Typical Behaviour of Random Interval Homeomorphisms

Singular Stationary Measures for Random Piecewise Affine Interval Homeomorphisms

“…More precisely, there exists δ > 0 such that for every (p − , p + ) with p − , p + < 1 2 + δ there is a non-empty open interval J p − ,p + ⊂ (1, 3 2 ), depending continuously on (p − , p + ), such that for γ ∈ J p − ,p + and a ∈ (0, a max ) for some a max = a max (γ) > 0,depending continuously on γ, we have…”

“…In [5, Theorem 10], they proved that for a generic system in G (in the Baire category sense under a natural topology), the unique non-atomic stationary measure is singular. This result was extended by Bradík and Roth in [3,Theorem 6.2], where they allowed the functions to be only differentiable at 0, 1, and showed that in addition to being singular, the stationary measure has typically full support. Similar results were obtained by Czernous [4] for minimal systems on the circle.…”

“…Similar results were obtained by Czernous [4] for minimal systems on the circle. However, as the finite-dimensional space of AM-systems is meagre as a subset of the spaces considered in [3,4,5], these results give no information on the singularity of stationary measures for typical AM-systems.…”

“…The dynamics of AM-systems and related ones has already gained some interest in recent years, being studied in e.g. [1,2,3,4,5,6,25]. In the class of uniformly contracting iterated function systems, the piecewise linear maps and the dimension of their attractors were studied recently in [22].…”

“…Remark 1.2. One should note that the question of Alsedà and Misiurewicz has been answered when considered within a much broader class of general random interval homeomorphisms with positive endpoint Lyapunov exponents [3,5] and minimal random homeomorphisms of the circle [4]. More precisely, Czernous and Szarek considered in [5] the closure G of the space G of all random systems ((g − , g + ), (p − , p + )) of absolutely continuous, increasing homeomorphisms g − , g + of [0, 1], taken with probabilities p − , p + , such that g − , g + are C 1 in some fixed neighbourhoods of 0 and 1, have positive endpoint Lyapunov exponents and satisfy g − (x) < x < g + (x) for x ∈ (0, 1).…”