Measure Of Maximal Entropy Research Articles

Maximal entropy measures in dimension zero

Maximal entropy measures in dimension zero

Bowen parameter and Hausdorff dimension for expanding rational semigroups

We estimate the Bowen parameters and the Hausdorff dimensions of the Julia sets of expanding finitely generated rational semigroups. We show that the Bowen parameter is larger than or equal to the ratio of the entropy of the skew product map $\tilde{f}$ and the Lyapunov exponent of $\tilde{f}$ with respect to the maximal entropy measure for $\tilde{f}$. Moreover, we show that the equality holds if and only if the generators are simultaneously conjugate to the form $a_{j}z^{\pm d}$ by a M\"{o}bius transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than $2$.

Rich phase transitions in step skew products

We present examples of partially hyperbolic and topologically transitive local diffeomorphisms defined as skew products over a horseshoe which exhibit rich phase transitions for the topological pressure. This phase transition follows from a gap in the spectrum of the central Lyapunov exponents. It is associated with the coexistence of two equilibrium states with positive entropy. The diffeomorphisms mix hyperbolic behaviour of different types. However, in some sense the expanding behaviour is not dominating which is indicated by the existence of a measure of maximal entropy with nonpositive central exponent.

Computability of Brolin-Lyubich Measure

Brolin-Lyubich measure $\lambda_R$ of a rational endomorphism $R:\riem\to\riem$ with $\deg R\geq 2$ is the unique invariant measure of maximal entropy $h_{\lambda_R}=h_{\text{top}}(R)=\log d$. Its support is the Julia set $J(R)$. We demonstrate that $\lambda_R$ is always computable by an algorithm which has access to coefficients of $R$, even when $J(R)$ is not computable. In the case when $R$ is a polynomial, Brolin-Lyubich measure coincides with the harmonic measure of the basin of infinity. We find a sufficient condition for computability of the harmonic measure of a domain, which holds for the basin of infinity of a polynomial mapping, and show that computability may fail for a general domain.

On some coding and mixing properties for a class of chaotic systems

We study certain ergodic properties of equilibrium measures of hyperbolic non-invertible maps f on basic sets with overlaps Λ. We prove that if the equilibrium measure \({\mu_\phi}\) of a Holder potential \({\phi}\) , is 1-sided Bernoulli, then f is expanding from the point of view of a pointwise section dimension of \({\mu_\phi}\) . If the measure of maximal entropy μ0 is 1-sided Bernoulli, then f is shown to be distance expanding on Λ; and if \({\mu_\phi}\) is 1-sided Bernoulli for f expanding, then \({\mu_\phi}\) must be the measure of maximal entropy. These properties are very different from the case of hyperbolic diffeomorphisms. Another result is about the non 1-sided Bernoullicity for certain equilibrium measures for hyperbolic toral endomorphisms. We also prove the non-existence of generating Rokhlin partitions for measure-preserving endomorphisms in several cases, among which the case of hyperbolic non-expanding toral endomorphisms with Haar measure. Nevertheless the system \({(\Lambda, f, \mu_\phi)}\) is shown to have always exponential decay of correlations on Holder observables and to be mixing of any order.

Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets

We give approximations for the Gibbsstates of arbitrary Hölder potentials $\phi$,with the help of weighted sums of atomic measures on preimage sets, in the case of smooth non-invertible maps hyperbolic onfolded basic sets $\Lambda$. The endomorphism may have also stable directions on$\Lambda$ and is non-expanding in general. Folding of the phase space means that we do not have a foliation structure for the local unstable manifolds(instead they depend on the whole past and may intersect each other both inside and outside $\Lambda$). We consider here simultaneously all $n$-preimages in $\Lambda$ of a point, instead of the usual way of taking only the consecutive preimages from some given prehistory. We thus obtain the weighted distribution of consecutive preimage sets, with respect to various equilibrium measures on the saddle-type folded set $\Lambda$. In particular we obtain the distribution of preimage sets on $\Lambda$, with respect to the measure of maximal entropy. Our result is not a direct application of Birkhoff Ergodic Theorem on the inverse limit $\hat \Lambda$, since the set of prehistories of a point is uncountable in general, and the speed of convergence may vary for different prehistories in $\hat \Lambda$. For hyperbolic toral endomorphisms, we obtain the distribution of the consecutive preimage sets towards an inverse SRB measure, for Lebesgue-almost all points.

Discontinuity of topological entropy for Lozi maps

AbstractRecently, Buzzi [Maximal entropy measures for piecewise affine surface homeomorphisms. Ergod. Th. & Dynam. Sys.29 (2009), 1723–1763] showed in the compact case that the entropy map f→htop(f) is lower semi-continuous for all piecewise affine surface homeomorphisms. We prove that topological entropy for Lozi maps can jump from zero to a value above 0.1203 as one crosses a particular parameter and hence it is not upper semi-continuous in general. Moreover, our results can be extended to a small neighborhood of this parameter showing the jump in the entropy occurs along a line segment in the parameter space.

Uniqueness of equilibrium states for some partially hyperbolic horseshoes

In this note, we consider a partially hyperbolic horseshoe and prove uniqueness of equilibrium states for a class of potentials. In particular we obtain that there exists a unique maximal entropy measure.

