers. In particular, up to taking a power of $\phi$, we may suppose that $\phi$ has a representative $\Phi_x$ which preserves $\Stab(x)$. Since $B_k \neq B_{\ell}$ and $B_k,B_{\ell} \subseteq \Stab(x)$, the automorphism $\Phi_x|_{\Stab(x)}$ is exponentially growing. By~\cite{GabLev95}, the rank of $\Stab(x)$ is less than $N$. An inductive argument replacing $F_N$ and $\phi$ by $\Stab(x)$ and the outer class of $\Phi_x|_{\Stab(x)}$ concludes the proof of Assertion~\eqref{lemm4.1.2}.

The proof of Assertion~\eqref{lemm4.1.2} is identical to the one of Assertion~\eqref{lemm4.1.1} replacing the fact that $B_k \neq B_{\ell}$ by the fact that $g \notin B_{k}$.
\end{proof}

\begin{lemm}\label{Lem Kpg moved by elements}
Let $N \geq 3$, let $\clF$ be a sporadic free factor system of $F_{N}$ and let $H$ be a subgroup of $\IA_{N}(\ZZ/3\ZZ) \cap \Out(F_{N},\clF)$ which is irreducible with respect to $\clF \leq \{[F_{N}]\}$. Suppose that there exists $\phi \in H$ such that $\Poly(\phi|_{\clF})=\Poly(H|_{\clF})$. If $\Poly(\phi) \neq \Poly(H)$, there exists an infinite subset $X \subseteq H$ such that for all distinct $\psi_1,\psi_2 \in X$, we have $\psi_1(K_{PG}(\phi)) \cap \psi_2(K_{PG}(\phi))=\varnothing$.
\end{lemm}

\begin{proof}
Let $\clF\wedge \clA(\phi)=\{[A_1],\,\ldots,\,[A_r]\}$. Since
\[
\Poly(\phi|_{\clF})=\Poly(H|_{\clF}) \subseteq \Poly(H) \subsetneq \Poly(\phi),
\]
we have $\clA(\phi) \neq \clF \wedge \clA(\phi)$. By~\cite[Lemma~5.18$\MK$(7)]{Guerch2021NorthSouth}, one of the following holds.
\begin{enumerate}\romanenumi
\item\label{lemm4.2.i} There exist distinct $i,j \in \{1,\ldots,r\}$ such that, up to replacing $A_i$ by a conjugate, we have $\clA(\phi)=(\clF\wedge \clA(\phi)-\{[A_i],[A_j]\}) \cup \{[A_i \ast A_j]\}$.

%\medskip

\item\label{lemm4.2.ii} There exist $i \in \{1,\ldots,r\}$ and an element $g \in F_{N}$ such that $\clA(\phi)=(\clF\wedge \clA(\phi)-\{[A_i]\}) \cup \{[A_i \ast \left\langle g \right\rangle] \}$.

%\medskip

\item\label{lemm4.2.iii} There exists $g \in F_{N}$ such that $\clA(\phi)=\clF\wedge\clA(\phi) \cup \{[\left\langle g \right\rangle]\}$.
\end{enumerate}

%\medskip

By Definition~\ref{Defi sporadic}, Assertion~\eqref{lemm4.2.ii} only occurs when the extension $\clF \leq \{[F_{N}]\}$ is an HNN extension over the trivial group. In particular, we have $\clF=\{[A]\}$ for some subgroup $A$ of $F_{N}$ and, up to changing the representative of $[A]$, we have $F_{N}=A \ast \langle g \rangle$ and $A_i \subseteq A$.

%\noindent{\bf Case~1 }
\begin{case}\label{case1}
Suppose that there exist distinct $i,j \in \{1,\,\ldots,\,r\}$ such that
\[
\clA(\phi)=\left(\clF\wedge \clA(\phi)-\{[A_i],[A_j]\}\right) \cup \{[A_i \ast A_j]\}.
\]

%\medskip

Since $\Poly(\phi|_{\clF})=\Poly(H|_{\clF})$ and $\Poly(\phi) \neq \Poly(H)$, there exists $\psi \in H$ such that, for every $n \in \NN^*$, the element $\psi^n$ does not preserve $[A_i \ast A_j]$ while preserving $[A_i]$ and $[A_j]$. Hence there exist a representative $\Psi$ of $\psi$ such that, for every $n \in \NN^*$, there exists $g_n \in F_N-A_i \ast A_j$ such that $\Psi^n(A_i)=A_i$ and $\Psi^n(A_j)=g_nA_jg_n^{-1}$. Note that
\[
g\Psi^n\left(A_i \ast A_j\right)g^{-1}=gA_ig^{-1} \ast gg_nA_jg_n^{-1}g^{-1}.
\]

%\medskip

%\noindent{\bf Claim~1. }
\begin{claim}\label{claim1}
For every $n \in \NN^*$ and every $g \in F_{N}$, there exist $t=t(g,n) \in F_N$ and $s=s(g,n) \in \{i,j\}$ such that
\[
\left(A_i \ast A_j\right) \cap \left(g\Psi^n(A_i \ast A_j)g^{-1}\right) \subseteq tA_st^{-1}.
\]
\end{claim}
%\medskip

\begin{proof}
Let $n \in \NN^*$. Note that, since $g_n \in F_N-A_i \ast A_j$ and since $A_i \ast A_j$ is a malnormal subgroup of $F_N$, $A_i \ast A_j$ is distinct from $gg_n (A_i \ast A_j)g_n^{-1}g^{-1}$ or from $g(A_i \ast A_j)g^{-1}$. Therefore, we can apply Lemma~\ref{Lem construction r tree}$\MK$\eqref{lemm4.1.1} to $\phi$ and the polynomial subgroups $A_i \ast A_j$, $gg_n (A_i \ast A_j)g_n^{-1}g^{-1}$ and $g(A_i \ast A_j)g^{-1}$. Thus, there exist a subgroup $B'$ of $F_{N}$ containing the subgroups $A_i \ast A_j$, $gg_n (A_i \ast A_j)g_n^{-1}g^{-1}$ and $g(A_i \ast A_j)g^{-1}$ and an $\RR$-tree $T'$ equipped with a minimal, isometric action of $B'$ with trivial arc stabilizers and such that the subgroups $A_i \ast A_j$, $gg_n (A_i \ast A_j)g_n^{-1}g^{-1}$ and $g(A_i \ast A_j)g^{-1}$ are elliptic but do not have a common fixed point. Let $x_1$ be the point in $T'$ fixed by $A_i \ast A_j$, let $x_2$ be the point fixed by $g(A_i \ast A_j)g^{-1}$ and let $x_3$ be the point fixed by $gg_n (A_i \ast A_j)g_n^{-1}g^{-1}$.

Let
\[
G=g\Psi^n\left(A_i \ast A_j\right)g^{-1}= gA_ig^{-1} \ast gg_nA_jg_n^{-1}g^{-1}.
\]

Suppose first that $x_2=x_3$. Then $x_1 \neq x_2$ by hypothesis. Note that the group $G \cap (A_i \ast A_j)$ fixes both $x_1$ and $x_2$. Since arc stabilizers are trivial, the intersection $G \cap (A_i \ast A_j)$ is trivial.

Thus, we may suppose that $x_2 \neq x_3$. Since arc stabilizers are trivial, by a standard ping pong argument, the points in $T'$ fixed by elements of $G$ are in the orbits of $x_2$ and $x_3$. Since arc stabilizers are trivial, and since $G$ is the free product of $gA_ig^{-1}$ and $gg_nA_jg_n^{-1}g^{-1}$, we see that $G \cap \Stab(x_2)=gA_ig^{-1}$ and $G \cap \Stab(x_3)=gg_nA_jg_n^{-1}g^{-1}$. Thus, elliptic elements in $G$ are contained in conjugates of $gA_ig^{-1}$ and conjugates of $gg_nA_jg_n^{-1}g^{-1}$. Since the intersection of $G$ with $A_i \ast A_j$ is elliptic, it is contained in a conjugate of $A_i$ or a conjugate of $A_j$. This proves Claim~\ref{claim1}.
\end{proof}

Claim~\ref{claim1} implies that, for all distinct $m,n \in \NN$ and every element $x \in F_{N}$, there exist $t=t(x,m,n) \in F_N$ and $s=s(x,m,n) \in \{i,j\}$ such that
\[
\Psi^n\left(A_i \ast A_j\right) \cap \left(x\Psi^m(A_i \ast A_j)x^{-1}\right) \subseteq tA_st^{-1}.
\]

