On the existence of a σ-finite acim for a random iteration of intermittent Markov maps with uniformly contractive part

Abstract

For an annealed type random dynamical system arising from non-uniformly expanding maps which admits uniformly contractive branches, we establish the existence of an absolutely continuous [Formula: see text]-finite invariant measure. We also show when the invariant measure is infinite.

Invariant measures for random piecewise convex maps

“…, n satisfy the conditions on S β from (1) and ( 2), then the strongest contracting property in {X i } n i =1 would dominate the statistical laws of the random system. (II) In the condition (1), the assumption that τ α and S β are C 1 can be relaxed to the following condition: there are families of countable open subintervals {I L n } n and {I R n } n , with the closure of their union being X, such that, for ν A -almost every α ∈ A and ν B -almost every β ∈ B, τ α and S β are C 1 on I L n and I R n , respectively for each n. Hence some (but not all) examples from [30] are also in sight of the present paper. (III) In the above conditions (1) and (2), we do not exclude Recall that for a Markov operator P, a measure µ over (X, B) is called an absolutely continuous (resp.…”

“…Hence we cannot expect so-called spectral decomposition method from the Lasota-Yorke type inequality any longer. We refer to [30] for similar arguments and some background.…”

Invariant measures for random piecewise convex maps

“…(II) In the condition (1), the assumption that τ α and S β are C 1 can be relaxed to the following condition: there are families of countable open subintervals {I L n } n and {I R n } n , with the closure of their union being X, such that, for ν A -almost every α ∈ A and ν B -almost every β ∈ B, τ α and S β are C 1 on I L n and I R n , respectively for each n. Hence some (but not all) examples from [29] are also in sight of the present paper.…”

“…Hence we cannot expect so-called spectral decomposition method from the Lasota-Yorke type inequality any longer. We refer to [29] for similar arguments and some background.…”

Singular Stationary Measures for Random Piecewise Affine Interval Homeomorphisms

“…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].…”