Measure Of Maximal Entropy Research Articles

Flexibility of measure-theoretic entropy of boundary maps associated to Fuchsian groups

AbstractGiven a closed, orientable, compact surface S of constant negative curvature and genus $g \geq 2$ , we study the measure-theoretic entropy of the Bowen–Series boundary map with respect to its smooth invariant measure. We obtain an explicit formula for the entropy that only depends on the perimeter of the $(8g-4)$ -sided fundamental polygon of the surface S and its genus. Using this, we analyze how the entropy changes in the Teichmüller space of S and prove the following flexibility result: the measure-theoretic entropy takes all values between 0 and a maximum that is achieved on the surface that admits a regular $(8g-4)$ -sided fundamental polygon. We also compare the measure-theoretic entropy to the topological entropy of these maps and show that the smooth invariant measure is not a measure of maximal entropy.

Fluctuations of Time Averages Around Closed Geodesics in Non-Positive Curvature

We consider the geodesic flow for a rank one non-positive curvature closed manifold. We prove an asymptotic version of the Central Limit Theorem for families of measures constructed from regular closed geodesics converging to the Bowen-Margulis-Knieper measure of maximal entropy. The technique expands on ideas of Denker, Senti and Zhang, who proved this type of asymptotic Lindeberg Central Limit Theorem on periodic orbits for expansive maps with the specification property. We extend these techniques from the uniform to the non-uniform setting, and from discrete-time to continuous-time. We consider H\"older observables subject only to the Lindeberg condition and a weak positive variance condition. If we assume a natural strengthened positive variance condition, the Lindeberg condition is always satisfied. Our results extend to dynamical arrays of H\"older observables, and to weighted periodic orbit measures which converge to a unique equilibrium state.

There are no deviations for the ergodic averages of Giulietti–Liverani horocycle flows on the two-torus

AbstractWe show that the ergodic integrals for the horocycle flow on the two-torus associated by Giulietti and Liverani with an Anosov diffeomorphism either grow linearly or are bounded; in other words, there are no deviations. For this, we use the topological invariance of the Artin–Mazur zeta function to exclude resonances outside the open unit disc. Transfer operators acting on suitable spaces of anisotropic distributions and their Ruelle determinants are the key tools used in the proof. As a bonus, we show that for any $C^\infty $ Anosov diffeomorphism F on the two-torus, the correlations for the measure of maximal entropy and $C^\infty $ observables decay with a rate strictly smaller than $e^{-h_{\mathrm {top}}(F)}$ . We compare our results with very recent related work of Forni.

Uniqueness of the measure of maximal entropy for the standard map

In this paper we prove that for sufficiently large parameters the standard map has a unique measure of maximal entropy (m.m.e.). Moreover, we prove: the m.m.e. is Bernoulli, and the periodic points with Lyapunov exponents bounded away from zero equidistribute with respect to the m.m.e.We prove some estimates regarding the Hausdorff dimension of the m.m.e. and about the density of the support of the measure on the manifold. For a generic large parameter, we prove that the support of the m.m.e. has Hausdorff dimension 2. We also obtain the C^2 -robustness of several of these properties.

On subshifts with slow forbidden word growth

AbstractIn this work, we treat subshifts, defined in terms of an alphabet $\mathcal {A}$ and (usually infinite) forbidden list $\mathcal {F}$ , where the number of n-letter words in $\mathcal {F}$ has ‘slow growth rate’ in n. We show that such subshifts are well behaved in several ways; for instance, they are boundedly supermultiplicative in the sense of Baker and Ghenciu [Dynamical properties of S-gap shifts and other shift spaces. J. Math. Anal. Appl.430(2) (2015), 633–647] and they have unique measures of maximal entropy with the K-property and which satisfy Gibbs bounds on large (measure-theoretically) sets. The main tool in our proofs is a more general result, which states that bounded supermultiplicativity and a sort of measure-theoretic specification property together imply uniqueness of the measure of maximum entropy and our Gibbs bounds. We also show that some well-known classes of subshifts can be treated by our results, including the symbolic codings of $x \mapsto \alpha + \beta x$ (the so-called $\alpha $ - $\beta $ shifts of Hofbauer [Maximal measures for simple piecewise monotonic transformations. Z. Wahrsch. verw. Geb.52(3) (1980), 289–300]) and the bounded density subshifts of Stanley [Bounded density shifts. Ergod. Th. & Dynam. Sys.33(6) (2013), 1891–1928].

