scholarly journals Intrinsic ergodicity via obstruction entropies

2013 ◽  
Vol 34 (6) ◽  
pp. 1816-1831 ◽  
Author(s):  
VAUGHN CLIMENHAGA ◽  
DANIEL J. THOMPSON
Keyword(s):  
Topological Entropy ◽  
Maximal Entropy ◽  
Unique Measure ◽  
Measure Of Maximal Entropy

AbstractBowen showed that a continuous expansive map with specification has a unique measure of maximal entropy. We show that the conclusion remains true under weaker non-uniform versions of these hypotheses. To this end, we introduce the notions of obstructions to expansivity and specification, and show that if the entropy of such obstructions is smaller than the topological entropy of the map, then there is a unique measure of maximal entropy.

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

2011 ◽  
Vol 32 (1) ◽  
pp. 63-79 ◽  
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.

Algorithmic complexity of points in dynamical systems

1993 ◽  
Vol 13 (4) ◽  
pp. 807-830 ◽  
Author(s):  
Homer S. White
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Entropy ◽  
Invariant Probability Measure ◽  
Algorithmic Complexity ◽  
Maximal Entropy ◽  
Property A ◽  
Open Set ◽  
Unique Measure ◽  
Set Of Points

AbstractThis work is based on the author's dissertation. We examine the algorithmic complexity (in the sense of Kolmogorov and Chaitin) of the orbits of points in dynamical systems. Extending a theorem of A. A. Brudno, we note that for any ergodic invariant probability measure on a compact dynamical system, almost every trajectory has a limiting complexity equal to the entropy of the system. We use these results to show that for minimal dynamical systems, and for systems with the tracking property (a weaker version of specification), the set of points whose trajectories have upper complexity equal to the topological entropy is residual. We give an example of a topologically transitive system with positive entropy for which an uncountable open set of points has upper complexity equal to zero. We use techniques from universal data compression to prove a recurrence theorem: if a compact dynamical system has a unique measure of maximal entropy, then any point whose lower complexity is equal to the topological entropy is generic for that unique measure. Finally, we discuss algorithmic versions of the theorem of Kamae on preservation of the class of normal sequences under selection by sequences of zero Kamae-entropy.

scholarly journals Entropy properties of rational endomorphisms of the Riemann sphere

1983 ◽  
Vol 3 (3) ◽  
pp. 351-385 ◽  
Author(s):  
M. Ju. Ljubich
Keyword(s):  
Riemann Sphere ◽  
Periodic Points ◽  
Maximal Entropy ◽  
Unique Measure ◽  
Measure Of Maximal Entropy

AbstractIn this paper the existence of a unique measure of maximal entropy for rational endomorphisms of the Riemann sphere is established. The equidistribution of pre-images and periodic points with respect to this measure is proved.

scholarly journals Excursions to the cusps for geometrically finite hyperbolic orbifolds and equidistribution of closed geodesics in regular covers

2021 ◽  
pp. 1-47
Author(s):  
RON MOR
Keyword(s):  
Maximal Entropy ◽  
Closed Geodesics ◽  
Geometrically Finite ◽  
Unique Measure ◽  
Entropy Criterion ◽  
Measure Of Maximal Entropy

Abstract We give a finitary criterion for the convergence of measures on non-elementary geometrically finite hyperbolic orbifolds to the unique measure of maximal entropy. We give an entropy criterion controlling escape of mass to the cusps of the orbifold. Using this criterion, we prove new results on the distribution of collections of closed geodesics on such an orbifold, and as a corollary, we prove the equidistribution of closed geodesics up to a certain length in amenable regular covers of geometrically finite orbifolds.

Geodesic Flows Modelled by Expansive Flows

2018 ◽  
Vol 62 (1) ◽  
pp. 61-95 ◽  
Author(s):  
Katrin Gelfert ◽  
Rafael O. Ruggiero
Keyword(s):  
Geodesic Flow ◽  
Maximal Entropy ◽  
Geodesic Flows ◽  
Focal Points ◽  
Local Product ◽  
Unique Measure ◽  
Higher Genus ◽  
Expansive Flow ◽  
Expansive Flows ◽  
Measure Of Maximal Entropy

AbstractGiven a smooth compact surface without focal points and of higher genus, it is shown that its geodesic flow is semi-conjugate to a continuous expansive flow with a local product structure such that the semi-conjugation preserves time parametrization. It is concluded that the geodesic flow has a unique measure of maximal entropy.

scholarly journals Flexibility of Lyapunov exponents with respect to two classes of measures on the torus

2021 ◽  
pp. 1-40
Author(s):  
ALENA ERCHENKO
Keyword(s):  
Lyapunov Exponent ◽  
Probability Measure ◽  
Lebesgue Measure ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Maximal Entropy ◽  
Anosov Diffeomorphism ◽  
Dynamical Invariants ◽  
Area Preserving ◽  
Measure Of Maximal Entropy

Abstract We consider a smooth area-preserving Anosov diffeomorphism $f\colon \mathbb T^2\rightarrow \mathbb T^2$ homotopic to an Anosov automorphism L of $\mathbb T^2$ . It is known that the positive Lyapunov exponent of f with respect to the normalized Lebesgue measure is less than or equal to the topological entropy of L, which, in addition, is less than or equal to the Lyapunov exponent of f with respect to the probability measure of maximal entropy. Moreover, the equalities only occur simultaneously. We show that these are the only restrictions on these two dynamical invariants.

Non-uniqueness of measures of maximal entropy for subshifts of finite type

1994 ◽  
Vol 14 (2) ◽  
pp. 213-235 ◽  
Author(s):  
Robert Burton ◽  
Jeffrey E. Steif
Keyword(s):  
Finite Type ◽  
Higher Dimensions ◽  
One Dimension ◽  
Maximal Entropy ◽  
Subshift Of Finite Type ◽  
Unique Measure ◽  
Extremal Measures ◽  
Subshifts Of Finite Type ◽  
Measures Of Maximal Entropy ◽  
Measure Of Maximal Entropy

AbstractIt is known that in one dimension an irreducible subshift of finite type has a unique measure of maximal entropy, the so-called Parry measure. Here we give a counterexample to this in higher dimensions. For this example, we also describe the geometric structure of the measures of maximal entropy and show that there are exactly two extremal measures.

Entropy and volume growth

2000 ◽  
Vol 20 (1) ◽  
pp. 77-84 ◽  
Author(s):  
KURT COGSWELL
Keyword(s):  
Growth Rate ◽  
Probability Measure ◽  
Topological Entropy ◽  
Compact Manifold ◽  
Volume Growth ◽  
Maximal Entropy ◽  
Measure Of Maximal Entropy

We consider a $C^{1+1}$ diffeomorphism $f$ of a compact manifold $M$ which preserves an ergodic probability measure $\mu$. We conclude that $\mu$-a.e. $x \in M$ is contained in a disk $D_x \subset W^u(x)$, with $D_x$ open in the $W^u(x)$ topology, which exhibits an exponential volume growth rate greater than or equal to the measure-theoretic entropy of $f$ with respect to $\mu$. Drawing on results of Newhouse and Yomdin, we then find that when $f$ is $C^\infty$ and $\mu$ is a measure of maximal entropy, this exponential volume growth rate equals the topological entropy of $f$ for $\mu$-a.e. $x$.

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.

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