Description of tempered exponential dichotomies by admissibility with no Lyapunov norms

TL;DR

This paper characterizes tempered exponential dichotomies in random systems using admissibility of function classes without Lyapunov norms, extending the concept beyond MET-systems and analyzing stability under parameter changes.

Contribution

It provides a new measurable admissibility-based description of tempered exponential dichotomies applicable to general random systems, not limited to MET-systems, and simplifies the characterization for MET-systems.

Findings

Describes tempered exponential dichotomies without Lyapunov norms.

Extends the theory to non-MET random systems.

Proves roughness and Hölder continuity of projections under parameter variation.

Abstract

Tempered exponential dichotomy formulates the nonuniform hyperbolicity for random dynamical systems. It was described by admissibility of a pair of function classes defined with Lyapunov norms, For MET-systems (systems satisfying the assumptions of multiplicative ergodic theorem (abbreviated as MET)), it can be described by admissibility of a pair without a Lyapunov norm. However, it is not known how to choose a suitable Lyapunov norms before a tempered exponential dichotomy is given, and there are examples of random systems which are not MET-systems but have a tempered exponential dichotomy. In this paper we give a description of tempered exponential dichotomy for general random systems, which may not be MET-systems, purely by measurable admissibility of three pairs of function classes with no Lyapunov norms. Further, restricting to the MET-systems, we obtain a simpler description of only one pair with no Lyapunov norms. Finally, we use our results to prove the roughness of tempered exponential dichotomies for parametric random systems and give a H\"older continuous dependence of the associated projections on the parameter.