By for instance~\cite[Fact~I.1.2]{HandelMosher20}, for any subgroups $A$ and $B$ of $F_N$, we have the equalities $(\partial_{\infty} A) \cap (\partial_{\infty} B) =\partial_{\infty} (A \cap B)$ and $(\partial^2 A) \cap (\partial^2 B) =\partial^2 (A \cap B)$. Thus, for all distinct $m,n \in \NN$ and every $x \in F_{N}$, we have
\begin{multline*}
\partial^2 \left(\Psi^n(A_i \ast A_j)\right) \cap \partial^2 \left(x\Psi^m(A_i \ast A_j)x^{-1}\right)\\
\begin{aligned}
&=\partial^2\left(\Psi^n(A_i \ast A_j) \cap x\Psi^m(A_i \ast A_j)x^{-1}\right) \\
&\subseteq \partial^2(tA_st^{-1}) \\
&\subseteq \overline{\bigcup_{y\,\in\,F_{N}} \big(\partial^2 \left(yA_iy^{-1}\right) \cup\partial^2 \left(yA_jy^{-1}\right)\big)}.
\end{aligned}
\end{multline*}
By definition of $K_{PG}(\phi)$, we have $[\mu] \in K_{PG}(\phi)$ if and only if
\[
\Supp(\mu) \subseteq \partial^2\clA(\phi) \cap \partial^2(F_{N},\clF\wedge \clA(\phi)) =\partial^2 \{[A_i \ast A_j]\}\cap \partial^2(F_{N},\clF\wedge \clA(\phi)).
\]
Moreover, if $n \in \NN$ and if $[\mu] \in \psi^n(K_{PG}(\phi))$, then
\[
\Supp(\mu) \subseteq \partial^2\psi^n(\clA(\phi)) \cap \partial^2(F_{N},\clF\wedge \clA(\phi)) =\partial^2 \{[A_i \ast g_nA_jg_n^{-1}]\}\cap \partial^2(F_{N},\clF\wedge \clA(\phi)).
\]
Let $n,m \in \NN$ be distinct. Suppose towards a contradiction that
\[
\psi^n(K_{PG}(\phi)) \cap \psi^m(K_{PG}(\phi)) \neq \varnothing
\]
and let $[\mu] \in \psi^n(K_{PG}(\phi)) \cap \psi^m(K_{PG}(\phi))$. Thus, the support of $\mu$ is contained in
\begin{multline*}
\left(\bigcup_{x,y\,\in\,F_N} \big(\partial^2 (x(A_i \ast g_nA_jg_n^{-1})x^{-1})\big) \cap \big(\partial^2 \left(y(A_i \ast g_mA_jg_m^{-1})y^{-1}\right)\big)\right)\\
\cap \partial^2(F_{N},\clF\wedge \clA(\phi))
\end{multline*}
and there exist $x,y \in F_N$ such that $\mu$ gives positive measure to
\[
\Big(\partial^2\left(x\left(A_i \ast g_nA_jg_n^{-1}\right)x^{-1}\right) \cap \partial^2 \left(y\left(A_i \ast g_mA_jg_m^{-1}\right)y^{-1}\right)\Big) \cap \partial^2(F_{N},\clF\wedge \clA(\phi)).
\]
By $F_{N}$-invariance of $\mu$, there exists $x \in F_{N}$ such that $\mu$ gives positive measure to
\begin{multline*}
\partial^2 \left(A_i \ast g_nA_jg_n^{-1}\right) \cap \partial^2 \left(x\left(A_i \ast g_mA_jg_m^{-1}\right)x^{-1}\right) \cap\partial^2\left(F_{N},\clF\wedge \clA(\phi)\right) \\
\subseteq \overline{\left(\bigcup_{y\,\in\,F_{N}} \partial^2 \left(yA_iy^{-1}\right) \cup \partial^2 \left(yA_jy^{-1}\right)\right)} \cap \partial^2(F_{N},\clF\wedge \clA(\phi))
\end{multline*}
and the last intersection is empty by the definition of the relative boundary, a contradiction.
\end{case}
%\medskip

%\noindent{\bf Case~2 }
\begin{case}\label{case2}
Suppose that either there exist $i \in \{1,\,\ldots,\,r\}$ and an element $g \in F_{N}$ such that $\clA(\phi)=(\clF\wedge \clA(\phi)-\{[A_i]\}) \cup \{[A_i \ast \langle g \rangle] \}$ or there exists $g \in F_{N}$ such that $\clA(\phi)=\clF\wedge\clA(\phi) \cup \{[\langle g\rangle]\}$.

%\medskip

In order to treat both cases simultaneously, in the case that there exists $g \in F_{N}$ such that $\clA(\phi)=\clF\wedge\clA(\phi) \cup \{[\left\langle g \right\rangle]\}$, we fix $A_i=\{e\}$.

Recall that we have $\clF=\{[A]\}$ for some subgroup $A$ of $F_{N}$ and, up to changing the representative of $[A]$, we have $F_{N}=A \ast \left\langle g \right\rangle$ and $A_i \subseteq A$. In particular, since $H$ preserves the extension $\clF \leq \{[F_{N}]\}$, for every $\psi \in H$, there exist a unique representative $\Psi_0$ of $\psi$ and $g_{\psi} \in A$ such that $\Psi_0(A)=A$ and $\Psi_0(g)=gg_{\psi}$.

%\medskip

%\noindent{\bf Claim~2. }
\begin{claim}\label{claim2}
There exists $\psi \in H$ such that, for every $n \in \NN^*$, we have $g_{\psi^n} \linebreak\notin A_i$.
\end{claim}
%\medskip

\begin{proof} 
First note that, since $H$ is irreducible with respect to $\clF \leq \{[F_{N}]\}$, the subgroup $H$ does not preserve the free factor system $\clF \cup \{[\langle g\rangle]\}$. Thus, there exists $\psi' \in H$ such that $g_{\psi'} \neq 1$.

Let $S$ be the subset of $H$ consisting in every element $\psi' \in H$ such that $g_{\psi'} \neq 1$. Note that, since $H \subseteq \IA_N(\ZZ/3\ZZ)$, for every $m \in \NN^*$ and every $\psi' \in S$, we have $g_{\psi'^m} \neq 1$ as $\psi'^m$ cannot fix the conjugacy class of $g$. Hence $S$ is stable under taking powers. In particular, if $A_i$ is trivial, any $\psi \in S$ satisfies the assertion of Claim~\ref{claim2}. Similarly, the complement of $S$ is stable under taking powers.

Note also that for every $\psi' \in S$, the elements $g$ and $g_{\psi'}$ are contained in distinct factors of $A \ast \langle g \rangle$.

We now claim that there exists $\theta \in S$ such that one of the following holds:
\begin{enumerate}\romanenumi
\item\label{proofclaim2.i} for any distinct $m,n \in \NN^*$, we have $\Theta_0^n(A_i) \cap \Theta_0^m(A_i)=\{e\}$ (this is equivalent to the fact that, for all $m \neq n$, we have $\Theta_0^n(A_i) \neq \Theta_0^m(A_i)$);

\item\label{proofclaim2.ii} for every $n \in \NN^*$, we have $g_{\theta^n} \notin A_i$.
\end{enumerate}

Indeed, for every element $\psi' \in S$, the automorphism $\Psi_0'$ acts naturally on the set of conjugates of $A_i$. If there exists $\psi' \in S$ such that $A_i$ has an infinite orbit, then we may take $\theta=\psi'$, which satisfies~\eqref{proofclaim2.i}.

Thus, we may suppose that, for every element $\psi \in S$, the element $\Psi_0$ has a power which preserves $A_i$. We now construct an element $\theta \in S$ which satisfies Assertion~\eqref{proofclaim2.ii}.

Since $\Poly(H) \neq \Poly(\phi)$, there exists $\psi' \in H$ such that $ A_i \ast \langle g \rangle \nsubseteq \Poly(\psi')$. We distinguish between two cases, according to whether $\psi' \in S$ or not.

If $\psi' \in S$, up to taking a power of $\psi'$, we have $\Psi_0'(A_i)=A_i$ and $ A_i \ast \langle g \rangle \nsubseteq \Poly(\psi')$.

Note that $A_i$ is then contained in the polynomial subgroup of the automorphism $\Psi_0'$. As $ A_i \ast \langle g \rangle \nsubseteq \Poly(\psi')$, for every $n \in \NN^*$, we have $g_{\psi'^n} \notin A_i$. Thus, we may take $\theta=\psi'$.

So we may suppose that $\psi' \notin S$ and, for every $\theta' \in S$, that $A_i \ast \langle g \rangle \subseteq \Poly(\theta')$. Thus, there exists $\theta' \in S$ such that $\Theta_0'(A_i)=A_i$ and $g_{\theta'} \in A_i$. Moreover, we have $\Psi_0'(g)=g$ and, since $A_i \ast \langle g \rangle \nsubseteq \Poly(\psi')$, the subgroup $A_i$ has an infinite orbit under iteration of $\Psi_0'$.

Then, for every $n \in \NN^*$, we have
\[
\Theta_0'\Psi_0'^n\Theta_0'^{-1}(g)=gg_{\theta'\psi'^n \theta'^{-1}}=gg_{\theta'}\Theta_0'\left(\Psi_0'^n(g_{\theta'^{-1}})\right).
\]
Since $g_{\theta'^{-1}} \in A_i$, we have $\Psi_0'^n(g_{\theta'^{-1}}) \notin A_i$ and $\Theta_0'(\Psi_0'^n(g_{\theta'^{-1}})) \notin A_i$. Since $g_{\theta'} \in A_i$, we have $g_{\theta'\psi'^n \theta'^{-1}} \notin A_i$. Therefore, the element $\theta'\psi' \theta'^{-1} \in S$ satisfies Assertion~\eqref{proofclaim2.ii}. Hence we may take $\theta=\theta'\psi' \theta'^{-1}$. This proves the existence of $\theta$.

Suppose first that $\theta$ satisfies Assertion~\eqref{proofclaim2.ii}. Then we may set $\psi=\theta$, so that $\psi$ satisfies the assertion of Claim~\ref{claim2}. Otherwise, $\theta$ satisfies~\eqref{proofclaim2.i} and, up to taking a power of $\theta$, we may suppose that $g_{\theta} \in A_i$.

We claim that $\theta^2$ satisfies the assertion of Claim~\ref{claim2}. Indeed, note that, for every $n \in \NN^*$, we have
\[
g_{\theta^{2n}}=h_0\ldots h_{2n-1},
\]
where, for every $j \in \{0,\,\ldots,\,2n-1\}$, the element $h_j$ is a nontrivial element of $\Theta_0^j(A_i)$, the fact that $h_j$ is nontrivial following from the fact that $\theta \in S$.

Thus, in order to show that $\theta^2$ satisfies the assertion of Claim~$2$, it suffices to show that, for every $m \in \NN$, we have
\begin{equation}\label{Equation theta}
\left\langle \Theta_0^j(A_i)\right\rangle_{j\,\in\, \{0,\,\ldots,\, m\}}= A_i \ast \ldots \ast \Theta_0^{m}(A_i).
\end{equation}

We prove Equation~\eqref{Equation theta} by induction on $m$, the result being trivial when $m=0$.

Since $\theta$ satisfies Assertion~$(i)$, for any distinct $j,k \in \{0,\,\ldots,\,m\}$, we have
\[
\Theta_0^j(A_i) \neq \Theta_0^k(A_i).
\]
In particular, we can apply Lemma~\ref{Lem construction r tree}~$(1)$ to the outer class $[\Phi_0|_A] \in \Out(A)$ and the set $\{\Theta_0^j(A_i)\}_{j\,\in\,\{0,\,\ldots,\,m\}}$ of polynomial subgroups of $[\Phi_0|_A]$. Thus, there exists a subgroup $B'$ of $A$ containing $\{ \Theta_0^j(A_i)\}_{j\,\in\,\{0,\,\ldots,\,m\}}$ and an $\RR$-tree $T'$ equipped with a minimal, isometric action of $B'$ with trivial arc stabilizers, such that, for every $j \in \{0,\,\ldots,\,m\}$, the group $\Theta_0^j(A_i)$ fixes a point $x_j$ and there exist distinct $k_1,k_2$ such that $x_{k_1} \neq x_{k_2}$.

