Equilibrium states for interval maps: the potential $-tlog\left|Df\right|$

Bruin, Henk; Todd, Mike

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 4, pp. 559-600.

Soit $f:I\to I$ une application multimodale de classe $C^2$ dont les dérivées le long des orbites des points critiques sont à croissance polynomiale, où $I$ est un intervalle. Nous démontrons l’existence et l’unicité d’un état d’équilibre pour le potentiel ${\phi }_t:x\mapsto -tlog\left|Df\left(x\right)\right|$ lorsque $t$ est proche de $1$, et que la fonction de pression $t\mapsto P\left({\phi }_t\right)$ est analytique sur un intervalle approprié près de $t=1$.

Let $f:I\to I$ be a $C^2$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential ${\phi }_t:x\mapsto -tlog\left|Df\left(x\right)\right|$ for $t$ close to $1$, and also that the pressure function $t\mapsto P\left({\phi }_t\right)$ is analytic on an appropriate interval near $t=1$.

Classification : 37D35, 37D25, 37E05
Keywords: equilibrium states, thermodynamic formalism, interval maps, non-uniform hyperbolicity
Mot clés : États d'équilibre, formalisme thermodynamique, applications de l'intervalle, hyperbolicité non-uniforme

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