Non-uniformly expanding maps with additive noise and a fully-positive-Lyapunov stationary measure are asserted to have dense random horseshoes, a random version of Smale's horseshoe.
Horseshoes for a class of nonuniformly expanding random dynamical systems on the circle
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abstract
We propose a notion of random horseshoe for one-dimensional random dynamical systems. We prove the abundance of random horseshoes for a class of circle endomorphisms subject to additive noise, large enough to make the Lyapunov exponent positive. In particular, we provide conditions which guarantee that given any pair of disjoint intervals, for almost every noise realization, there exists a positive density sequence of return times to these intervals such that the induced dynamics are the full shift on two symbols.
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Non uniform expansion and additive noise imply random horseshoe
Non-uniformly expanding maps with additive noise and a fully-positive-Lyapunov stationary measure are asserted to have dense random horseshoes, a random version of Smale's horseshoe.