Since $T'$ has trivial arc stabilizers, the groups $\Stab(x_0),\,\ldots,\,\Stab(x_m)$ generate their free product. Since there exist $k_1,k_2$ such that $x_{k_1} \neq x_{k_2}$, for every $\ell \in \{0,\,\ldots,\,m\}$, the group $\Stab(x_{\ell})$ contains at most $m-1$ elements of the set $\{\Theta_0^j(A_i)\}_{j\,\in\,\{0,\,\ldots,\,m\}}$. Thus, we can apply the induction hypothesis to conclude the proof of Equation~\eqref{Equation theta} and thus the proof of Claim~\ref{claim2}.
\end{proof}
%\medskip

Let $\psi \in H$ and $g_{\psi}$ be as in the claim. We claim that, for every $n \in \NN^*$, the conjugacy class $[gg_{\psi^n}]$ has exponential growth under iteration of $\phi$. Indeed, recall the construction of $\Phi_0$ above Claim~\ref{claim2}. Since $g_{\phi}, g_{\psi^n} \in A$ and since $\Phi_0(A)=A$, for every $m \in \NN$, the element $\Phi_0^m(gg_{\psi^n})$ is cyclically reduced. Hence $[gg_{\psi^n}]$ has exponential growth under iteration of $\phi$ if and only if $gg_{\psi^n}$ has exponential growth under iteration of $\Phi_0$. But the polynomial subgroup of $\Phi_0$ is $A_i \ast \langle g \rangle$. Since $g_{\psi^n} \notin A_i$, the element $gg_{\psi^n}$ has exponential growth under iteration of $\Phi_0$. This proves the claim. In particular, for every $n \in \NN^*$, no conjugate of $gg_{\psi^n}$ is contained in $A_i \ast \langle g \rangle$.

Let $\Psi \in \psi$ be such that, for every $n \in \NN^*$, there exists $h_{\psi^n} \in A$ with $\Psi^n(A_i)=A_i$ and $\Psi^n(g)=h_{\psi^n}gg_{\psi^n}h_{\psi^n}^{-1}$. Note that, for every $n \in \NN^*$, we have
\[
\Psi^n\left(A_i \ast \langle g \rangle\right) = A_i \ast h_{\psi^n} \langle gg_{\psi^n}\rangle h_{\psi^n}^{-1}.
\]

%\medskip

%\noindent{\bf Claim~3. }
\begin{claim}\label{claim3}
For every $n \in \NN^*$ and every $a\in F_{N}$, there exists $t=t(n,a)$ such that
\[
\left(a\Psi^n(A_i \ast \left\langle g \right\rangle)a^{-1}\right) \cap (A_i \ast \left\langle g \right\rangle) \subseteq tA_it^{-1}.
\]
\end{claim}
%\medskip

\begin{proof}
Let $n \in \NN^*$ and let $a \in F_{N}$. First note that $ah_{\psi^n} gg_{\psi^n}h_{\psi^n}^{-1}a^{-1} \notin a (A_i \ast \langle g \rangle) a^{-1}$. Indeed, since $F_{N}=A \ast \langle g \rangle$, the element $h_{\psi^n} gg_{\psi^n}h_{\psi^n}^{-1}$ can be written uniquely as a reduced product of elements in $A$ and elements in $\langle g \rangle$. Since $h_{\psi^n}, g_{\psi^n} \in A$, if we have $h_{\psi^n} gg_{\psi^n}h_{\psi^n}^{-1} \in A_i \ast \langle g \rangle$, then $h_{\psi^n} \in A_i$ and $g_{\psi^n}h_{\psi^n}^{-1} \in A_i$. Therefore, we have $g_{\psi^n} \in A_i$, a contradiction. Thus, we have $ah_{\psi^n} gg_{\psi^n}h_{\psi^n}^{-1}a^{-1} \notin a (A_i \ast \langle g \rangle)a^{-1}$.

In particular, we can apply Lemma~\ref{Lem construction r tree}$\MK$\eqref{lemm4.1.2} to $\phi$, the polynomial subgroups $A_i \ast \langle g \rangle$, $a(A_i \ast \langle g \rangle) a^{-1}$ and the element $ah_{\psi^n}gg_{\psi^{n}}h_{\psi^n}^{-1}a^{-1}$. This shows that there exist a subgroup $B'$ of $F_{N}$ containing $A_i \ast \langle g \rangle$, $a(A_i \ast \langle g \rangle) a^{-1}$ and $ah_{\psi^n}gg_{\psi^{n}}h_{\psi^n}^{-1}a^{-1}$, and an $\RR$-tree $T'$ equipped with a minimal, isometric action of $B'$ with trivial arc stabilizers and such that $A_i \ast \langle g \rangle$ fixes a point $x_1$ in $T'$, $a(A_i \ast \langle g \rangle) a^{-1}$ fixes a point $x_2=ax_1$ in $T'$ and if $x_1=x_2$, then $ah_{\psi^n}gg_{\psi^n}h_{\psi^n}^{-1}a^{-1}$ does not fix $x_1$.

Let $G= (a\Psi^n(A_i \ast \langle g \rangle)a^{-1}) \cap (A_i \ast \langle g \rangle)$. The group $G$ fixes $x_1$. Let $h \in G$. Since we have $h \in a\Psi^n(A_i \ast \langle g \rangle)a^{-1}$, the element $h$ can be written as a product of elements $s_0a_1b_1\ldots a_kb_ks_0^{-1}$ where the element $s_0$ is in $a\Psi^n(A_i \ast \langle g \rangle)a^{-1}$ and, for every $i \in \{1,\,\ldots,\,k\}$, we have $a_i \in aA_ia^{-1}$ and $ b_i \in \langle ah_{\psi^n}gg_{\psi^n}h_{\psi^n}^{-1}a^{-1}\rangle$. We suppose that $a_1b_1\ldots a_kb_k$ is a cyclic reduction of $h$ when written in the free product $aA_ia^{-1} \ast \langle ah_{\psi^n}gg_{\psi^n}h_{\psi^n}^{-1}a^{-1}\rangle$. We will prove that $h$ is a conjugate of $a_1$.

Suppose first that $ah_{\psi^n}gg_{\psi^n}h_{\psi^n}^{-1}a^{-1}$ fixes a point $x$ in $T'$. We distinguish between two cases, according to $x$.

Suppose that $x=x_2$. Then $x_1 \neq x_2$. Recall that
\[
a\Psi^n(A_i \ast \left\langle g \right\rangle)a^{-1} = a\left(A_i \ast h_{\psi^n} \langle gg_{\psi^n}\rangle h_{\psi^n}^{-1}\right)a^{-1}.
\]
Thus $a\Psi^n(A_i \ast \langle g \rangle)a^{-1}$ fixes $x_2$ and $h$ fixes both $x_1$ and $x_2$. Since $T'$ has trivial arc stabilizers, we see that $h=e$.

Suppose now that $x \neq x_2$. Then the minimal tree in $T'$ of the subgroup of $F_{N}$ generated by $\Stab(x)$ and $\Stab(x_2)$ is simplicial and its vertex stabilizers are conjugates of $\Stab(x)$ and $\Stab(x_2)$. Recall that $a\Psi^n(A_i \ast \left\langle g \right\rangle)a^{-1}$ is a free product with one factor fixing $x$ and the other factor fixing $x_2$. Thus, since arc stabilizers in $T'$ are trivial, elliptic elements of $a\Psi^n(A_i \ast \langle g \rangle)a^{-1}$ are contained in conjugates of $A_i$ or in conjugates of $h_{\psi^n} \langle gg_{\psi^n}\rangle h_{\psi^n}^{-1}$. Since $h$ is elliptic in $T'$, we see that $h$ is conjugate to either $a_1$ or $b_k$.

Recall that we proved above Claim~\ref{claim3} that $gg_{\psi^n} \notin \Poly(\phi)$. Thus, no conjugate of $gg_{\psi^n}$ is contained in $A_i \ast \langle g \rangle$. Since $h \in A_i \ast \langle g \rangle$, the element $h$ is conjugate to $a_1$.

Finally, suppose that $ah_{\psi^n}gg_{\psi^n}h_{\psi^n}^{-1}a^{-1}$ is loxodromic. Then the minimal tree in $T'$ of $\langle \Stab(x_2),ah_{\psi^n}gg_{\psi^n}h_{\psi^n}^{-1}a^{-1} \rangle$ is simplicial and its vertex stabilizers are either trivial or conjugates of $\Stab(x_2)$. Note that $a\Psi^n(A_i \ast \langle g \rangle)a^{-1}$ is a free product with one factor, $A_i$, fixing $x_2$ and the other factor being cyclic, generated by the loxodromic element $ah_{\psi^n}gg_{\psi^n}h_{\psi^n}^{-1}a^{-1}$. Thus, since arc stabilizers in $T'$ are trivial, elliptic elements of the group $a\Psi^n(A_i \ast \langle g \rangle)a^{-1}$ are contained in conjugates of $A_i$. Since $h$ fixes $x_1$, it is contained in a conjugate of $A_i$. Thus, in all cases, $h$ is contained in a conjugate of~$A_i$.