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

We show that the measure of maximal entropy for the hereditary closure of a $\mathscr {B}$-free subshift has the Gibbs property if and only if the Mirsky measure of the subshift is purely atomic. This answers an open question asked by Peckner. Moreover, w

Measures of maximal entropy on subsystems of topological suspension semiflows

Given a compact topological dynamical system $(X, f)$ with positive entropy and upper semicontinuous entropy map, and any closed invariant subset $Y \subset X$ with positive entropy, we show that there exists a continuous roof function such that the set o

Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics

We give an alternative proof of a result of Cantat and Dupont, showing that any automorphism of a K3 surface with measure of maximal entropy in the Lebesgue class must be a Kummer example. Our method exploits the existence of Ricci-flat metrics on K3s and also covers the non-projective case.

Regularity of entropy, geodesic currents and entropy at infinity

In this work, we introduce the notion of entropy at infinity, and define a wide class of noncompact manifolds with negative curvature, those which admit a critical gap between entropy at infinity and topological entropy. We call them strongly positively recurrent manifolds (SPR), and provide many examples. We show that dynamically, they behave as compact manifolds. In particular, they admit a finite measure of maximal entropy. Using the point of view of currents at infinity, we show that on these SPR manifolds the topological entropy of the geodesic flow varies in a C 1 -way along (uniformly) C 1 -perturbations of the metric. This result generalizes former work of Katok (1982) and Katok-Knieper-Weiss (1991) in the compact case.

Maximal entropy measures of diffeomorphisms of circle fiber bundles

We characterize the maximal entropy measures of partially hyperbolic C^2 diffeomorphisms whose center foliations form circle bundles, by means of suitable finite sets of saddle points, that we call skeletons. In the special case of 3-dimensional nilmanifolds other than the torus, this entails the following dichotomy: either the diffeomorphism is a rotation extension of an Anosov diffeomorphism -- in which case there is a unique maximal measure, with full support and zero center Lyapunov exponents -- or there exist exactly two ergodic maximal measures, both hyperbolic and whose center Lyapunov exponents have opposite signs. Moreover, the set of maximal measures varies continuously with the diffeomorphism.

Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points

We prove that for closed surfaces M with Riemannian metrics without conjugate points and genus ≥2 the geodesic flow on the unit tangent bundle T1M has a unique measure of maximal entropy. Furthermore, this measure is fully supported on T1M, is the limiting distribution of closed orbits, and the flow is mixing with respect to this measure. We formulate conditions under which this result extends to higher dimensions.

Entropy rigidity for 3D conservative Anosov flows and dispersing billiards

Given an integer $$k \ge 5$$ , and a $$C^k$$ Anosov flow $$\Phi $$ on some compact connected 3-manifold preserving a smooth volume, we show that the measure of maximal entropy is the volume measure if and only if $$\Phi $$ is $$C^{k-\varepsilon }$$ -conjugate to an algebraic flow, for $$\varepsilon >0$$ arbitrarily small. Moreover, in the case of dispersing billiards, we show that if the measure of maximal entropy is the volume measure, then the Birkhoff Normal Form of regular periodic orbits with a homoclinic intersection is linear.

Anosov flows, growth rates on covers and group extensions of subshifts

The aim of this paper is to study growth properties of group extensions of hyperbolic dynamical systems, where we do not assume that the extension satisfies the symmetry conditions seen, for example, in the work of Stadlbauer on symmetric group extensions and of the authors on geodesic flows. Our main application is to growth rates of periodic orbits for covers of an Anosov flow: we reduce the problem of counting periodic orbits in an amenable cover X to counting in a maximal abelian subcover Xab\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X^{\\mathrm {ab}}$$\\end{document}. In this way, we obtain an equivalence for the Gurevič entropy: h(X)=h(Xab)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h(X)=h(X^{\\mathrm {ab}})$$\\end{document} if and only if the covering group is amenable. In addition, when we project the periodic orbits for amenable covers X to the compact factor M, they equidistribute with respect to a natural equilibrium measure — in the case of the geodesic flow, the measure of maximal entropy.

