Weighted Topological Entropy of Random Dynamical Systems

TL;DR

This paper introduces a weighted topological entropy for random dynamical systems, establishes a variational principle linking it to measure-theoretic entropy, and extends classical entropy formulas to the random setting.

Contribution

It provides a new variational principle for weighted topological entropy in random dynamical systems, generalizing existing results and answering an open question.

Findings

Proves the measurability of weighted entropy in the base space.

Establishes a variational principle relating weighted topological and measure-theoretic entropy.

Extends classical entropy formulas like Shannon-McMillan-Breiman and Brin-Katok to the weighted random setting.

Abstract

Let $f_{i},i=1,2$ be continuous bundle random dynamical systems over an ergodic compact metric system $(\Omega,\mathcal{F},\mathbb{P},\vartheta)$. Assume that ${\bf a}=(a_{1},a_{2})\in\mathbb{R}^{2}$ with $a_{1}>0$ and $a_{2}\geq0$, $f_{2}$ is a factor of $f_{1}$ with a factor map $\Pi:\Omega\times X_{1}\rightarrow\Omega\times X_{2}$. We define the ${\bf a}$-weighted Bowen topological entropy of $h^{{\bf a}}(\omega,f_{1},X_{1})$ of $f_{1}$ with respect to $\omega\in \Omega$. It is shown that the quality $h^{{\bf a}}(\omega,f_{1},X_{1})$ is measurable in $\Omega$, and denoted that $h^{{\bf a}}(f_{1},\Omega\times X_{1})$ is the integration of $h^{{\bf a}}(\omega,f_{1},X_{1})$ against $\mathbb{P}$. We prove the following variational principle: \begin{align*} h^{{\bf a}}(f_{1},\Omega\times X_{1})=\sup\left\{a_{1}h_{\mu}^{(r)}(f_{1})+a_{2}h_{\mu\circ\Pi^{-1}}^{(r)}(f_{2})\right\}, \end{align*} where the supremum is taken over the set of all $\mu\in\mathcal{M}_{\mathbb{P}}^{1}(\Omega\times X_{1},f_{1})$. In the case of random dynamical systems with an ergodic and compact driving system, this gives an affirmative answer to the question posed by Feng and Huang [Variational principle for weighted topological pressure, J. Math. Pures Appl. 106 (2016), 411-452]. It also generalizes the relativized variational principle for fiber topological entropy, and provides a topological extension of Hausdorff dimension of invariant sets and random measures on the $2$-torus $\mathbb{T}^{2}$. In addition, the Shannon-McMillan-Breiman theorem, Brin-Katok local entropy formula and Katok entropy formula of weighted measure-theoretic entropy for random dynamical systems are also established in this paper.