Therefore, every element of $G$ is contained in a conjugate of $A_i$. Recall that $A_i \ast \langle g \rangle$ is a malnormal subgroup of $F_N$, so that every conjugate of $A_i$ intersecting $A_i \ast \langle g \rangle$ nontrivially is a conjugate of $A_i$ whose conjugator is in $A_i \ast \langle g \rangle$. Thus every element of $G$ fixes a point in the Bass-Serre tree $S$ of $A_i \ast \langle g \rangle$ associated with $A_i$. Since edge stabilizers in $S$ are trivial, this implies that the group $G$ fixes a point in $S$, hence is contained in a conjugate of $A_i$. This proves Claim~\ref{claim3}.
\end{proof}
%\medskip

Claim~\ref{claim3} implies that, for all distinct $n,m \in \NN^*$ and every $x \in F_{N}$, there exists $t=t(m,n,x)$ such that we have
\[
\Psi^n(A_i \ast \left\langle g \right\rangle) \cap x\Psi^m(A_i \ast \left\langle g \right\rangle)x^{-1}\subseteq tA_it^{-1}
\]
By~\cite[Fact~I.1.2]{HandelMosher20}, we have
\[
\partial^2\Psi^n(A_i \ast \left\langle g \right\rangle) \cap \partial^2\left(x\Psi^m(A_i \ast \left\langle g \right\rangle)x^{-1}\right)\subseteq \partial^2 \left(tA_it^{-1}\right) \subseteq \overline{\bigcup_{y\,\in\,F_{N}} \partial^2 \left(yA_iy^{-1}\right)}.
\]
The rest of the proof is then similar to the one of Case~\ref{case1}.
\end{case}\let\qed\relax
\end{proof}

\begin{lemm}\label{Lem preparation ping pong}
Let $N \geq 3$, let $\clF$ and $\clF_1=\{[A_1],\,\ldots,\,[A_k]\}$ be two free factor systems of $F_{N}$ with $\clF \leq \clF_1$ such that the extension $\clF \leq \clF_1$ is sporadic. Let $H$ be a subgroup of $\Out(F_{N},\clF,\clF_1) \cap \IA_{N}(\ZZ/3\ZZ)$ such that $H$ is irreducible with respect to $\clF \leq \clF_1$. Suppose that there exists $\phi \in H$ such that $\Poly(\phi|_{\clF})=\Poly(H|_{\clF})$. Suppose that
 $\Poly(\phi|_{\clF_1}) \neq \Poly(H|_{\clF_1})$. There exists $\psi \in H$ such that for every $i \in \{1,\,\ldots,\,k\}$, we have $\psi(K_{PG}([\Phi_i|_{A_i}])) \cap K_{PG}([\Phi_i|_{A_i}])=\varnothing$, where $[\Phi_i|_{A_i}]$ is
defined in Equation~\eqref{Equation defi phi i} of Section~\ref{Section currents associated automorphism} and
\[
\Delta_+([A_i],\phi) \cap \psi(\Delta_-([A_i],\phi))=\Delta_-([A_i], \phi) \cap \psi(\Delta_+([A_i], \phi))=\varnothing.
\]
\end{lemm}

\begin{proof}
The proof follows~\cite[Lemma~5.1]{clay2019atoroidal}. Recall that, since the extension $\clF \leq \clF_1$ is sporadic, there exists $\ell \in \{1,\,\ldots,\,k\}$ such that, for every $i \in \{1,\,\ldots,\,k\}-\{\ell\}$, we have $[A_i] \in \clF$. By Lemma~\ref{Lem Kpg moved by elements} applied to the image of $H$ in $\Out(A_{\ell})$ (which is contained in $\IA(A_{\ell},\ZZ/3\ZZ)$), there exists an infinite subset $X \subseteq H$ such that, for any distinct $h_1,h_2 \in X$, we have
\[
h_1(K_{PG}\big([\Phi_{\ell}|_{A_{\ell}}])\big) \cap h_2(K_{PG}([\Phi_{\ell}|_{A_{\ell}}]))=\varnothing.
\]

We now prove that there exist $h_1,h_2 \in X$ such that $h_2^{-1}h_1$ satisfies the assertion of Lemma~\ref{Lem preparation ping pong}. Note that, for any distinct $h_1,h_2 \in X$, we have
\[
h_2^{-1}h_1(K_{PG}([\Phi_{\ell}|_{A_{\ell}}])) \cap K_{PG}([\Phi_{\ell}|_{A_{\ell}}])=\varnothing.
\]
Hence it suffices to find two distinct $h_1,h_2 \in X$ such that $\psi=h_2^{-1}h_1$ satisfies the second assertion of Lemma~\ref{Lem preparation ping pong}.

Let $i \in \{1,\,\ldots,\,k\}$ and let $[\mu]$ be an extremal point of $\Delta_+([A_i],\phi)$ or $\Delta_-([A_i],\phi)$. By~\cite[Lemma~4.13]{Guerch2021NorthSouth}, the support $\Supp(\mu)$ contains the support of \emph{only} finitely many projective currents $[\mu_1],\,\ldots,\,[\mu_s] \in \PCurr(F_N,\clF\wedge \clA(\phi))$ such that, for every $t \in \{1,\,\ldots,\,s\}$, the support of $\mu_t$ is uniquely ergodic.

Let $E_{\mu}=\{[\mu_1],\,\ldots,\,[\mu_s]\}$. Let $E_{\phi}=\bigcup E_{\mu}$, where the union is taken over all $i$ in $\{1,\,\ldots,\,k\}$ and extremal points of $\Delta_+([A_i],\phi)$ and $\Delta_-([A_i],\phi)$. The set $E_{\phi}$ is finite by~\cite[Lemma~4.7]{Guerch2021NorthSouth}.

Since the set $E_{\phi}$ is finite, up to taking an infinite subset of $X$, we may suppose that, for every $s \in E_{\phi}$, either $h_1s=h_2s$ for every $h_1,h_2 \in X$ or for every distinct $h_1,h_2 \in X$, we have $h_1s \neq h_2s$. Let $E_1 \subseteq E_{\phi}$ be the subset for which the first alternative occurs and let $E_{\infty}=E_{\phi}-E_1$.

Let $h_1 \in X$ and, for every $s \in E_{\infty}$, let
\[
X_s=\left\{h \in X \;\middle|\; h_1s=hs' \text{ for some }s' \in
E_{\infty}\right\}.
\]
Note that $X_s$ is a finite set. Let $h_2 \in X-\bigcup_{s \in E_{\infty}} X_s$. For every $s,s' \in E_{\infty}$, we have $h_1s \neq h_2 s'$. If there exists $s' \in E_1$ such that $h_1s=h_2s'$, then $s=h_1^{-1}h_2s'=s'$, contradicting the fact that $s \in E_{\infty}$. Thus, for every $s \in E_{\infty}$, we have $h_2^{-1}h_1s \notin E_{\phi}$ and for every $s \in E_1$, we have $h_2^{-1}h_1s=s$. Let $\psi=h_2^{-1}h_1$. Then, for every $s \in E_{\phi}$, either $\psi(s)=s$ or $\psi(s) \notin E_{\phi}$. Moreover, by construction of $X$, for every $i \in \{1,\,\ldots,\,k\}$, we have $\psi(K_{PG}([\Phi_i|_{A_i}])) \cap K_{PG}([\Phi_i|_{A_i}])=\varnothing$. Thus, $\psi$ satisfies the first assertion of Lemma~\ref{Lem preparation ping pong}.

We now prove that $\psi$ satisfies the second assertion. Let $i \in \{1,\,\ldots,\,k\}$, let $[\mu] \in \Delta_-([A_i], \phi)$ and suppose for a contradiction that we have $\psi([\mu]) \in \Delta_+([A_i],\phi)$. There exist extremal measures $\mu_1^-,\,\ldots,\,\mu_m^-$ of $\Delta_-([A_i],\phi)$ and $\lambda_1,\,\ldots,\,\lambda_m \in \RR_+$ such that $\mu=\sum_{j=1}^m \lambda_j\mu_j^-$. Similarly, there exist extremal measures $\mu_1^+,\,\ldots,\,\mu_n^+$ of $\Delta_+([A_i],\phi)$ and $\alpha_1,\,\ldots,\,\alpha_n \in \RR_+$ such that $\psi(\mu)=\sum_{j=1}^n \alpha_j\mu_j^+$.

Thus, we have
\[
\sum_{j=1}^m \lambda_j\psi(\mu_j^-)=\psi(\mu)=\sum_{j=1}^n \alpha_j\mu_j^+.
\]
In particular, we have
\[
\bigcup_{j=1}^m\Supp\left(\psi(\mu_j^-)\right)=\bigcup_{j=1}^n\Supp(\mu_j^+).
\]
Let $\Lambda \subseteq \Supp(\mu_1^-)$ be the uniquely ergodic support of a current in $E_{\phi}$. Let $\Psi$ be a representative of $\psi$ and let $\partial^2\Psi$ be the homeomorphism of $\partial^2 F_{N}$ induced by $\Psi$. Since uniquely ergodic laminations are minimal, there exists $j \in \{1,\,\ldots,\,n\}$ such that we have $\partial^2\Psi(\Lambda) \subseteq \Supp(\mu_j^+)$. Thus, we have $\psi([\mu_1^-|_{\Lambda}])=[\mu_j^+|_{\Lambda}]$. This contradicts the fact that $[\mu_1^-|_{\Lambda}]$ and $[\mu_j^+|_{\Lambda}]$ are distinct elements of $E_{\phi}$ since $\Delta_+([A_i],\phi) \cap \Delta_-([A_i],\phi)=\varnothing$.
\end{proof}

\begin{prop}\label{Prop ping pong}
Let $N \geq 3$, let $\clF$ and $\clF_1=\{[A_1],\,\ldots,\,[A_k]\}$ be two free factor systems of $F_{N}$ with $\clF \leq \clF_1$ such that the extension $\clF \leq \clF_1$ is sporadic. Let $H$ be a subgroup of $\IA_{N}(\ZZ/3\ZZ) \cap \Out(F_{N},\clF,\clF_1)$ such that $H$ is irreducible with respect to $\clF \leq \clF_1$. Suppose that there exists $\phi \in H$ such that $\Poly(\phi|_{\clF})=\Poly(H|_{\clF})$. Suppose that $\Poly(\phi|_{\clF_1}) \neq \Poly(H|_{\clF_1})$. There exist $\psi \in H$ and a constant $M>0$ such that, for all $m,n \geq M$, if $\theta=\psi\phi\psi^{-1}$, we have $\Poly(\theta^m\phi^n|_{\clF_1})=\Poly(H|_{\clF_1})$.
\end{prop}

\begin{proof}
The proof follows~\cite[Proposition~5.2]{clay2019atoroidal}. Let $\psi \in H$ be an element given by Lemma~\ref{Lem preparation ping pong} and let $\theta=\psi\phi\psi^{-1}$. For every $i \in \{1,\,\ldots,\,k\}$, let $\Theta_i$ be a representative of $\theta$ such that $\Theta_i(A_i)=A_i$ and $\Phi_i$ be a representative of $\phi$ such that $\Phi_i(A_i)=A_i$. Note that, since for every $i \in \{1,\,\ldots\,,k\}$, $[\Phi_i|_{A_i}]$ is almost atoroidal relative to $\clF$, so is $[\Theta_i|_{A_i}]$. Moreover, for every $i \in \{1,\,\ldots,\,k\}$, we have $K_{PG}([\Theta_i|_{A_i}])=[\Psi_i|_{A_i}](K_{PG}([\Phi_i|_{A_i}]))$.

