Nous étudions comment les mesures physiques varient avec la dynamique sous-jacente, dans la classe ouverte des difféomorphismes , , fortement partiellement hyperboliques pour lesquelles les exposants de Lyapunov centraux de tout u-état de Gibbs sont positifs. Lorsque transitifs, de tels difféomorphismes possédent une unique mesure physique qui persiste et varie continûment avec la dynamique.
Un des ingrédients principaux de la preuve est un nouveau lemme de type Pliss qui, appliqué dans le contexte adéquat, implique que la fréquence des temps hyperboliques est proche de un. Une autre nouveauté est l'introduction d'une nouvelle caractérisation des cu-états de Gibbs. Chacun de ses deux aspects ayant leur propre intérêt.
Le cas non transitif est aussi traité : dans ce contexte, le nombre de mesures physiques est une fonction semi-continue supérieure du difféomorphisme, et les mesures physiques varient continûment sous des hypothèses naturelles.
We study how physical measures vary with the underlying dynamics in the open class of , , strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs u-state is positive. If transitive, such a diffeomorphism has a unique physical measure that persists and varies continuously with the dynamics.
A main ingredient in the proof is a new Pliss-like Lemma which, under the right circumstances, yields frequency of hyperbolic times close to one. Another novelty is the introduction of a new characterization of Gibbs cu-states. Both of these may be of independent interest.
The non-transitive case is also treated: here the number of physical measures varies upper semi-continuously with the diffeomorphism, and physical measures vary continuously whenever possible.
Mots-clés : Partial hyperbolicity, Lyapunov exponents, SRB measures, Stable ergodicity, Statistical stability
@article{AIHPC_2020__37_6_1245_0,
author = {Andersson, Martin and V\'asquez, Carlos H.},
title = {Statistical stability of mostly expanding diffeomorphisms},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1245--1270},
publisher = {Elsevier},
volume = {37},
number = {6},
year = {2020},
doi = {10.1016/j.anihpc.2020.04.007},
mrnumber = {4168916},
zbl = {1477.37041},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.anihpc.2020.04.007/}
}
TY - JOUR AU - Andersson, Martin AU - Vásquez, Carlos H. TI - Statistical stability of mostly expanding diffeomorphisms JO - Annales de l'I.H.P. Analyse non linéaire PY - 2020 SP - 1245 EP - 1270 VL - 37 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2020.04.007/ DO - 10.1016/j.anihpc.2020.04.007 LA - en ID - AIHPC_2020__37_6_1245_0 ER -
%0 Journal Article %A Andersson, Martin %A Vásquez, Carlos H. %T Statistical stability of mostly expanding diffeomorphisms %J Annales de l'I.H.P. Analyse non linéaire %D 2020 %P 1245-1270 %V 37 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2020.04.007/ %R 10.1016/j.anihpc.2020.04.007 %G en %F AIHPC_2020__37_6_1245_0
Andersson, Martin; Vásquez, Carlos H. Statistical stability of mostly expanding diffeomorphisms. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 6, pp. 1245-1270. doi : 10.1016/j.anihpc.2020.04.007. http://www.numdam.org/articles/10.1016/j.anihpc.2020.04.007/
[1] Nonuniform hyperbolicity for -generic diffeomorphisms, Isr. J. Math., Volume 183 (Jun 2011) no. 1, pp. 1 | MR | Zbl
[2] Strong statistical stability of non-uniformly expanding maps, Nonlinearity, Volume 17 (2004) no. 4, pp. 1193–1215 | MR | Zbl
[3] SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., Volume 140 (2000) no. 2, pp. 351–398 | MR | Zbl
[4] Statistical stability and continuity of SRB entropy for systems with Gibbs-Markov structures, Commun. Math. Phys., Volume 296 (2010) no. 3, pp. 739–767 | MR | Zbl
