Absolutely continuous invariant measures for non-uniformly expanding maps

Regularity of Absolutely Continuous Invariant Measures for Piecewise Expanding Unimodal Maps

Let f : [0, 1]→ [0, 1] be a piecewise expanding unimodal map of class C, with k ≥ 1, and μ = ρdx the (unique) SRB measure associated to it. We study the regularity of ρ. In particular, points N where ρ is not differentiable has zero Hausdorff dimension, but is uncountable if the critical orbit of f is dense. This improves on a work of Szewc (1984). We also obtain results about higher order diff...

Absolutely continuous invariant measures for piecewise real-analytic expanding maps on the plane

We prove the existence of absolutely continuous invariant measures for piecewise real-analytic expanding maps on bounded regions in the plane.

Absolutely Continuous Invariant Measures That Are Maximal

Let A be a certain irreducible 0-1 matrix and let t denote the family of piecewise linear Markov maps on [0,1] which are consistent with A. The main result of this paper characterizes those maps in t whose (unique) absolutely continuous invariant measure is maximal, and proves that for "most" of the maps that are consistent with A, the absolutely continuous invariant measure is not maximal.

Invariant Measures for Real Analytic Expanding Maps

Let X be a compact connected subset of Rd with non-empty interior, and T : X → X a real analytic full branch expanding map with countably many branches. Elements of a thermodynamic formalism for such systems are developed, including criteria for compactness of transfer operators acting on spaces of bounded holomorphic functions. In particular a new sufficient condition for the existence of a T ...

Random Perturbations of Non-Uniformly Expanding Maps