Measure Of Maximal Entropy Research Articles

Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy

We classify complex projective surfaces X with an automorphism f of positive entropy for which the unique measure of maximal entropy is absolutely continuous with respect to the Lebesgue measure. As a byproduct, if X is a K3 surface and is not a Kummer surface, the periodic points of f are equidistributed with respect to a probability measure which is singular with respect to the canonical volume of X . The proof is based on complex algebraic geometry and Hodge theory, Pesin’s theory and renormalization techniques. A crucial argument relies on a new compactness property of entire curves parametrizing the invariant manifolds of the automorphism.

On the measure of maximal entropy for finite horizon Sinai Billiard maps

The Sinai billiard map T T on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition h ∗ h_* for the topological entropy of T T . We prove that h ∗ h_* is not smaller than the value given by the variational principle, and that it is equal to the definitions of Bowen using spanning or separating sets. Under a mild condition of sparse recurrence to the singularities, we get more: First, using a transfer operator acting on a space of anisotropic distributions, we construct an invariant probability measure μ ∗ \mu _* of maximal entropy for T T (i.e., h μ ∗ ( T ) = h ∗ h_{\mu _*}(T)=h_* ), we show that μ ∗ \mu _* has full support and is Bernoulli, and we prove that μ ∗ \mu _* is the unique measure of maximal entropy and that it is different from the smooth invariant measure except if all nongrazing periodic orbits have multiplier equal to h ∗ h_* . Second, h ∗ h_* is equal to the Bowen–Pesin–Pitskel topological entropy of the restriction of T T to a noncompact domain of continuity. Last, applying results of Lima and Matheus, as upgraded by Buzzi, the map T T has at least C e n h ∗ C e^{nh_*} periodic points of period n n for all n ∈ N n \in \mathbb {N} .

Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds

<p style="text-indent:20px;">We study the geodesic flow of a class of 3-manifolds introduced by Benoist which have some hyperbolicity but are non-Riemannian, not CAT(0), and with non-<inline-formula><tex-math id="M1">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> geodesic flow. The geometries are nonstrictly convex Hilbert geometries in dimension three which admit compact quotient manifolds by discrete groups of projective transformations. We prove the Patterson–Sullivan density is canonical, with applications to counting, and construct explicitly the Bowen–Margulis measure of maximal entropy. The main result of this work is ergodicity of the Bowen–Margulis measure.

Ergodic frequency measures for random substitutions

We construct a family of ergodic measures on random substitution subshifts (RS-subshifts) associated to a primitive random substitution. In particular, the word frequencies of every finite legal word exist for almost every element of the random substitution subshift with respect to these measures. As an application, we show that for a certain class of random substitutions the measures of maximal entropy are frequency measures.

Fluctuations of ergodic sums on periodic orbits under specification

<p style='text-indent:20px;'>We study the fluctuations of ergodic sums using global and local specifications on periodic points. We obtain Lindeberg-type central limit theorems in both situations. As an application, when the system possesses a unique measure of maximal entropy, we show weak convergence of ergodic sums to a mixture of normal distributions. Our results suggest decomposing the variances of ergodic sums according to global and local sources.

Isomorphism and bi-Lipschitz equivalence between the univoque sets

In this paper, we consider a class of self-similar sets, denoted by \begin{document}$ \mathcal{A} $\end{document} , and investigate the set of points in the self-similar sets having unique codings. We call such set the univoque set and denote it by \begin{document}$ U_1 $\end{document} . We analyze the isomorphism and bi-Lipschitz equivalence between the univoque sets. The main result of this paper, in terms of the dimension of \begin{document}$ U_1 $\end{document} , is to give several equivalent conditions which describe that the closure of two univoque sets, under the lazy maps, are measure theoretically isomorphic with respect to the unique measure of maximal entropy. Moreover, we prove, under the condition \begin{document}$ U_1 $\end{document} is closed, that isomorphism and bi-Lipschitz equivalence between the univoque sets have resonant phenomenon.

Entropy of partially hyperbolic flows with center dimension two

In this article we study the regularity of the topological and metric entropy of partially hyperbolic flows with two-dimensional center direction. We show that the topological entropy is upper semicontinuous with respect to the flow, and we give an example where the lower semicontinuity fails. We also show that if such a flow has no fixed points, then it is entropy expansive, and consequentely the metric entropy function is upper semicontinuous, there exist equilibrium states (and measures of maximal entropy), and principal symbolic extensions.

Transitive topological Markov chains of given entropy and period with or without measure of maximal entropy

We show that, for every positive real number h and every positive integer p, there exist oriented graphs G, G' (with countably many vertices) that are strongly connected, of period p, of Gurevich entropy h, such that G is positive recurrent (thus the topological Markov chain on G admits a measure of maximal entropy) and G' is transient (thus the topological Markov chain on G' admits no measure of maximal entropy).

