Continuous singular measures with absolutely continuous convolution squares

Indistinguishability of Absolutely Continuous and Singular Distributions

It is shown that there are no consistent decision rules for the hypothesis testing problem of distinguishing between absolutely continuous and purely singular probability distributions on the real line. In fact, there are no consistent decision rules for distinguishing between absolutely continuous distributions and distributions supported by Borel sets of Hausdorff dimension 0. It follows that...

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.

Singular Limits of Absolutely Continuous Invariant Measures for Families of Transitive Maps

We investigate the dependence on the parameters of absolutely continuous invariant measures for a family of piecewise linear piecewise expanding maps. We construct an example to show that the transitivity of the maps does not imply the convergence of those measures to the absolutely continuous invariant measure for the limit map.

On Absolutely Continuous Transformations

Non-absolutely Continuous Foliations

We consider a partially hyperbolic diffeomorphism of a compact smooth manifold preserving a smooth measure. Assuming that the central distribution is integrable to a foliation with compact smooth leaves we show that this foliation fails to have the absolute continuity property provided that the sum of Lyapunov exponents in the central direction is not zero on a set of positive measure. We also ...