Let $i \in \{1,\,\ldots,\,k\}$. Let $\clF \wedge \{[A_i]\}$ be the free factor system of $A_i$ induced by $\clF$: it is the free factor system of $A_i$ consisting in the intersection of $A_i$ with every subgroup $A$ of $F_{N}$ such that $[A] \in \clF$. It is well-defined by for instance~\cite[Theorem~3.14]{ScoWal79}.

%\medskip

%\noindent{\bf Claim. }
\begin{enonce*}[plain]{Claim}
We have
\[
\widehat{\Delta}_+([A_i],\phi) \cap \psi\left(\widehat{\Delta}_-([A_i],\phi)\right)=\varnothing\text{ and }\widehat{\Delta}_-([A_i],\phi) \cap \psi\left(\widehat{\Delta}_+([A_i],\phi)\right)=\varnothing.
\]
\end{enonce*}
%\medskip

\begin{proof}
We prove the first equality, the other one being similar. By Lemma~\ref{Lem preparation ping pong}, we have $\Delta_+([A_i],\phi) \cap \psi(\Delta_-([A_i],\phi))=\varnothing$ and $\psi(K_{PG}([\Phi_i|_{A_i}])) \cap K_{PG}([\Phi_i|_{A_i}])=\varnothing$.

Let $[\mu] \in \widehat{\Delta}_+([A_i],\phi) \cap \psi(\widehat{\Delta}_-([A_i],\phi))$. By definition, there exist $[\mu_1] \in \Delta_+([A_i],\phi)$, $[\nu_1] \in K_{PG}([\Phi_i|_{A_i}])$, $t \in [0,1]$, and $[\mu_2] \in \psi(\Delta_-([A_i],\phi)$, $[\nu_2] \in \psi(K_{PG}([\Phi_i|_{A_i}]))$ and $s \in [0,1]$ such that
\[
[\mu]=[t\mu_1+(1-t)\nu_1]=[s\mu_2+(1-s)\nu_2].
\]
Note that
\[
\partial^2 (\clF \wedge \{[A_i]\}) \cap \partial^2\clA(\phi) \cap \partial^2(A_i, \clF \wedge \{[A_i]\}\wedge \clA(\phi))=\varnothing.
\]
Moreover, since $\Poly(\phi|_{\clF})=\Poly(H|_{\clF})$, we have $\Poly(\theta|_{\clF})=\Poly(H|_\clF)$. Therefore, we see that $\clF\wedge\clA(\phi)=\clF\wedge \psi(\clA(\phi))$. Thus, we have
\[
\partial^2 (\clF \wedge \{[A_i]\}) \cap \psi\left(\partial^2\clA(\phi)\right) \cap \partial^2(A_i, \clF \wedge \{[A_i]\}\wedge \clA(\phi))=\varnothing.
\]
Recall that, by Proposition~\ref{Prop Existence and properties of Delta}, the supports of the currents in
\[
\Delta_+([A_i],\phi) \cup \psi(\Delta_-([A_i],\phi))
\]
are contained in $\partial^2(\clF\wedge \{[A_i]\})$. Thus, we have
\begin{multline*}
\mu_1\left(\partial^2\clA(\phi) \cap \partial^2(A_i, \clF \wedge \{[A_i]\}\wedge \clA(\phi))\right)\\
= \mu_1\left(\partial^2 (\clF \wedge \{[A_i]\}) \cap \partial^2\clA(\phi) \cap \partial^2(A_i, \clF \wedge \{[A_i]\}\wedge \clA(\phi))\right) =0.
\end{multline*}
Since $\clF\wedge\clA(\phi)=\clF\wedge \psi(\clA(\phi))$, we also have
\begin{multline*}
\mu_2\left(\partial^2\clA(\phi) \cap \partial^2(A_i, \clF \wedge \{[A_i]\}\wedge \clA(\phi))\right) \\
\begin{aligned}
&=\mu_2\left(\partial^2 (\clF \wedge \{[A_i]\}) \cap \partial^2\clA(\phi) \cap \partial^2(A_i, \clF \wedge \{[A_i]\}\wedge \clA(\phi))\right) \\
\;&= \mu_2\left(\partial^2 (\clF \wedge \{[A_i]\}) \cap \psi(\partial^2\clA(\phi)) \cap \partial^2(A_i, \clF \wedge \{[A_i]\}\wedge \clA(\phi))\right)= 0.
\end{aligned}
\end{multline*}

Thus, if $B$ is a measurable subset contained in $\partial^2\clA(\phi) \cap \partial^2(A_i, \clF \wedge \{[A_i]\}\wedge \clA(\phi))$ and if $s,t<1$, we have: $\mu(B)>0$ if and only if $\nu_1(B)>0$ if and only if $\nu_2(B)>0.$

By definition, the supports of currents in $K_{PG}([\Phi_i|_{A_i}])$ are contained in the subset $\partial^2\clA(\phi) \cap \partial^2(A_i, \clF \wedge \{[A_i]\}\wedge \clA(\phi))$ and the supports of currents in $\psi(K_{PG}([\Phi_i|_{A_i}]))$ are contained in $\psi(\partial^2\clA(\phi)) \cap \partial^2(A_i, \clF \wedge \{[A_i]\}\wedge \clA(\phi))$. Hence the support of $\nu_1$ is contained in the support of $\nu_2$. By definition of $\psi(K_{PG}([\Phi_i|_{A_i}]))$, this implies that
\[
\nu_1 \in K_{PG}([\Phi_i|_{A_i}]) \cap \psi(K_{PG}([\Phi_i|_{A_i}]))=\varnothing.
\]
Thus, we necessarily have $t=1$.

Thus, we have $[\mu]=[\mu_1]$ and the support of $\mu$ is contained in $\partial^2(\clF \wedge \{[A_i]\})$. Since the support of $\nu_2$ is contained in $\psi(\partial^2\clA(\phi)) \cap \partial^2(A_i, \clF \wedge \{[A_i]\}\wedge \clA(\phi))$ which is disjoint from $\partial^2(\clF \wedge \{[A_i]\})$, we also have $s=1$. This implies that $[\mu_1]=[\mu_2]$ and that $\Delta_+([A_i],\phi) \cap \psi(\Delta_-([A_i],\phi))\neq \varnothing,$ a contradiction.
\end{proof}

%\medskip

By the claim, there exist subsets $U,V, \widehat{U}, \widehat{V}$ of $\PCurr(A_i,(\clF \wedge \{[A_i]\}) \wedge \clA(\phi))$ such that:

%\medskip
\begin{enumerate}
\item\label{1p.23} $\Delta_+([A_i],\phi) \subseteq U$, $\widehat{\Delta}_+([A_i],\phi) \subseteq \widehat{U}$, $\Delta_-([A_i],\phi) \subseteq V$, $\widehat{\Delta}_-([A_i],\phi) \subseteq \widehat{V}$;

\item\label{2p.23} $U \subseteq \widehat{U}$, $V \subseteq \widehat{V}$ and $U \cap K_{PG}(\phi)=V \cap K_{PG}(\phi)=\varnothing$;

\item\label{3p.23} $\widehat{U} \cap \psi (\widehat{V})=\varnothing$ and $\widehat{V} \cap \psi (\widehat{U})=\varnothing$.
\end{enumerate}

Note that Assertion~\eqref{2p.23} implies that $U \subsetneq \widehat{U}$ (resp. $V \subsetneq \widehat{V}$) since $K_{PG}(\phi) \subseteq \widehat{U}$ (resp. $K_{PG}(\phi) \subseteq \widehat{V}$). Let $\frkB$ and $C>0$ be respectively the basis of $F_{N}$ and the constant given by Proposition~\ref{Prop moving exponential1}$\MK$\eqref{prop2.11.1}. Let $M_0(\phi)$ (resp. $M_0(\theta^{-1})$) be the constant associated with $\phi$, $U$ and $\widehat{V}$ (resp $\theta^{-1}$, $\psi(V)$ and $\psi(\widehat{U})$) given by Theorem~\ref{Theo North-South dynamics almost atoroidal}. Let $M_1(\phi)$ and $L_1(\phi)$, (resp. $M_1(\theta)$ and $L_1(\theta)$) be the constants associated with $[\Phi_i|_{A_i}]$ and $\widehat{V}$ (resp. $[\Theta_i|_{A_i}]$ and $\psi(\widehat{V})$) given by Proposition~\ref{Prop moving exponential}. Similarly, let $M_1(\phi^{-1})$ and $L_1(\phi^{-1})$, (resp. $M_1(\theta^{-1})$ and $L_1(\theta^{-1})$) be the constants associated with $[\Phi_i|_{A_i}^{-1}]$ and $\widehat{U}$ (resp. $[\Theta_i|_{A_i}^{-1}]$ and $\psi(\widehat{U})$) given by Proposition~\ref{Prop moving exponential}. Let
\begin{align*}
M(i) &=\max
\left\{M_0(\phi),M_0(\theta^{-1}),M_1(\phi),M_1(\theta),
M_1(\phi^{-1}),M_1(\theta^{-1})\right\}
\\
\intertext{and let}
L &=\min
\left\{L_1(\phi),L_1(\theta),L_1(\phi^{-1}),L_1
(\theta^{-1})\right\} >0.
\end{align*}

Let $M(i)'$ be such that $3^{M(i)'}L^2>1$. Let
\[
M=\max_{i\,\in\,\{1,\,\ldots,\,k\}} M(i)\quad\text{and}\quad M'=\max_{i\,\in\,\{1,\,\ldots,\,k\}} M(i)'.
\]

Let $m,n \geq M+M'$ and let $\mu \in \Curr(A_i,\clF \wedge \{[A_i]\} \wedge \clA(\phi))$ be a nonzero current. We will prove that $[\mu] \notin K_{PG}(\theta^m\phi^n)$. This will imply that for every element $g \in F_N$ such that $\eta_{[g]} \in \Curr(A_i,\clF \wedge \{[A_i]\} \wedge \clA(\phi))$, we have $g \notin \Poly(\theta^m\phi^n)$. The proof is in two steps according to whether $[\mu] \in \widehat{V}$ or not.

