Metastability, Lyapunov exponents, escape rates, and topological entropy in random dynamical systems

TL;DR

This paper investigates metastability in random dynamical systems by analyzing Lyapunov exponents, escape rates, and topological entropy, providing bounds and decompositions for complex stochastic systems.

Contribution

It introduces a novel connection between Lyapunov spectra of random cocycles and system decompositions, offering rigorous bounds on escape rates and entropy.

Findings

Decomposition of systems into disjoint parts with bounds on escape rates.

Bounds on topological entropy for random shifts of finite type.

Use of Lyapunov spectrum to analyze metastability.

Abstract

We explore the concept of metastability in random dynamical systems, focussing on connections between random Perron-Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The Lyapunov spectrum of the random Perron-Frobenius cocycle and the random adjacency matrix cocycle is used to decompose the random system into two disjoint random systems with rigorous upper and lower bounds on (i) the escape rate in the setting of random maps, and (ii) topological entropy in the setting of random shifts of finite type, respectively.