Polynomial diffeomorphisms ofC 2. IV: The measure of maximal entropy and laminar currents

Abstract We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy

Journal of the European Mathematical Society ◽

10.4171/jems/946 ◽

2020 ◽

Vol 22 (4) ◽

pp. 1289-1351

Author(s):

Serge Cantat ◽

Christophe Dupont

Keyword(s):

Maximal Entropy ◽

Measure Of Maximal Entropy

THE SCENERY FLOW FOR HYPERBOLIC JULIA SETS

Proceedings of the London Mathematical Society ◽

10.1112/s002461150201362x ◽

2002 ◽

Vol 85 (2) ◽

pp. 467-492 ◽

Cited By ~ 8

Author(s):

TIM BEDFORD ◽

ALBERT M. FISHER ◽

MARIUSZ URBAŃSKI

Keyword(s):

Julia Set ◽

Density Theorem ◽

Finite Union ◽

Maximal Entropy ◽

Rational Maps ◽

Rational Map ◽

Open Set ◽

Almost All ◽

Scenery Flow ◽

Measure Of Maximal Entropy

We define the scenery flow space at a point z in the Julia set J of a hyperbolic rational map $T : \mathbb{C} \to \mathbb{C}$ with degree at least 2, and more generally for T a conformal mixing repellor.We prove that, for hyperbolic rational maps, except for a few exceptional cases listed below, the scenery flow is ergodic. We also prove ergodicity for almost all conformal mixing repellors; here the statement is that the scenery flow is ergodic for the repellors which are not linear nor contained in a finite union of real-analytic curves, and furthermore that for the collection of such maps based on a fixed open set U, the ergodic cases form a dense open subset of that collection. Scenery flow ergodicity implies that one generates the same scenery flow by zooming down towards almost every z with respect to the Hausdorff measure $H^d$, where d is the dimension of J, and that the flow has a unique measure of maximal entropy.For all conformal mixing repellors, the flow is loosely Bernoulli and has topological entropy at most d. Moreover the flow at almost every point is the same up to a rotation, and so as a corollary, one has an analogue of the Lebesgue density theorem for the fractal set, giving a different proof of a theorem of Falconer.2000 Mathematical Subject Classification: 37F15, 37F35, 37D20.

Conformally maximal polynomial-like dynamics and invariant harmonic measure

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338579706104x ◽

1997 ◽

Vol 17 (1) ◽

pp. 1-27 ◽

Cited By ~ 2

Author(s):

ZOLTAN BALOGH ◽

IRINA POPOVICI ◽

ALEXANDER VOLBERG

Keyword(s):

Dynamical Systems ◽

Harmonic Measure ◽

Julia Set ◽

Maximal Entropy ◽

Generalized Polynomial ◽

Necessary And Sufficient ◽

Totally Disconnected ◽

Measure Of Maximal Entropy

Let $f : V \to U$ be a (generalized) polynomial-like map. Suppose that harmonic measure $\omega= \omega(\cdot,\infty)$ on the Julia set $J_{f}$ is equal to the measure of maximal entropy $m$ for $f : J_{f} \hookleftarrow$. Then the dynamics $(f,V,U)$ is called maximal. We are going to give a criterion for the dynamics to be conformally equivalent to a maximal one, that is to be conformally maximal. In the second part of this paper we construct an invariant ‘harmonic’ measure $\mu$ such that ${d\mu}/{d\omega}$ is Hölder for certain dynamics. This allows us to prove in this class of dynamical systems that $\omega\approx m$ is necessary and sufficient for $(f,V,U)$ to be conformally maximal. In the particular case when $f$ is expanding and $J_{f}$ is a circle, our result becomes a theorem of Shub and Sullivan; so throughout the paper we are dealing with an analog of a theorem of Shub and Sullivan on ‘wild’ (e.g. totally disconnected) $J_{f}$ and for certain non-expanding $f$. We also construct (under certain assumptions) invariant harmonic measure on $J_{f}$. In this respect, our work stems from one of the works of Carleson.

surface diffeomorphisms with no maximal entropy measure

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2013.25 ◽

2013 ◽

Vol 34 (6) ◽

pp. 1770-1793 ◽

Cited By ~ 3

Author(s):

JÉRÔME BUZZI

Keyword(s):

Maximal Entropy ◽

Entropy Measure ◽

Surface Diffeomorphisms ◽

Measure Of Maximal Entropy

AbstractFor any $1\leq r\lt \infty $, we build on the disk, and therefore on any manifold, a ${C}^{r} $-diffeomorphism with no measure of maximal entropy.

On the measure of maximal entropy for finite horizon Sinai Billiard maps

Journal of the American Mathematical Society ◽

10.1090/jams/939 ◽

2020 ◽

Vol 33 (2) ◽

pp. 381-449 ◽

Cited By ~ 1

Author(s):

Viviane Baladi ◽

Mark F. Demers

Keyword(s):

Finite Horizon ◽

Maximal Entropy ◽

Sinai Billiard ◽

Measure Of Maximal Entropy

Hyperbolic measure of maximal entropy for generic rational maps of \mathbb{P}^k

Annales de l’institut Fourier ◽

10.5802/aif.2861 ◽

2014 ◽

Vol 64 (2) ◽

pp. 645-680 ◽

Cited By ~ 1

Author(s):

Gabriel Vigny

Keyword(s):

Maximal Entropy ◽

Rational Maps ◽

Measure Of Maximal Entropy

Hereditary subshifts whose measure of maximal entropy does not have the Gibbs property

Colloquium Mathematicum ◽

10.4064/cm8223-11-2020 ◽

2021 ◽

Author(s):

Joanna Kułaga-Przymus ◽

Michał D. Lemańczyk

Keyword(s):

Maximal Entropy ◽

Measure Of Maximal Entropy

Continuity of the measure of maximal entropy for unimodal maps on the interval

Qualitative Theory of Dynamical Systems ◽

10.1007/bf02972823 ◽

2003 ◽

Vol 4 (1) ◽

pp. 67-76 ◽

Cited By ~ 3

Author(s):

Peter Raith

Keyword(s):

Maximal Entropy ◽

Unimodal Maps ◽

Measure Of Maximal Entropy

Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385710000854 ◽

2011 ◽

Vol 32 (1) ◽

pp. 63-79 ◽

Cited By ~ 22

Author(s):

J. BUZZI ◽

T. FISHER ◽

M. SAMBARINO ◽

C. VÁSQUEZ

Keyword(s):

Hopf Bifurcation ◽

Maximal Entropy ◽

Open Set ◽

Entropy Measures ◽

Unique Measure ◽

Anosov Systems ◽

Anosov System ◽

Partially Hyperbolic ◽

Structurally Stable ◽

Measure Of Maximal Entropy

AbstractWe show that a class of robustly transitive diffeomorphisms originally described by Mañé are intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have the following stability with respect to their entropy. Their topological entropy is constant and they each have a unique measure of maximal entropy with respect to which periodic orbits are equidistributed. Moreover, equipped with their respective measure of maximal entropy, these diffeomorphisms are pairwise isomorphic. We show that the method applies to several classes of systems which are similarly derived from Anosov, i.e. produced by an isotopy from an Anosov system, namely, a mixed Mañé example and one obtained through a Hopf bifurcation.