Weighted Topological Entropy of Random Dynamical Systems

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.