scholarly journals Properties of the maximal entropy measure and geometry of Hénon attractors

2019 ◽  
Vol 21 (8) ◽  
pp. 2233-2299 ◽  
Author(s):  
Pierre Berger
Keyword(s):  
Maximal Entropy ◽  
Entropy Measure

scholarly journals Statistical properties of the maximal entropy measure for partially hyperbolic attractors

2016 ◽  
Vol 37 (4) ◽  
pp. 1060-1101 ◽  
Author(s):  
ARMANDO CASTRO ◽  
TEÓFILO NASCIMENTO
Keyword(s):  
Probability Measure ◽  
Hyperbolic Systems ◽  
Maximal Entropy ◽  
Entropy Measure ◽  
Hölder Continuous ◽  
Hyperbolic Diffeomorphisms ◽  
The Central Limit Theorem ◽  
Exponential Decay Of Correlations ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms

We show the existence and uniqueness of the maximal entropy probability measure for partially hyperbolic diffeomorphisms which are semiconjugate to non-uniformly expanding maps. Using the theory of projective metrics on cones, we then prove exponential decay of correlations for Hölder continuous observables and the central limit theorem for the maximal entropy probability measure. Moreover, for systems derived from a solenoid, we also prove the statistical stability for the maximal entropy probability measure. Finally, we use such techniques to obtain similar results in a context containing partially hyperbolic systems derived from Anosov.

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.

scholarly journals Large entropy implies existence of a maximal entropy measure for interval maps

2006 ◽  
Vol 14 (4) ◽  
pp. 673-688 ◽  
Author(s):  
Jérôme Buzzi ◽  
◽  
Sylvie Ruette ◽  
Keyword(s):  
Maximal Entropy ◽  
Entropy Measure ◽  
Interval Maps

The lap-counting function for linear mod one transformations II: the Markov chain for generalized lap numbers

1997 ◽  
Vol 17 (1) ◽  
pp. 123-146 ◽  
Author(s):  
LEOPOLD FLATTO ◽  
JEFFREY C. LAGARIAS
Keyword(s):  
Markov Chain ◽  
Unit Disk ◽  
Counting Function ◽  
Unit Interval ◽  
Maximal Entropy ◽  
Entropy Measure

Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is $L_{\beta,\alpha}(z)=\sum_{n=1}^{\infty} L_{n}z^{n}$, where $L_{n}$ essentially counts the number of monotonic pieces of the $n$th iterate $f_{\beta,\alpha}^{n}$. Part I showed that the function $L_{\beta,\alpha}(z)$ is meromorphic on the unit disk $|z|<1$ and analytic on $|z|<1/\beta$. This paper shows that the singularities of $L_{\beta,\alpha}(z)$ on the circle $|z|=1/\beta$ are contained in the set $\{(1/\beta)\exp (2\pi il/N):0\le l\le N-1\}$, for some integer $N\ge 1$. Here $N$ can be taken to be the period $N_{\beta,\alpha}$ of a certain Markov chain $\Sigma_{\beta,\alpha}$ which encodes information about generalized lap numbers $L_{n}(i,j)$ of $f_{\beta,\alpha}$, where $L_{n}(i,j)$ counts monotonic pieces of $f_{\beta,\alpha}^{n}$ whose image is $[f^{i}(0),f^{j}(1^{-}))$. We show that $N_{\beta,\alpha}=1$ whenever $\beta>2$. Finally, we give the criterion that $N_{\beta,\alpha}=1$ if and only if for all $n\ge 1$ the map $f_{\beta,\alpha}^{n}$ is ergodic with respect to the maximal entropy measure of $f_{\beta,\alpha}$.

scholarly journals The maximal entropy measure of Fatou boundaries

2018 ◽  
Vol 38 (9) ◽  
pp. 4421-4431
Author(s):  
Jane Hawkins ◽  
◽  
Michael Taylor
Keyword(s):  
Maximal Entropy ◽  
Entropy Measure

scholarly journals Effectivity of uniqueness of the maximal entropy measure on -adic homogeneous spaces

2015 ◽  
Vol 36 (6) ◽  
pp. 1972-1988 ◽  
Author(s):  
RENE RÜHR
Keyword(s):  
Dynamical System ◽  
Probability Measure ◽  
Algebraic Group ◽  
Homogeneous Spaces ◽  
Regular Representation ◽  
Invariant Probability Measure ◽  
Linear Algebraic Group ◽  
Maximal Entropy ◽  
Entropy Measure ◽  
Effective Version

We consider the dynamical system given by an $\text{Ad}$-diagonalizable element $a$ of the $\mathbb{Q}_{p}$-points $G$ of a unimodular linear algebraic group acting by translation on a finite volume quotient $X$. Assuming that this action is exponentially mixing (e.g. if $G$ is simple) we give an effective version (in terms of $K$-finite vectors of the regular representation) of the following statement: If ${\it\mu}$ is an $a$-invariant probability measure with measure-theoretical entropy close to the topological entropy of $a$, then ${\it\mu}$ is close to the unique $G$-invariant probability measure of $X$.

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