The bifurcation measure has maximal entropy

Let A be a complex manifold and let $${({f_\lambda})_{\lambda \in {\rm{\Lambda}}}}$$ be a holomorphic family of rational maps of degree d ≥ 2of ℙ1. We define a natural notion of entropy of bifurcation, mimicking the classical definition of entropy, by the parametric growth rate of critical orbits. We also define a notion of a measure-theoretic bifurcation entropy for which we prove a variational principle: the measure of bifurcation is a measure of maximal entropy. We rely crucially on a generalization of Yomdin’s bound of the volume of the image of a dynamical ball. Applying our results to complex dynamics in several variables, we notably define and compute the entropy of the trace measure of the Green currents of a holomorphic endomorphism of ℙk.

On the Patterson-Sullivan measure for geodesic flows on rank 1 manifolds without focal points

<p style='text-indent:20px;'>In this article, we consider the geodesic flow on a compact rank <inline-formula><tex-math id="M2">\begin{document}$ 1 $\end{document}</tex-math></inline-formula> Riemannian manifold <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> without focal points, whose universal cover is denoted by <inline-formula><tex-math id="M4">\begin{document}$ X $\end{document}</tex-math></inline-formula>. On the ideal boundary <inline-formula><tex-math id="M5">\begin{document}$ X(\infty) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M6">\begin{document}$ X $\end{document}</tex-math></inline-formula>, we show the existence and uniqueness of the Busemann density, which is realized via the Patterson-Sullivan measure. Based on the the Patterson-Sullivan measure, we show that the geodesic flow on <inline-formula><tex-math id="M7">\begin{document}$ M $\end{document}</tex-math></inline-formula> has a unique invariant measure of maximal entropy. We also obtain the asymptotic growth rate of the volume of geodesic spheres in <inline-formula><tex-math id="M8">\begin{document}$ X $\end{document}</tex-math></inline-formula> and the growth rate of the number of closed geodesics on <inline-formula><tex-math id="M9">\begin{document}$ M $\end{document}</tex-math></inline-formula>. These results generalize the work of Margulis and Knieper in the case of negative and nonpositive curvature respectively.

Intrinsic ergodicity for factors of -shifts

We show that every subshift factor of the ()-shift is intrinsically ergodic, when and the ()-expansion of 1 is not periodic with odd period. Moreover, the unique measure of maximal entropy satisfies a certain Gibbs property. This is an application of the technique established by Climenhaga and Thompson to prove intrinsic ergodicity for symbolic systems beyond specification. We also prove that there exists a subshift factor of the ()-shift which is not intrinsically ergodic in the cases other than the above.

Periodic expansiveness of smooth surface diffeomorphisms and applications

We prove that periodic asymptotic expansiveness introduced in [13] implies the equidistribution of periodic points along measures of maximal entropy. Then following Yomdin's approach [50] we show by using semi-algebraic tools that C^\infty interval maps and C^\infty surface diffeomorphisms satisfy this expansiveness property respectively for repelling and saddle hyperbolic points with Lyapunov exponents uniformly away from zero.

MAXENT3D_PID: An Estimator for the Maximum-Entropy Trivariate Partial Information Decomposition

Partial information decomposition (PID) separates the contributions of sources about a target into unique, redundant, and synergistic components of information. In essence, PID answers the question of “who knows what” of a system of random variables and hence has applications to a wide spectrum of fields ranging from social to biological sciences. The paper presents MaxEnt3D_Pid, an algorithm that computes the PID of three sources, based on a recently-proposed maximum entropy measure, using convex optimization (cone programming). We describe the algorithm and its associated software utilization and report the results of various experiments assessing its accuracy. Moreover, the paper shows that a hierarchy of bivariate and trivariate PID allows obtaining the finer quantities of the trivariate partial information measure.

The structure of the space of ergodic measures of transitive partially hyperbolic sets

We provide examples of transitive partially hyperbolic dynamics (specific but paradigmatic examples of homoclinic classes) which blend different types of hyperbolicity in the one-dimensional center direction. These homoclinic classes have two disjoint parts: an “exposed” piece which is poorly homoclinically related with the rest and a “core” with rich homoclinic relations. There is an associated natural division of the space of ergodic measures which are either supported on the exposed piece or on the core. We describe the topology of these two parts and show that they glue along nonhyperbolic measures. Measures of maximal entropy are discussed in more detail. We present examples where the measure of maximal entropy is nonhyperbolic. We also present examples where the measure of maximal entropy is unique and nonhyperbolic, however in this case the dynamics is nontransitive.

