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On the cardinality of measures of maximal relative entropy for smooth skew products

Matheus M. Castro, Gary Froyland

TL;DR

This work studies how many ergodic measures of maximal $\mathbb{P}$-relative entropy can occur for smooth skew-product diffeomorphisms $\Theta$ over a transitive Anosov base. The authors combine a Sarig-type countable Markov coding for hyperbolic measures with Petersen–Quas–Shin finiteness arguments for factor maps to show that there are at most countably many hyperbolic ergodic measures maximizing $h_\mu(\Theta|\mathbb{P})$, and establish a corresponding countability result in the surface-fiber case when the relative entropy is positive. They also provide an application to random perturbations of the standard map, demonstrating the existence (and countability) of maximal $\mathbb{P}$-relative entropy measures in that setting. Overall, the paper extends countability phenomena from the classical entropy theory to the relative, random, skew-product context by integrating symbolic coding with relative-entropy finiteness methods, offering new tools for relative equilibrium states in random dynamics.

Abstract

Let $Ω$ and $M$ be compact smooth manifolds and let $Θ:Ω\times M\toΩ\times M$ be a $\mathcal C^{1+α}$ skew-product diffeomorphism over a transitive Anosov base. We show that $Θ$ has at most countably many ergodic hyperbolic measures of maximal relative entropy. When $\dim M=2$, if $Θ$ has positive relative topological entropy, then $Θ$ has at most countably many ergodic measures of maximal relative entropy.

On the cardinality of measures of maximal relative entropy for smooth skew products

TL;DR

This work studies how many ergodic measures of maximal -relative entropy can occur for smooth skew-product diffeomorphisms over a transitive Anosov base. The authors combine a Sarig-type countable Markov coding for hyperbolic measures with Petersen–Quas–Shin finiteness arguments for factor maps to show that there are at most countably many hyperbolic ergodic measures maximizing , and establish a corresponding countability result in the surface-fiber case when the relative entropy is positive. They also provide an application to random perturbations of the standard map, demonstrating the existence (and countability) of maximal -relative entropy measures in that setting. Overall, the paper extends countability phenomena from the classical entropy theory to the relative, random, skew-product context by integrating symbolic coding with relative-entropy finiteness methods, offering new tools for relative equilibrium states in random dynamics.

Abstract

Let and be compact smooth manifolds and let be a skew-product diffeomorphism over a transitive Anosov base. We show that has at most countably many ergodic hyperbolic measures of maximal relative entropy. When , if has positive relative topological entropy, then has at most countably many ergodic measures of maximal relative entropy.

Paper Structure

This paper contains 11 sections, 18 theorems, 189 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Measures of maximal relative entropy
  3. Main results
  4. An application of Corollary \ref{['cor:B']}
  5. Hyperbolic measures and Borel homoclinic classes
  6. Hyperbolic measures
  7. Borel homoclinic classes
  8. Topological Markov shifts
  9. Measures of maximal relative entropy on topological Markov shifts
  10. Petersen-Quas-Shin construction
  11. Proofs of Theorem A, Corollary B and Proposition \ref{['prop:example']}

Key Result

Theorem A

Let $\Omega$ and $M$ be compact smooth manifolds, and be a $\mathcal{C}^{1+\alpha}$ diffeomorphism, for some $\alpha>0$. Assume that $\theta:\Omega\to\Omega$ is a transitive Anosov diffeomorphism and $\mathbb P$ is a $\theta$-invariant ergodic probability measure with full support. Then, $\Theta$ admits at most countably many hyperbolic ergodic measure

Figures (2)

  • Figure 1: Illustration of a rectangle $R$ (shown as green dots) as the intersection of segments of stable and unstable manifolds (blue and red curves, respectively).
  • Figure 2: Illustration of the graph transform in Steps 1–2 for $k=4$ and $\omega_{0}=0$. The grey strips indicate $S_1$ (left) and $S_2$ (right). The orange curves are the restrictions $\gamma_{1}=\gamma|_{J_{1}}\in\mathcal{G}_1$ and $\gamma_{2}=\gamma|_{J_{2}}\in\mathcal{G}_2$ of $\gamma(t)=(t,t/2)$, while the blue curves are their images $T_{\omega_{0}}\circ\gamma_{1}$ and $T_{\omega_{0}}\circ\gamma_{2}$.

Theorems & Definitions (45)

  • Definition 1.1: Metric entropy
  • Definition 1.2: Relative metric entropy
  • Definition 1.3: Measures of maximal $\mathbb P$-relative entropy
  • Definition 1.4
  • Theorem A
  • Corollary B
  • Proposition 1.5
  • proof
  • Proposition 1.6
  • Remark 1.7
  • ...and 35 more