Synchronisation of almost all trajectories of a random dynamical system

Julian Newman

doi:10.3934/dcds.2020176

Article Contents

2020, Volume 40, Issue 7: 4163-4177. Doi: 10.3934/dcds.2020176

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Synchronisation of almost all trajectories of a random dynamical system

Julian Newman

Department of Mathematics, University of Exeter, Exeter, EX4 4QF, United Kingdom

Received: April 2019

Revised: December 2019

Published online: July 2020

The author wishes gratefully to acknowledge funding from an EPSRC Doctoral Training Account and an EPSRC Doctoral Prize, both at Imperial College London.

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Abstract

It has been shown by Le Jan that, given a memoryless-noise random dynamical system together with an ergodic distribution for the associated Markov transition probabilities, if the support of the ergodic distribution admits locally asymptotically stable trajectories, then there is a random attracting set consisting of finitely many points, whose basin of forward-time attraction includes a random full measure open set. In this paper, we present necessary and sufficient conditions for this attracting set to be a singleton. Our result does not require the state space to be compact, but holds on general Lusin metric spaces (in both discrete and continuous time).

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Mathematics Subject Classification: Primary: 37H05; Secondary: 60F15.

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