Rotation sets and rotational entropy for random dynamical systems on the torus

Abstract

In this paper, we consider the rotation theory for a random dynamical system φ on the m-dimensional torus Tm(m≥1). Here φ is generated by random composition of a collection of continuous maps φω, ω∈Ω, each of which is homotopic to the identity. Let Φω be the fundamental lift of φω,ω∈Ω, and assume that the periodic maps Φω−id,ω∈Ω, are uniformly bounded on Rm. Firstly, the notions of rotation set, pointwise rotation set and measure rotation set for φ are introduced and investigated. It is shown that the rotation set is a random compact and connected set. Secondly, the notion of rotational entropy, which connects the topological entropy and the rotation theory, of φ is introduced. It is shown that the rotational entropy is zero, however, it can be positive, even infinite without the above assumption.