\noindent{$\bullet$ } Suppose first that $[\mu] \notin \widehat{V}$. Then by Theorem~\ref{Theo North-South dynamics almost atoroidal}, we have $\phi^n(\mu) \in U$. By Proposition~\ref{Prop moving exponential}, we have $\lVert \phi^n(\mu) \rVert_{\clF} \geq 3^{n-M}L\lVert \mu \rVert_{\clF}$. Since $U \cap \psi(\widehat{V})=\varnothing$, by Proposition~\ref{Prop moving exponential}, we have
\[
\left\lVert \theta^m\phi^n(\mu) \right\rVert_{\clF} \geq 3^{m-M}L\left\lVert \phi^n(\mu) \right\rVert_{\clF} \geq 3^{m+n-2M}L^2\lVert \mu \rVert_{\clF}.
\]

Note that, by Theorem~\ref{Theo North-South dynamics almost atoroidal} applied to $\theta$ and the open subsets $\psi(V)$ $\psi(U)$, $\psi(\widehat{V})$ and $\psi(\widehat{U})$, we have $\theta^m\phi^n([\mu]) \in \psi(U) \subseteq \psi(\widehat{U})$. Since $\widehat{V} \cap \psi(\widehat{U})=\varnothing$, we have $\theta^m\phi^n([\mu]) \notin \widehat{V}$. Therefore, we can apply the same arguments replacing $\mu$ by $\theta^m\phi^n(\mu)$ and an inductive argument shows that, for every $n' \in \NN^*$, we have
\[
\left\lVert \left(\theta^m\phi^n\right)^{n'}(\mu) \right\rVert_{\clF} \geq 3^{n'(m+n-2M-M')}\left(3^{M'}L^2\right)^{n'}\lVert \mu \rVert_{\clF}.
\]
Therefore, if $\mu$ is the current associated with an $\clF \wedge \{[A_i]\} \wedge \clA(\phi)$-nonperipheral element $g \in A_i$ with $[\mu] \notin \widehat{V}$, for every $n' \geq 1$, by Proposition~\ref{Prop moving exponential1}$\MK$\eqref{prop2.11.1} we have
\[
\ell_{\frkB}\big((\theta^m\phi^n)^{n'}([g])\big) \geq 3^{n'\left(m+n-2M-M'\right)}\left(3^{M'}L^2\right)^{n'}C\lVert \mu \rVert_{\clF} \geq 3^{n'\left(m+n-2M-M'\right)}C.
\]
Hence we have $g \notin \Poly([\Theta_i^m\Phi_i^n|_{A_i}])$.

\vspace*{4pt}
\noindent{$\bullet$ } Suppose now that $[\mu] \in \widehat{V}$. As in the first case, This implies that $[\mu] \notin \psi(\widehat{U})$ and, by Theorem~\ref{Theo North-South dynamics almost atoroidal}, that $\theta^{-m}([\mu]) \in \psi(V)$. By Proposition~\ref{Prop moving exponential}, we have $\lVert \theta^{-m}(\mu) \rVert_{\clF} \geq 3^{m-M}L\lVert \mu \rVert_{\clF}$. Since $\psi(V) \cap \widehat{U}=\varnothing$, we have $\theta^{-m}([\mu]) \notin \widehat{U}$ and
\[
\left\lVert \phi^{-n}\theta^{-m}(\mu)\right\rVert_{\clF} \geq 3^{n-M}L\left\lVert \theta^{-m}(\mu) \right\rVert_{\clF} \geq 3^{n+m-2M-M'}\left(3^{M'}L^2\right)\lVert \mu \rVert_{\clF}.
\]
By Theorem~\ref{Theo North-South dynamics almost atoroidal}, we have $\phi^{-n}\theta^{-m}(\mu) \in V$. As in the first case, since $\widehat{V} \cap \psi(\widehat{U})=\varnothing$, we have $\phi^{-n}\theta^{-m}(\mu) \notin \psi(\widehat{U})$ and, for every $n' \in \NN^*$, we have
\[
\left\lVert \left(\phi^{-n}\theta^{-m}\right)^{n'}(\mu) \right\rVert_{\clF} \geq 3^{n'(m+n-2M-M')}\left(3^{M'}L^2\right)^{2n'}\lVert \mu \rVert_{\clF}.
\]

Therefore as in the first case, replacing $\mu$ by the rational current associated with an $\clF \wedge \{[A_i]\} \wedge \clA(\phi)$-nonperipheral element $g \in A_i$ with $[\mu] \in \widehat{V}$, we see that
\[
g \notin \Poly\left([\Phi_i^{-n}\Theta_i^{-m}|_{A_i}]\right)=\Poly\left([\Theta_i^m\Phi_i^n|_{A_i}]\right).
\]

Therefore, $\theta^m\phi^n|_{\clF_1}$ is expanding relative to $\clF \wedge \clA(\phi)$. Thus, we have
\[
\Poly(\theta^m\phi^n|_{\clF_1})=\Poly(\phi|_{\clF})=\Poly(H|_{\clF}) \subseteq \Poly(H|_{\clF_1}).
\]
Since $\Poly(H|_{\clF_1}) \subseteq \Poly(\theta^m\phi^n|_{\clF_1})$, we have in fact $\Poly(H|_{\clF_1}) = \Poly(\theta^m\phi^n|_{\clF_1})$. This concludes the proof of Proposition~\ref{Prop ping pong}.
\end{proof}

\begin{prop}\label{Prop pushing sporadic extensions}
Let $N \geq 3$ and let $H$ be a subgroup of $\IA_{N}(\ZZ/3\ZZ)$. Let
\[
\varnothing=\clF_0 < \clF_1 <\ldots < \clF_k=\{[F_{N}]\}
\]
be a maximal $H$-invariant sequence of free factor systems. Let $2 \leq i \leq k$. Suppose that $\clF_{i-1} \leq \clF_i$ is sporadic. Suppose that there exists $\phi \in H$ such that
\begin{enumerate}\alphenumi
\item\label{prop4.5.a} $\Poly(H|_{\clF_{i-1}})=\Poly(\phi|_{\clF_{i-1}})$;

\item\label{prop4.5.b} for every $j \in \{1,\,\ldots,\,k\}$, if the extension $\clF_{j-1} \leq \clF_j$ is nonsporadic, then $\phi|_{\clF_j}$ is fully irreducible relative to $\clF_{j-1}$ and if $H|_{\clF_{j}}$ is atoroidal relative to $\clF_{j-1}$, so is $\phi|_{\clF_j}$.
\end{enumerate}
Then there exists $\widehat{\phi} \in H$ such that:
\begin{enumerate}
\item\label{prop4.5.1} $\Poly(H|_{\clF_{i}})=\Poly(\widehat{\phi}|_{\clF_{i}})$;

\item\label{prop4.5.2} for every $j \in \{1,\,\ldots,\,k\}$, if the extension $\clF_{j-1} \leq \clF_j$ is nonsporadic, then $\widehat{\phi}|_{\clF_j}$ is fully irreducible relative to $\clF_{j-1}$ and if $H|_{\clF_{j}}$ is atoroidal relative to $\clF_{j-1}$, so is $\widehat{\phi}|_{\clF_j}$.
\end{enumerate}
\end{prop}

\begin{proof}
The proof follows~\cite[Proposition~5.3]{clay2019atoroidal}. If $\Poly(H|_{\clF_{i}})=\Poly(\phi|_{\clF_{i}})$, we may take $\widehat{\phi}=\phi$.

Otherwise, by Proposition~\ref{Prop ping pong}, there exist $\psi \in H$ and a constant $M>0$ such that, for every $m,n \geq M$, if $\theta=\psi\phi\psi^{-1}$, we have $\Poly(\theta^m\phi^n|_{\clF_i})=\Poly(H|_{\clF_i})$. Therefore, for every $m,n \geq M$, the element $\widehat{\phi}=\theta^m\phi^n$ satisfies~\eqref{prop4.5.1}.

It remains to show that there exist $m,n \geq M$ such that $\theta^m\phi^n$ satisfies~\eqref{prop4.5.2}. Let
\[
S=\left\{j\;\middle|\; \text{the extension } \clF_{j-1} \leq \clF_j \text{ is nonsporadic}\right\}
\]
and let $j \in S$.

Let $X_j$ be the Gromov hyperbolic space equipped with an isometric action of $H$ constructed in the proof of Theorem~\ref{Theo nonsporadic extension}. Then $\psi \in H$ is a loxodromic element of $X_j$ for every $j \in S$ if and only if $\psi$ satisfies Hypothesis~\eqref{prop4.5.b} of Proposition~\ref{Prop pushing sporadic extensions}. In particular, the elements $\phi$ and $\theta$ are loxodromic elements of $X_j$.

Recall that two loxodromic isometries of a Gromov-hyperbolic space $X$ are \emph{independent} if their fixed point sets in $\partial_{\infty} X$ are disjoint and are \emph{dependent} otherwise. Let $I \subseteq S$ be the subset of indices where for every $j \in I$, the elements $\phi$ and $\theta$ are independent and let $D=S-I$. By standard ping pong arguments (see for instance~\cite[Proposition~4.2, Theorem~3.1]{ClayUya2018}), there exist constants $m,n_0 \geq M$ such that for every $n \geq n_0$ and every $j \in I$, the element $\theta^m\phi^n$ acts loxodromically on $X_j$. By~\cite[Proposition~3.4]{ClayUya2018}, there exists $n \geq n_0$ such that, for every $j \in D$ and every $j \in I$, the element $\theta^m\phi^n$ acts loxodromically on $X_j$.

Thus, for every $j \in S$ and every $n \geq n_0$, the element $\theta^m\phi^n$ satisfies Hypothesis~\eqref{prop4.5.b}. 