Quantile‐induced uncertain heavy ordered weighted averaging operator and the application in incentive evaluation problems

To uncertain evaluation problems, we integrate incentive management into the aggregation process and propose an aggregation operator called quantile-induced uncertain heavy ordered weighted averaging (QI-UHOWA) operator, which is an extension of the quantile-induced heavy ordered weighted averaging (QI-HOWA) operator. We provide an approach for determining the quantile order-inducing variables by using the technique for order preference by similarity to ideal solution method and the Hamming distance. In this case, the quantile values are measurements of relative developments of alternatives. Furthermore, we analyze the main properties of the operator including commutativity, boundedness, and monotonicity with uniform development space. The QI-UHOWA weighting vector is calculated using the maximum entropy measure with a given level of incentive attitude. We further expand the weighting method to the case of hierarchical stimulation. Moreover, the QI-UHOWA operator is generalized using the quasi-arithmetic mean. Finally, a numerical example regarding the selection of the optimal candidate(s) is given. The aggregation results are compared with those of the UOWA and QI-UOWA operator to illustrate the validity of the QI-UHOWA operator.

Extender sets and measures of maximal entropy for subshifts

For countable amenable finitely generated torsion-free G, we prove inequalities relating μ ( v ) and μ ( w ) for any measure of maximal entropy μ on a G-subshift and any words v , w where the extender set of v is contained in the extender set of w. Our main results are two generalizations of a theorem of Meyerovitch (Ergodic Theory Dynam. Systems 33 (2013) 934–953): the first applies to all such v , w when G = Z , and the second to v , w with the same shape for any G. As a consequence of our results we give new and simpler proofs of several facts about synchronized subshifts (including a result from Thomsen, Ergodic Theory Dynam. Systems 26 (2006) 1235–1256) and we answer a question of Climenhaga.

Multiscale sparse microcanonical models

We study approximations of non-Gaussian stationary processes having long range correlations with microcanonical models. These models are conditioned by the empirical value of an energy vector, evaluated on a single realization. Asymptotic properties of maximum entropy microcanonical and macrocanonical processes and their convergence to Gibbs measures are reviewed. We show that the Jacobian of the energy vector controls the entropy rate of microcanonical processes. Sampling maximum entropy processes through MCMC algorithms require too many operations when the number of constraints is large. We define microcanonical gradient descent processes by transporting a maximum entropy measure with a gradient descent algorithm which enforces the energy conditions. Convergence and symmetries are analyzed. Approximations of non-Gaussian processes with long range interactions are defined with multiscale energy vectors computed with wavelet and scattering transforms. Sparsity properties are captured with \mathbf{l}^1 norms. Approximations of Gaussian, Ising and point processes are studied, as well as image and audio texture synthesis.

Measures of maximal entropy for suspension flows over the full shift

We consider suspension flows with continuous roof function over the full shift $$\Sigma $$ on a finite alphabet. For any positive entropy subshift of finite type $$Y \subset \Sigma $$, we explicitly construct a roof function such that the measure(s) of maximal entropy for the suspension flow over $$\Sigma $$ are exactly the lifts of the measure(s) of maximal entropy for Y. In the case when Y is transitive, this gives a unique measure of maximal entropy for the flow which is not fully supported. If Y has more than one transitive component, all with the same entropy, this gives explicit examples of suspension flows over the full shift with multiple measures of maximal entropy. This contrasts with the case of a Holder continuous roof function where it is well known the measure of maximal entropy is unique and fully supported.

Properties of the maximal entropy measure and geometry of Hénon attractors

We consider an abundant class of non-uniformly hyperbolic C^2 -Hénon like diffeomorphisms called strongly regular and which corresponds to Benedicks–Carleson parameters. We prove the existence of m &gt; 0 such that for any such diffeomorphism f , every invariant probability measure of f has a Lyapunov exponent greater than m , answering a question of L. Carleson. Moreover, we show the existence and uniqueness of a measure of maximal entropy, which answers a question of M. Lyubich and Y. Pesin. We also prove that the maximal entropy measure is equidistributed on the periodic points and is finitarily Bernoulli, which gives an answer to a question of J.-P. Thouvenot. Finally, we show that the maximal entropy measure is exponentially mixing and satisfies the central limit theorem. The proof is based on a new construction of a Young tower for which the first return time coincides with the symbolic return time, and whose orbit is conjugate to a strongly positive recurrent Markov shift.

Entropy Spectrum of Lyapunov Exponents for Nonhyperbolic Step Skew-Products and Elliptic Cocycles

We study the fiber Lyapunov exponents of step skew-product maps over a complete shift of N, $${N\ge2}$$ , symbols and with C1 diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and zero exponents. Examples of such systems arise from the projective action of $${2\times 2}$$ matrix cocycles and our results apply to an open and dense subset of elliptic $${\mathrm{SL}(2,\mathbb{R})}$$ cocycles. We derive a multifractal analysis for the topological entropy of the level sets of Lyapunov exponent. The results are formulated in terms of Legendre–Fenchel transforms of restricted variational pressures, considering hyperbolic ergodic measures only, as well as in terms of restricted variational principles of entropies of ergodic measures with a given exponent. We show that the entropy of the level sets is a continuous function of the Lyapunov exponent. The level set of the zero exponent has positive, but not maximal, topological entropy. Under the additional assumption of proximality, as for example for skew-products arising from certain matrix cocycles, there exist two unique ergodic measures of maximal entropy, one with negative and one with positive fiber Lyapunov exponent.