| Abstract: |
| In this talk, we study local stable and unstable sets for two-sided continuous bundle random dynamical systems with positive entropy, which serves as natural substitutes for invariant manifolds when $C^{1+\alpha}$ smoothness of the system is unavailable. For uniformly equicontinuous systems and ergodic invariant measures with positive entropy, we prove a lower bound for the Hausdorff dimension of local unstable sets in terms of the ratio of entropy to the maximal Lyapunov exponent. A symmetric statement for local stable sets is obtained by considering the reversed dynamics. Our results apply, in particular, to random dynamical systems generated by random differential equations with globally Lipschitz nonlinearities and to discrete-time systems arising from i.i.d. compositions of homeomorphisms drawn from a precompact subset of $C(X,X)$, where $X$ is a compact metric space. This is a joint work with Xiao Ma and Xiaomin Zhou. |
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