A family of rotation numbers for discrete random dynamics on the circle
Christian S. Rodrigues and Paulo R. C. Ruffino
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Submission date: 19. Jul. 2013 (revised version: August 2013)
published in: Stochastics and dynamics, 15 (2015) 3, art-no. 1550021
DOI number (of the published article): 10.1142/S0219493715500215
MSC-Numbers: 37C40, 37E45, 37E10, 37A05
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We revisit the problem of well-defining rotation numbers for discrete random dynamical systems on S1. We show that, contrasting with deterministic systems, the topological (i.e. based on Poincaré lifts) approach does depend on the choice of lifts (e.g. continuously for nonatomic randomness). Furthermore, the winding orbit rotation number does not agree with the topological rotation number. Existence and conversion formulae between these distinct numbers are presented. Finally, we prove a sampling in time theorem which recover the rotation number of continuous Stratonovich stochastic dynamical systems on S1 out of its time discretisation of the flow.