%Hence for every $j \in S_1$, the element $\theta^m\phi^n|_{\clF_j}$ is fully irreducible atoroidal relative to $\clF_{j-1}$ and for every $j \in S_2$, the element $\theta^m\phi^n|_{\clF_j}$ is fully irreducible relative to $\clF_{j-1}$.
This concludes the proof of Proposition~\ref{Prop pushing sporadic extensions}.
\end{proof}

\section{Proof of the main result and applications}

We are now ready to complete the proof of our main theorem.

\begin{theo}\label{Theo poly growth}
Let $N \geq 3$ and let $H$ be a subgroup of $\Out(F_{N})$. There exists $\phi \in H$ such that $\Poly(\phi)=\Poly(H)$.
\end{theo}

\begin{proof}
Since $\IA_{N}(\ZZ/3\ZZ)$ is a finite index subgroup of $\Out(F_{N})$ and since for every $\psi \in H$ and every $n \in \NN^*$, we have $\Poly(\psi^k)=\Poly(\psi)$, we see that
\[
\Poly(H)=\Poly(H \cap \IA_{N}(\ZZ/3\ZZ)).
\]
Hence we may suppose that $H$ is a subgroup of $\IA_{N}(\ZZ/3\ZZ)$.

Let
\[
\varnothing=\clF_0 < \clF_1 <\ldots < \clF_k=\{[F_{N}]\}
\]
be a maximal $H$-invariant sequence of free factor systems. By Theorem~\ref{Theo nonsporadic extension}, there exists $\phi \in H$ such that for every $j \in \{1,\,\ldots,\,k\}$ such that the extension $\clF_{j-1} \leq \clF_j$ is nonsporadic, the element $\phi|_{\clF_j}$ is fully irreducible relative to $\clF_{j-1}$ and if $H|_{\clF_j}$ is atoroidal relative to $\clF_{j-1}$, so is $\phi|_{\clF_{j-1}}$.

We now prove by induction on $i \in \{0,\,\ldots,\,k\}$ that for every $i \in \{0,\,\ldots,\,k\}$, there exists $\phi_i \in H$ such that

%\medskip
\begin{enumerate}\alphenumi
\item\label{prooftheo5.1.a} $\Poly(\phi_i|_{\clF_i})=\Poly(H|_{\clF_i})$;
\item\label{prooftheo5.1.b} for every $j \in \{1,\,\ldots,\,k\}$ such that the extension $\clF_{j-1} \leq \clF_j$ is nonsporadic, the element $\phi_i|_{\clF_j}$ is fully irreducible relative to $\clF_{j-1}$ and if $H|_{\clF_j}$ is atoroidal relative to $\clF_{j-1}$, so is $\phi_i|_{\clF_{j-1}}$.
\end{enumerate}

For the base case $i=0$, we set $\phi_0=\phi$.

Let $i \in \{1,\,\ldots,\,k\}$ and suppose that $\phi_{i-1} \in H$ has been constructed. We distinguish between two cases, according to the nature of the extension $\clF_{i-1} \leq \clF_i$.

Suppose first that the extension $\clF_{i-1} \leq \clF_i$ is nonsporadic. We set $\phi_i=\phi_{i-1}$. We claim that $\phi_i$ satisfies the hypotheses. Indeed, it clearly satisfies~\eqref{prooftheo5.1.b}.

For~\eqref{prooftheo5.1.a}, since $\Poly(\phi_{i-1}|_{\clF_{i-1}})=\Poly(H|_{\clF_{i-1}})$, it suffices to show that for every element $g \in F_{N}$ which is $\clF_i$-peripheral but $\clF_{i-1}$-nonperipheral, if $g \in \Poly(\phi_i|_{\clF_i})$, then $g \in \Poly(H|_{\clF_i})$.

Note that, if $\phi_i|_{\clF_i}$ is atoroidal relative to $\clF_{i-1}$, by Proposition~\ref{Prop properties fully irreducible}$\MK$\eqref{prop3.1.1}, we have $\Poly(\phi_i|_{\clF_i})=\Poly(\phi_i|_{\clF_{i-1}})$. Hence we have $\Poly(H|_{\clF_i})=\Poly(\phi_i|_{\clF_i})$. So we may suppose that $\phi_i|_{\clF_i}$ is not atoroidal relative to $\clF_{i-1}$.

Let $g \in \Poly(\phi_i|_{\clF_i})$ be an element which is $\clF_i$-peripheral but $\clF_{i-1}$-nonperipheral. By Proposition~\ref{Prop properties fully irreducible}$\MK$\eqref{prop3.1.1}, there exists at most one (up to taking inverse) $h \in F_{N}$ such that $g \in \left\langle h \right\rangle$ and $[h]$ is fixed by $\phi_i$. By Proposition~\ref{Prop properties fully irreducible}$\MK$\eqref{prop3.1.2.b}, the conjugacy class of $[h]$ is fixed by $H$. Hence the conjugacy class of $[g]$ is fixed by $H$ and $g \in \Poly(H|_{\clF_i})$.

Suppose now that $\clF_{i-1} \leq \clF_i$ is a sporadic extension. If $\Poly(\phi_{i-1}|_{\clF_i})=\Poly(H|_{\clF_i})$, we set $\phi_i=\phi_{i-1}$. Then $\phi_i$ satisfies $(a)$ and $(b)$. Suppose that $\Poly(\phi_{i-1}|_{\clF_i})\neq \Poly(H|_{\clF_i})$. By Proposition~\ref{Prop pushing sporadic extensions}, there exists $\widehat{\phi}_{i-1} \in H$ such that $\widehat{\phi}_{i-1}$ satisfies~\eqref{prop4.5.a}
 and~\eqref{prop4.5.b}. Then we set $\phi_i=\widehat{\phi}_{i-1}$. This completes the induction argument. In particular, we have $\Poly(\phi_m)=\Poly(H)$. This concludes the proof of Theorem~\ref{Theo poly growth}.
\end{proof}
%\bigskip

We now give some applications of Theorem~\ref{Theo poly growth}. The first one is a straightforward consequence using the fact that for every $\phi \in \Out(F_{N})$, there exists a natural malnormal subgroup system associated with $\Poly(\phi)$.

\begin{coro}\label{Coro malnormal subgroup system}
Let $N \geq 3$ and let $H$ be a subgroup of $\Out(F_N)$ such that $\Poly(H) \neq \{1\}$. There exist nontrivial maximal subgroups $A_1,\,\ldots,\,A_k$ of $F_{N}$ such that
\[
\Poly(H)=\bigcup_{i=1}^k\bigcup_{g\,\in\,F_{N}} gA_ig^{-1}
\]
and $\clA=\{[A_1],\,\ldots,\,[A_k]\}$ is a malnormal subgroup system.
\end{coro}
%\end{proof}\let\qed\relax

If $H$ is a subgroup of $\Out(F_{N})$ such that $\Poly(H)\neq \{1\}$, we denote by $\clA(H)$ the malnormal subgroup system given by Corollary~\ref{Coro malnormal subgroup system}. If $\Poly(H)=\{1\}$, we set $\clA(H)=\varnothing$.

%\bigskip

The following result is a generalization of~\cite[Theorem~A]{clay2019atoroidal} regarding fixed conjugacy classes. For a subgroup system $\clA$ of $F_N$, recall the definition of $\Out(F_N,\clA^{(t)})$ above Definition~\ref{Defi malnormal subgroup system}. If $\phi\in \IA_{N}(\ZZ/3\ZZ)$, we denote by $\Fix(\phi)$ the set of conjugacy classes of maximal subgroups $P$ of $F_{N}$ such that $\phi \in \Out(F_N,\{[P]\}^{(t)})$. Note that, if $P$ is a subgroup of $F_N$ such that $[P] \in \Fix(\phi)$, then $P \subseteq \Poly(\phi)$. Moreover, by~\cite[Lemma~1.5]{Levitt09}, if $\Poly(\phi) \neq \{1\}$, the set $\Fix(\phi)$ is nonempty. If $H$ is a subgroup of $\IA_{N}(\ZZ/3\ZZ)$, we denote by $\Fix(H)$ the set of conjugacy classes of maximal subgroups $P$ of $F_N$ such that $H \subseteq
\Out(F_N,\{[P]\}^{(t)})$. The following result is a corollary of the existence of the malnormal subgroup system $\clA(H)$ associated with a subgroup $H$ of $\Out(F_N)$ constructed in Corollary~\ref{Coro malnormal subgroup system}.


\begin{coro}\label{Coro alternative single elements}
Let $N \geq 3$ and let $H$ be a subgroup of $\IA_{N}(\ZZ/3\ZZ)$. One of the following (mutually exclusive) statements holds.
\begin{enumerate}
\item\label{coro5.3.1} There exist a (possibly empty) finite set $\clC$ of conjugacy classes of maximal cyclic subgroups of $F_N$ such that
\[
\Fix(H)=\clA(H)=\clC.
\]

\item\label{coro5.3.2} There exists a nonabelian free subgroup $P$ of $F_N$ such that 
\[
H \subseteq \Out\left(F_N,\{[P]\}^{(t)}\right).
\]
\end{enumerate}
\end{coro}

\begin{proof}
First assume that $H$ is finitely generated. Suppose that~\eqref{coro5.3.1} does not hold. Let $\clA(H)=\{[P_1],\,\ldots,\,[P_{\ell}]\}$, where for every $i \in \{1,\,\ldots,\,\ell\}$, $P_i$ is a malnormal subgroup of $F_{N}$. Note that, for every $i \in \{1,\,\ldots,\,\ell\}$, since $P_i$ is malnormal, we have a natural homomorphism $H \to \Out(P_i)$ whose image, denoted by $H|_{P_i}$, is contained in the set of polynomially growing outer automorphisms of $P_i$.

Note that, since Assertion~\eqref{coro5.3.1} does not hold, there exists $i \in \{1,\,\ldots,\,\ell\}$ such that the rank of $P_i$ is at least equal to $2$. From now on we focus on this $P_i$ and the subgroup $H|_{P_i}$ of $\Out(P_i)$.

Since $H$ is finitely generated, up to taking a finite index subgroup of $H$, we can apply the Kolchin theorem for $\Out(F_{N})$ (see~\cite[Theorem~1.1]{BesFeiHan05}): there exists a $H|_{P_i}$-invariant sequence of free factor systems of $P_i$
\[
\varnothing=\clF_0^{(i)} < \clF_1^{(i)} <\ldots < \clF_{k_i}^{(i)}=\{[P_i]\}
\]
such that, for every $j \in \{1,\,\ldots,\,k_i\}$, the extension $\clF_{j-1}^{(i)} \leq \clF_j^{(i)}$ is sporadic.