[5] Statistical stability for robust classes of maps with non-uniform expansion, Ergod. Theory Dyn. Syst., Volume 22 (2002) no. 1, pp. 1–32 | MR | Zbl
[6] SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. Éc. Norm. Supér. (4), Volume 33 (2000) no. 1, pp. 1–32 | Numdam | MR | Zbl
[7] Robust ergodic properties in partially hyperbolic dynamics, Trans. Am. Math. Soc., Volume 362 (2010) no. 4, pp. 1831–1867 | MR | Zbl
[8] On mostly expanding diffeomorphisms, Ergod. Theory Dyn. Syst., Volume 38 (2018) no. 8, pp. 2838–2859 | MR | Zbl
[9] Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms, Trans. Am. Math. Soc., Volume 364 (2012) no. 6, pp. 2883–2907 | MR | Zbl
[10] Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2007 | DOI | MR | Zbl
[11] Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective, Mathematical Physics, III, Encyclopaedia of Mathematical Sciences, vol. 102, Springer-Verlag, Berlin, 2005 | MR | Zbl
[12] SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Isr. J. Math., Volume 115 (2000), pp. 157–193 | MR | Zbl
[13] Stable ergodicity for partially hyperbolic attractors with negative central exponents, J. Mod. Dyn., Volume 2 (2008) no. 1, pp. 63–81 | MR | Zbl
[14] Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms, Invent. Math., Volume 201 (2015) no. 2, pp. 385–517 | MR | Zbl
[15] Limit theorems for partially hyperbolic systems, Trans. Am. Math. Soc., Volume 356 (2004) no. 4, pp. 1637–1689 | MR | Zbl
[16] On differentiability of SRB states for partially hyperbolic systems, Invent. Math., Volume 155 (2004) no. 2, pp. 389–449 | MR | Zbl
[17] Geometric and measure-theoretical structures of maps with mostly contracting center, Commun. Math. Phys., Volume 341 (2016) no. 3, pp. 991–1014 | MR | Zbl
[18] Partially hyperbolic dynamical systems, Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 1–55 | MR | Zbl
[19] Invariant Manifolds, Lecture Notes in Mathematics, vol. 583, Springer-Verlag, Berlin-New York, 1977 | DOI | MR | Zbl
[20] Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bull. Am. Math. Soc. (N.S.), Volume 31 (1994) no. 1, pp. 68–74 | MR | Zbl
[21] Ergodic Theory and Differentiable Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Results in Mathematics and Related Areas (3), vol. 8, Springer-Verlag, Berlin, 1987 (translated from the Portuguese by Silvio Levy) | DOI | MR | Zbl
[22] A note on partially hyperbolic systems with mostly expanding centers, Proc. Am. Math. Soc., Volume 27 (2017) no. 2, pp. 589–607 | MR | Zbl
[23] Gibbs measures for partially hyperbolic attractors, Ergod. Theory Dyn. Syst., Volume 2 (1982) no. 3–4, pp. 417–438 | MR | Zbl
[24] On a conjecture of Smale, Differ. Uravn., Volume 8 (1972), pp. 268–282 | MR | Zbl
[25] Ergodic attractors, Trans. Am. Math. Soc., Volume 312 (1989) no. 1, pp. 1–54 | MR | Zbl
[26] Pathological foliations and removable zero exponents, Invent. Math., Volume 139 (2000) no. 3, pp. 495–508 | MR | Zbl
[27] Statistical stability for diffeomorphisms with dominated splitting, Ergod. Theory Dyn. Syst., Volume 27 (2007) no. 1, pp. 253–283 | MR | Zbl
[28] Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents, J. Mod. Dyn., Volume 3 (2009) no. 2, pp. 233–251 | MR | Zbl
[29] Physical measures and absolute continuity for one-dimensional center direction, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 30 (2013) no. 5, pp. 845–877 | Numdam | MR | Zbl
[30] Jiagang Yang, Entropy along expanding foliations, arXiv e-prints, 2018. | MR
[31] Statistical properties of dynamical systems with some hyperbolicity, Ann. Math. (2), Volume 147 (1998) no. 3, pp. 585–650 | MR | Zbl
[32] Recurrence times and rates of mixing, Isr. J. Math., Volume 110 (1999), pp. 153–188 | MR | Zbl
Cité par Sources :
