ABSOLUTELY CONTINUOUS COMPENSATORS

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 ...

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...

The absolutely continuous spectrum of Jacobi matrices

I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schrödinger operators. These include an Oracle Theorem that predicts the potential and rather general results on the approach to certain limit potentials. In particular, we prove a Denisov-Rakhmanov type theorem for the general finite gap case. The main theme...

Some Absolutely Continuous Representations of Function Algebras

In this paper we study some absolutely continuous representations of function algebras, which are weak -spectral in the sense of [5] and [6], for a scalar > 0. Precisely we investigate certain conditions for the existence of a spectral -dilation of such representation. Among others we obtain di¤erent results which generalize the corresponding theorems of D. Gaşpar [3]. Full text