Since, for every $j \in \{1,\,\ldots,\,k_i\}$, the extension $\clF_{j-1}^{(i)} \leq \clF_j^{(i)}$ is sporadic, we have $k_i \geq 2$.

Let $j_0$ be the maximal integer such that $\clF_{j_0-1}^{(i)}$ consists only in conjugacy classes of cyclic subgroups of $P_i$. The existence of $j_0$ follows from the following facts. First, we have $\clF_{k_i}^{(i)}=\{[P_i]\}$ with $P_i$ a nonabelian free subgroup. Moreover, since the extension $\varnothing \leq \clF_{1}^{(i)}$ is sporadic, the free factor system $\clF_{1}^{(i)}$ consists in the conjugacy class of a cyclic subgroup of $P_i$.

Since the extension $\clF_{j_0-1}^{(i)} \leq \clF_{j_0}^{(i)}$ is sporadic, by maximality of $j_0$, there exists a subgroup $U_{j_0}$ of $P_i$ such that $[U_{j_0}] \in \clF_{j_0}^{(i)}$ and one of the following holds:
\begin{enumerate}\alphenumi
\item\label{proof5.3.a} there exist two subgroups $B_1$ and $B_2$ of $P_i$ such that $\rank(B_1)=\rank(B_2)=1$, $[B_1],[B_2] \in \clF_{j_0-1}$ and $U_{j_0}=B_1 \ast B_2$;

\item\label{proof5.3.b} there exists a subgroup $B$ of $P_i$ such that $\rank(B)=1$, $[B] \in \clF_{j_0-1}$ and $U_{j_0}$ is an HNN extension of $B$ over the trivial group.
\end{enumerate}
If Case~\eqref{proof5.3.a} occurs, then $H$ acts as the identity on $U_{j_0}$ since $\rank(U_{j_0})=2$ and since every element of $H$ fixes elementwise a set of conjugacy classes of generators of $U_{j_0}$ (recall that the abelianization homomorphism $F_2 \to \ZZ^2$ induces an isomorphism $\Out(F_2) \simeq \GL(2,\ZZ)$). Hence Assertion~\eqref{coro5.3.2} holds.

If Case~\eqref{proof5.3.b} occurs, let $b$ be a generator of $B$ and let $t \in U_{j_0}$ be such that $U_{j_0}=\left\langle b \right\rangle \ast \left\langle t \right\rangle$. Then, since $H \subseteq \IA_{N}(\ZZ/3\ZZ)$, for every element $\psi$ of $H$, there exist $\Psi \in \psi$ and $k \in \ZZ$ such that $\Psi(b)=b$ and $\Psi(t)=tb^k$. In particular, for every $\psi \in H$, the automorphism $\Psi$ fixes the group generated by $b$ and $tbt^{-1}$ and Assertion~\eqref{coro5.3.2} holds. This concludes the proof when $H$ is finitely generated.

Suppose now that $H$ is not finitely generated and let $(H_m)_{m\,\in\,\NN}$ be an increasing sequence of finitely generated subgroups of $H$ such that $H=\bigcup_{m\,\in\,\NN} H_m$. For every $m \in \NN$, we have $H_m \subseteq \Out(F_{N},\Fix(H_m)^{(t)})$ and for every $m_1,m_2 \in \NN$ such that $m_1 \leq m_2$, we have $\Fix(H_{m_2}) \subseteq \Fix(H_{m_1})$. By~\cite[Theorem~1.5]{GuirardelLevitt2015McCool}, there exists $N \in \NN$ such that, for every $m \geq N$, we have $\Out(F_{N},\Fix(H_m)^{(t)})=\Out(F_{N},\Fix(H_N)^{(t)})$. In particular, we have $\Fix(H_N)=\Fix(H)$. The result now follows from the finitely generated case.
\end{proof}
%\hfill\qedsymbol

%\bigskip

The following result might be folklore as it is a consequence of the JSJ decomposition of $F_N$ relative to a cyclic subgroup not contained in any free factor, but we did not find a precise statement in the literature. If $S$ is a compact, connected surface, we denote by $\Mod(S)$ the group of homotopy classes of homeomorphisms that preserve the boundary of $S$.

\begin{coro}\label{Coro pseudo Anosov subgroup Mod}
Let $N \geq 3$ and let $H$ be a subgroup of $\IA_{N}(\ZZ/3\ZZ)$. The following assertions are equivalent:
\begin{enumerate}
\item\label{coro5.4.1} $\clA(H)=\{[\left\langle g \right\rangle]\}$, where $g$ is an element of $F_{N}$ not contained in a proper free factor of $F_{N}$;

\item\label{coro5.4.2} there exist a connected, compact surface $S$ with exactly one boundary component and an identification of $\pi_1(S)$ with $F_{N}$ such that $H$ is identified with a subgroup of $\Mod(S)$ and $H$ contains a pseudo-Anosov element.
\end{enumerate}
\end{coro}


\begin{proof}
The implication \eqref{coro5.4.2}$\Rightarrow$\eqref{coro5.4.1} is well known and a proof can be found for instance in~\cite[Corollary~7.5.4]{Guerch}. Suppose that~\eqref{coro5.4.1} holds. Let $\phi \in H$ be an element given by Theorem~\ref{Theo poly growth}. Then $\clA(\phi)=\clA(H)=\{[\left\langle g \right\rangle]\}$. In particular, since $H \subseteq \IA_{N}(\ZZ/3\ZZ)$, the conjugacy class of $g$ is fixed by every element of $H$. Let $f \colon G \to G$ be a CT map representing a power of $\phi$ (see the definition in~\cite[Definition~4.7]{FeiHan06}).

%\medskip

\begin{enonce*}{Claim}
The graph $G$ consists in a single stratum and this stratum is an EG stratum.
\end{enonce*}
%\medskip

\begin{proof}
Let $H_r$ be the highest stratum in $G$. Note that, since $g$ is not contained in any proper free factor of $F_{N}$, the reduced circuit $\gamma_g$ in $G$ representing the conjugacy class of $g$ has height $r$ and is fixed by $f$.

We now prove that $H_r$ is an EG stratum. Indeed, $H_r$ is either a zero stratum, an EG stratum or an NEG stratum. The stratum $H_r$ cannot be a zero stratum by~\cite[Definition~4.7$\MK$(6)]{FeiHan06}. Moreover, $H_r$ cannot be a NEG stratum as otherwise by~\cite[Proposition~4.1]{clay2019atoroidal}, since $\gamma_g$ has height $r$, the element $g$ would be a basis element of $F_{N}$, contradicting the fact that $g$ is not contained in any proper free factor of $F_{N}$. Hence $H_r$ is an EG stratum.

By~\cite[Fact~I.2.3]{HandelMosher20}, the stratum $H_r$ is a geometric stratum in the sense of~\cite[Definition~I.2.1]{HandelMosher20}. By~\cite[Proposition~I.2.18]{HandelMosher20}, the element $\phi$ fixes elementwise a finite set $\clC=\{[g],[c_1],\ldots,[c_k]\}$ of conjugacy classes of elements of $F_{N}$. Since $G$ is connected, by the definition of a geometric stratum and by~\cite[Proposition~I.2.18$\MK$(5)]{HandelMosher20}, the stratum $H_r$ is glued on $G_{r-1}$ along closed paths in $G_{r-1}$ whose associated reduced circuits represent the conjugacy classes $[c_1],\,\ldots,\,[c_k]$. Thus, we have $k \geq 1$ whenever $G_{r-1}$ is not reduced to a point. This implies that $\clC=\{[g]\}$ if and only if $G_{r-1}$ is reduced to a point, that is, if and only if $G$ consists in the single stratum $H_r$.
\end{proof}

%\medskip

By the claim and~\cite[Fact~I.2.3]{HandelMosher20} (see also~\cite[Theorem~4.1]{BesHan92}), the outer automorphism $\phi$ is \emph{geometric}: there exist a connected, compact surface $S$ with exactly one boundary component and an identification of $\pi_1(S)$ with $F_{N}$ such that $\phi$ is identified with a pseudo-Anosov element of $\Mod(S)$. Moreover, the conjugacy class $[g]$ is identified with the conjugacy class in $\pi_1(S)$ of the element associated with the homotopy class of the boundary component of $S$. Since $[g]$ is fixed by every element of $H$, by the Dehn-Nielsen-Baer theorem (see for instance~\cite[Theorem~8.8]{FarMar12} and~\cite[Theorem~5.6.2]{ZieschangVogtColdewey1980} for the orientable case and~\cite[Section~3]{Fujiwara2002} for the nonorientable case), the group $H$ is identified with a subgroup of $\Mod(S)$.
\end{proof}
%\bigskip

We finally state a proposition, whose proof can be found in~\cite[Proposition~7.5.6]{Guerch} in a more general setting, which allows us to compute the malnormal subgroup system $\clA(H)$ associated with some subgroups $H$ of $\Out(F_N)$. The definitions and properties associated with JSJ decompositions of $F_N$ can be found for instance in~\cite{guirardel2016jsj}, especially~\cite[Definitions~2.14, 5.13]{guirardel2016jsj}.

\begin{prop}
[{\cite[Proposition~7.5.6]{Guerch}}]
Let $N \geq 3$ and let $P$ be a finitely generated subgroup of $F_N$ such that $F_N$ is one-ended relative to $P$. Let $T$ be the $JSJ$ tree of $F_N$ over cyclic groups relative to $P$. Suppose that $\Out(F_N,\{[P]\}^{(t)})$ is infinite. Every subgroup $Q$ of $F_N$ such that $[Q] \in \clA(\Out(F_N,\{[P]\}^{(t)}))$ is either generated by stabilizers of some rigid vertices of $T$ or is an extended boundary subgroup of the stabilizer of some flexible vertex of $T$.
\end{prop}

\subsection*{Acknowledgements}
I warmly thank my advisors, Camille Horbez and Frédéric Paulin, for their precious advices and for carefully reading the different versions of this article. I also thank the anonymous referee for his/her numerous very helpful remarks.


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