Mathematics

In this work we study mixing properties of discrete dynamical
systems and related to them geometric structure. In the first
chapter we show that the direct product of maps with Young towers
admits a Young tower whose return times decay at a rate which is
bounded above by the slowest of the rates of decay of the return
times of the component maps. An application of this result, together
with other results in the literature, yields various statistical
properties for the direct product of various classes of systems,
including Lorenz-like maps, multimodal maps, piecewise $C^2$
interval maps with critical points and singularities, H\'enon maps
and partially hyperbolic systems.

The second chapter is dedicated to the problem of decay of
correlations for continuous observables. First we show that if the
underlying system admits Young tower then the rate of decay of
correlations for continuous observables can be estimated in terms of
modulus of continuity and the decay rate of tail of Young tower. In
the rest of the second chapter we study the relations between the
rates of decay of correlations for smooth observables and continuous
observables. We show that if the rates of decay of correlations is
known for $C^r,$ observables ($r\ge 1$) then it is possible to
obtain decay of correlations for continuous observables in terms of
modulus of continuity.

Mathematics

MAT/05