On entropy and intrinsic ergodicity of coded subshifts

Any coded subshift X C X_C defined by a set C C of code words contains a subshift, which we call L C L_C , consisting of limits of single code words. We show that when C C satisfies the unique decipherability property, the topological entropy h ( X C ) h(X_C) of X C X_C is determined completely by h ( L C ) h(L_C) and the number of code words of each length. More specifically, we show that h ( X C ) = h ( L C ) h(X_C) = h(L_C) exactly when a certain infinite series is less than or equal to 1 1 , and when that series is greater than 1 1 , we give a formula for h ( X C ) h(X_C) . In the latter case, an immediate corollary (using a result from [Israel J. Math. 192 (2012), pp. 785–817] is that X C X_C has a unique measure of maximal entropy.

The unique measure of maximal entropy for a compact rank one locally CAT(0) space

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ X $\end{document}</tex-math></inline-formula> be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. We prove the Bowen-Margulis measure on the space of geodesics is the unique measure of maximal entropy for the geodesic flow, which has topological entropy equal to the critical exponent of the Poincaré series.

Measures of maximal entropy for suspension flows

We study suspension flows defined over sub-shifts of finite type with continuous roof functions. We prove the existence of suspension flows with uncountably many ergodic measures of maximal entropy. More generally, we prove that any suspension flow defined over a sub-shift of finite type can be perturbed (by an arbitrarily small perturbation) so that the resulting flow has uncountably many ergodic measures of maximal entropy, and that the same can be arranged so that the new flow has a unique measure of maximal entropy.

Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps

<p style='text-indent:20px;'>For a class of piecewise hyperbolic maps in two dimensions, we propose a combinatorial definition of topological entropy by counting the maximal, open, connected components of the phase space on which iterates of the map are smooth. We prove that this quantity dominates the measure theoretic entropies of all invariant probability measures of the system, and then construct an invariant measure whose entropy equals the proposed topological entropy. We prove that our measure is the unique measure of maximal entropy, that it is ergodic, gives positive measure to every open set, and has exponential decay of correlations against Hölder continuous functions. As a consequence, we also prove a lower bound on the rate of growth of periodic orbits. The main tool used in the paper is the construction of anisotropic Banach spaces of distributions on which the relevant weighted transfer operator has a spectral gap. We then construct our measure of maximal entropy by taking a product of left and right maximal eigenvectors of this operator.

On Rational Functions Sharing the Measure of Maximal Entropy

We show that describing rational functions $$f_1,$$ $$f_2,$$ $$\dots ,f_n$$ sharing the measure of maximal entropy reduces to describing solutions of the functional equation $$A\circ X_1=A\circ X_2=\dots =A\circ X_n$$ in rational functions. We also provide some results about solutions of this equation.

Generalized Active Information: Extensions to Unbounded Domains

In the last three decades, several measures of complexity have been proposed. Up to this point, most of such measures have only been developed for finite spaces. In these scenarios the baseline distribution is uniform. This makes sense because, among other things, the uniform distribution is the measure of maximum entropy over the relevant space. Active information traditionally assumes a finite interval universe of discourse but can be extended to other cases where maximum entropy is defined. Illustrating this is the purpose of this paper. Disequilibrium from maximum entropy, measured as active information, can be evaluated from baselines with unbounded support.

Weak Limits of the Measures of Maximal Entropy for Orthogonal Polynomials

In this paper we study the sequence of orthonormal polynomials $\{P_n(\mu; z)\}$ defined by a probability measure $\mu$ with non-polar compact support $S(\mu)\subset\mathbb C$. We show that the support of any weak* limit of the sequence of measures of maximal entropy $\omega_n$ for $P_n$ is contained in the polynomial-convex hull of $S(\mu)$. And for $n$-th root regular measures the $\omega_n$ converge weak* to the equilibrium measure on $S(\mu)$.