Singular Stationary Measures for Random Piecewise Affine Interval Homeomorphisms

Abstract

We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsedà and Misiurewicz and contributing to a question of Navas on the absolute continuity of stationary measures, considered in the setup of semigroups of piecewise affine circle homeomorphisms. We focus on the case of resonant boundary derivatives.

Alsedà–Misiurewicz systems with place-dependent probabilities*

“…Next, let f 1 be the interval homeomorphism defined by 1). Setting a 0 = y 0 x 0 and a 1 = 1−y 0 1−x 0 , we can write After [2], we call these systems the Alsedà-Misiurewicz systems (in [2] the only restriction for (x 0 , y 0 ) ∈ (0, 1) × (0, 1) is that it should be under diagonal). Further, fix two positive real functions p 0 , p 1 on [0, 1] with p 0 (x) + p 1 (x) = 1 for every x ∈ [0, 1].…”

Typical behaviour of random interval homeomorphisms

“…We can find another dense subset of RDS(I) using the tools from [5] for piecewise linear homeomorphisms on the unit interval. The space RDS(I) contains piecewise linear homeomorphisms and we utilize the so-called non-resonant case from [5] on a neighborhood of zero to approximate any given f ∈ RDS(I) to show that the systems with fully supported stationary measures form another dense subset of RDS(I).…”

“…As in [5], the proof is based on the notion of minimality. Our random dynamical systems on I are never minimal since the endpoint set is invariant, so we work instead on R where there are no endpoints.…”

“…A partial solution to this conjecture is given in [5], where the authors continue to work with piecewise linear homeomorphisms and prove singularity of µ in the so-called resonant case. Moreover, they give a specific example of a resonant system for which µ is not only singular, but has full support in [0, 1].…”

“…In one sense, we work with a much larger class of systems by dropping the requirements of piecewise linearity of [5], C 2 -smoothness of [4], and continuous differentiability in some fixed β-neighborhoods of the endpoints of [2] and [3]. We assume only differentiability at the endpoints, which is needed to test for positive Lyapunov exponents.…”