Weighted topological entropy of random dynamical systems

This is the authors' abstract. We don't add key points for this paper.

Let fᵢ, i = 1, 2, …, k be continuous bundle random dynamical systems over an ergodic compact metric system (Ω, F, P, ϑ). Assume that a = (a₁, a₂, …, aₖ)ᵏ with a₁>0 and aᵢ≥0, fᵢ₊₁ is a factor of fᵢ with a factor map πᵢ:Eᵢ→ Eᵢ₊₁. It is shown that the a-weighted Bowen topological entropy h^ a(ω, f₁, E1, ω) is measurable in Ω, and denoted that h^ a(f₁, E₁) is the integration of h^ a(ω, f₁, E1, ω) against P over Ω. We prove the following variational principle:align* h^ a(f₁, E₁) = \∑i = 1ᵏ aᵢh_μ∘τᵢ₋₁⁻¹⁽ʳ⁾(fᵢ)\, align*where the supremum is taken over the set of all μP¹(E₁, f₁). 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.