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Attractor-repeller pair, Morse decomposition and Lyapunov function for random dynamical systems

Zhenxin Liu·Shuguan Ji·Menglong Su
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Abstract

In the stability theory of dynamical systems, Lyapunov functions play a fundamental role. In this paper, we study the attractor-repeller pair decomposition and Morse decomposition for compact metric space in the random setting. In contrast to [8], by introducing slightly stronger definitions of random attractor and repeller, we characterize attractor-repeller pair decompositions and Morse decompositions for random dynamical systems through the existence of Lyapunov functions. These characterizations, we think, deserve to be known widely.

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Zhenxin Liu, Shuguan Ji, Menglong Su. 2006-06-09. Attractor-repeller pair, Morse decomposition and Lyapunov function for random dynamical systems. https://arxiv.org/abs/math/0606205

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