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

2014 ◽  
Vol 64 (2) ◽  
pp. 645-680 ◽  
Author(s):  
Gabriel Vigny
Keyword(s):  
Maximal Entropy ◽  
Rational Maps ◽  
Measure Of Maximal Entropy

scholarly journals THE SCENERY FLOW FOR HYPERBOLIC JULIA SETS

2002 ◽  
Vol 85 (2) ◽  
pp. 467-492 ◽  
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.

REMARKS ON LARGE DEVIATION FOR RATIONAL MAPS ON THE RIEMANN SPHERE

2007 ◽  
Vol 07 (03) ◽  
pp. 357-363 ◽  
Author(s):  
HONGQIANG XIA ◽  
XINCHU FU
Keyword(s):  
Large Deviation ◽  
Riemann Sphere ◽  
Maximal Entropy ◽  
Rational Maps ◽  
Measure Of Maximal Entropy

We consider rational maps of degree d ≥ 2 on the Riemann sphere and obtain large deviation results for Hölder observables under the measure of maximal entropy.

Equilibrium measures for rational maps

1986 ◽  
Vol 6 (3) ◽  
pp. 393-399 ◽  
Author(s):  
Artur Oscar Lopes
Keyword(s):  
Equilibrium Measure ◽  
Julia Set ◽  
Maximal Entropy ◽  
Rational Maps ◽  
Rational Map ◽  
Equilibrium Measures ◽  
Polynomial Map ◽  
Two Measures ◽  
Measure Of Maximal Entropy

AbstractFor a polynomial map the measure of maximal entropy is the equilibrium measure for the logarithm potential in the Julia set [1], [4].Here we will show that in the case where f is a rational map such that f(∞) = ∞ and the Julia set is bounded, then the two measures mentioned above are equal if and only if f is a polynomial.

scholarly journals Thermodynamics of smooth models of pseudo–Anosov homeomorphisms

2021 ◽  
pp. 1-43
Author(s):  
DOMINIC VECONI
Keyword(s):  
Thermodynamic Formalism ◽  
Equilibrium States ◽  
Maximal Entropy ◽  
Srb Measure ◽  
The Central Limit Theorem ◽  
Exponential Decay Of Correlations ◽  
Surface Homeomorphisms ◽  
Young Towers ◽  
Uniqueness Of Equilibrium ◽  
Measure Of Maximal Entropy

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.

Conformally maximal polynomial-like dynamics and invariant harmonic measure

1997 ◽  
Vol 17 (1) ◽  
pp. 1-27 ◽  
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

2013 ◽  
Vol 34 (6) ◽  
pp. 1770-1793 ◽  
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.

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