Weighted Topological Entropy of Random Dynamical Systems

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 $\mathbf{a}=(a_1,a_2)\in {\mathbb{R}}^2$ with $a_1>0$ and $a_2\ge 0$, $f_2$ is a factor of $f_1$ with a factor map $\Pi :\Omega \times X_1\to \Omega \times X_2$. We define the $\mathbf{a}$-weighted Bowen topological entropy of $h^{\mathbf{a}}(\omega ,f_1,X_1)$ of $f_1$ with respect to $\omega \in \Omega$. It is shown that the quality $h^{\mathbf{a}}(\omega ,f_1,X_1)$ is measurable in $\Omega$, and denoted that $h^{\mathbf{a}}(f_1,\Omega \times X_1)$ is the integration of $h^{\mathbf{a}}(\omega ,f_1,X_1)$ against $\mathbb{P}$. We prove the following variational principle: \begin{align*} h^{\bf a}(f_{1},Ω\times X_{1})=\sup\left\{a_{1}h_μ^{(r)}(f_{1})+a_{2}h_{μ\circΠ^{-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.

arXiv admin note: text overlap with arXiv:1412.0078 by other authors