Weighted topological entropy of random dynamical systems

Let $ f_{i}, i = 1, 2, \ldots, k $ 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}, \ldots, a_{k})\in\mathbb{R}^{k} $ with $ a_{1}>0 $ and $ a_{i}\geq0 $, $ f_{i+1} $ is a factor of $ f_{i} $ with a factor map $ \pi_{i}:\mathcal{E}_{i}\rightarrow \mathcal{E}_{i+1} $. It is shown that the $ {\bf a} $-weighted Bowen topological entropy $ h^{{\bf a}}(\omega, f_{1}, \mathcal{E}_{1, \omega}) $ is measurable in $ \Omega $, and denoted that $ h^{{\bf a}}(f_{1}, \mathcal{E}_{1}) $ is the integration of $ h^{{\bf a}}(\omega, f_{1}, \mathcal{E}_{1, \omega}) $ against $ \mathbb{P} $ over $ \Omega $. We prove the following variational principle:$ \begin{align*} h^{{\bf a}}(f_{1}, \mathcal{E}_{1}) = \sup\left\{\sum\limits_{i = 1}^k a_ih_{\mu\circ\tau_{i-1}^{-1}}^{(r)}(f_{i})\right\}, \end{align*} $where the supremum is taken over the set of all $ \mu\in\mathcal{M}_{\mathbb{P}}^{1}(\mathcal{E}_{1}, f_{1}) $. In the case of random dynamical systems with an ergodic compact driving system, this gives an affirmative answer to the question posed by Feng and Huang [J. Math. Pures Appl. 106 (2016), 411-452] for fiber entropy theory. It also generalizes Kenyon-Peres' variational principle on compact random sets and provides a Hausdorff dimensional result for general random McMullen-Bedford Sierpinski gaskets.