Estimation of topological entropy in random dynamical systems
The author deals with the estimation of fiber topological entropy of differentiable random dynamical systems (RDS). Let \((\Omega, \mathcal{F}, \mathbb{P},\Phi)\) be a metric dynamical system. Let \(M\) be a compact smooth Riemannian manifold without boundary, \(\operatorname{Diff}^r(M)\) be the space of \(C^r (r>1)\) diffeomorphisms on \(M\). Let \(F\) be a Borel measurable mapping from \(\Omega\) to \(\operatorname{Diff}^r(M)\) that satisfies \(\underset{\Omega }{\int }\log ^+\parallel DF_{\omega}\parallel_{C^2}d\mathbb{P}<+\infty\). The author proves that \[h_{{\mathrm{top}}}(F)\leq \int \underset{n\rightarrow \infty }{\lim \sup } \frac{1}{n}\log \left(\int \parallel(D_xF_{\omega}^n)^{\land }\parallel dx\right)d\mathbb{P}(\omega )<+\infty. \] Furthermore, if \(F:\Omega\rightarrow\)Diff\(^{\infty}(M)\) satisfies \(\underset{\Omega }{\int }\log ^+\parallel DF_{\omega}\parallel_{C^{\infty}}d\mathbb{P}<+\infty\), the author proves that \[h_{{\mathrm{top}}}(F)\leq \int \underset{n\rightarrow \infty }{\lim \sup } \frac{1}{n}\log \left(\int \parallel(D_x F_{\omega}^n)^{\land}\parallel dx\right)d\mathbb{P}<+\infty,\] where \(h_{\mathrm{top}}(F)\) is the fiber topological entropy of \(F\). By using a multiplicative ergodic theorem, Lyapunov charts and a variational principle, the author proves that the fiber topological entropy is bounded above by the growth rate of the integral formula.
scientific article; zbMATH DE number 7188435
scientific article