Existence and uniqueness of the measure of maximal entropy for the Teichmüller flow on the moduli space of Abelian differentials

The main result of the paper is the statement that the 'smooth' measure of Masur and Veech is the unique measure of maximal entropy for the Teichmüller flow on the moduli space of Abelian differentials. The proof is based on the symbolic representation of the flow in Veech's space of zippered rectangles.Bibliography: 29 titles.

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

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.

Subshifts of finite type which have completely positive entropy

Domino tilings have been studied extensively for both their statistical properties and their dynamical properties. We construct a subshift of finite type using matching rules for several types of dominos. We combine the previous results about domino tilings to show that our subshift of finite type has a measure of maximal entropy with which the subshift has completely positive entropy but is not isomorphic to a Bernoulli shift.

Nice Inducing Schemes and the Thermodynamics of Rational Maps

We study the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter t we study the existence of equilibrium states of f for the potential \({-t {\rm ln} \left|f^{\prime} \right|}\) , and the analytic dependence on t of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism for a large class of rational maps, including well known classes of non-uniformly hyperbolic rational maps, such as (topological) Collet-Eckmann maps, and much beyond. In fact, our results apply to all non-renormalizable polynomials without indifferent periodic points, to infinitely renormalizable quadratic polynomials with a priori bounds, and all quadratic polynomials with real coefficients. As an application, for these maps we describe the dimension spectrum for Lyapunov exponents, and for pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results. For polynomials as above, we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in \({\mathbb{R}}\) , with at most two exceptions.

Measures of maximal relative entropy with full support

AbstractLet π be a factor map from an irreducible shift of finite type X to a shift space Y. Let ν be an invariant probability measure on Y with full support. We show that every measure on X of maximal relative entropy over ν is fully supported. As a result, given any invariant probability measure ν on Y with full support, there is an invariant probability measure μ on X with full support that maps to ν under π. If ν is ergodic, μ can be chosen to be ergodic. These results can be generalized to the case of sofic shifts. We demonstrate that the results do not extend to general shift spaces by providing counterexamples.

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

Let f be an endomorphism of \({\mathbb{C}\mathbb{P}^k}\) and ν be an f-invariant measure with positive Lyapunov exponents (λ1, . . . , λk). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by \({{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}\), which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of fn in \({\mathbb{C}\mathbb{P}^k}\) . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \({\mathbb{C}\mathbb{P}^k}\) and a normalization theorem for the ν-generic inverse branches of fn.

The maximal entropy measure detects non-uniform hyperbolicity

We characterize two of the most studied non-uniform hyperbolicity conditions for rational maps, semi-hyperbolicity and the topological Collet-Eckmann condition, in terms of the maximal entropy measure. Using the same tools in the proof of these results we give an extension of a result of Carleson, Jones and Yoccoz, that semi-hyperbolicity characterizes those polynomial maps whose basin of attraction of infinity is a John domain, to rational maps having a completely invariant attracting basin.

Entropy of meromorphic maps and dynamics of birational maps

We study the dynamics of meromorphic maps for a compact Kahler manifold X. More precisely, we give a simple criterion that allows us to produce a measure of maximal entropy. We can apply this result to bound the Lyapunov exponents. Then, we study the particular case of a family of generic birational maps of ℙ κ for which we construct the Green currents and the equilibrium measure. We use for that the theory of super-potentials. We show that the measure is mixing and gives no mass to pluripolar sets. Using the criterion we get that the measure is of maximal entropy. It implies finally that the measure is hyperbolic.

Dynamics of meromorphic maps with small topological degree I: from cohomology to currents

We consider the dynamics of a meromorphic map on a compact Kahler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which iterates of the map expand the cohomology class of a Kahler form. Our goal in this article and its sequels is to carry out a program for constructing and analyzing a natural measure of maximal entropy for each such map. Here we take the first step, using the linear action of the map on cohomology to construct and analyze invariant currents with special geometric structure. We also give some examples and consider in more detail the special cases where the surface is irrational or the self-intersections of the invariant currents vanish.

Generalized Chebyshev maps of $\mathbf{C}^{2}$ and their perturbations

The Chebyshev map is a typical chaotic map. We consider generalized Chebyshev maps $T_{k}$ of $\mathbf{C}^{2}$. The support of the maximal entropy measure of $T_{k}$ is connected. We perturb $T_{k}$ in a certain direction. Then we can show the support of the maximal entropy measure of this map is a Cantor set.

Abelian Sandpiles and the Harmonic Model

We present a construction of an entropy-preserving equivariant surjective map from the $d$-dimensional critical sandpile model to a certain closed, shift-invariant subgroup of $\mathbb{T}^{\mathbb{Z}^d}$ (the `harmonic model'). A similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the Bernoulli property of the measure of maximal entropy for that model.

Maximal entropy measures for piecewise affine surface homeomorphisms

AbstractWe study the dynamics of piecewise affine surface homeomorphisms from the point of view of their entropy. Under the assumption of positive topological entropy, we establish the existence of finitely many ergodic and invariant probability measures maximizing entropy and prove a multiplicative lower bound for the number of periodic points. This is intended as a step towards the understanding of surface diffeomorphisms. We proceed by building a jump transformation, using not first returns but carefully selected ‘good’ returns to dispense with Markov partitions. We control these good returns through some entropy and ergodic arguments.