Skew products and random walks on intervals

Details Skew products and random walks on intervals Skew products and random walks on intervals

2011-10-10

arxiv.org/abs/1110.2117v2

Mathematics

Dynamical Systems

26 pages

Scientific paper

In this paper we consider step skew products over a transitive subshift of finite type (topological Markov chain) with an interval fiber. For an open and dense set of such skew products we give a full description of dynamics. Namely, there exist only finite collection of alternating attractors and repellers; we also give an upper bound for their number. Any of them is a graph of a continuous map from the base to the fiber defined almost everywhere w.r.t. any ergodic Markov measure in the base. The orbits starting between the adjacent attractor and repeller tend to the attractor as $t \to +\infty$, and to the repeller as $t \to -\infty$. The attractors support ergodic hyperbolic SRB measures. There is a natural way to associate a random dynamical system to a step skew product. We show that any generic random dynamical system of this form has finitely many ergodic stationary measures. Each measure has negative Lyapunov exponent.

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