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arXiv · 2206.06778

Description of tempered exponential dichotomies by admissibility with no Lyapunov norms

Davor Dragičević·Weinian Zhang·Linfeng Zhou
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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.

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Davor Dragičević, Weinian Zhang, Linfeng Zhou. 2022-06-14. Description of tempered exponential dichotomies by admissibility with no Lyapunov norms. https://arxiv.org/abs/2206.06778

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