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{\'e}non maps and partially hyperbolic systems.

}, issn = {1078-0947}, doi = {10.3934/dcds.2016.36.1465}, url = {http://aimsciences.org//article/id/18d4526e-470d-467e-967a-a0345ad4c642}, author = {Stefano Luzzatto and Marks Ruziboev} } @article {doi:10.1080/14689367.2015.1046816, title = {Decay of correlations for invertible maps with non-H{\"o}lder observables}, journal = {Dynamical Systems}, volume = {30}, number = {3}, year = {2015}, pages = {341-352}, publisher = {Taylor \& Francis}, abstract = {An invertible dynamical system with some hyperbolic structure is considered. Upper estimates for the correlations of continuous observables are given in terms of modulus of continuity. The result is applied to certain H{\'e}non maps and Solenoid maps with intermittency.

}, doi = {10.1080/14689367.2015.1046816}, url = {https://doi.org/10.1080/14689367.2015.1046816}, author = {Marks Ruziboev} } @mastersthesis {2015, title = {Gibbs-Markov-Young Structures and Decay of Correlations}, year = {2015}, school = {SISSA}, abstract = {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.}, keywords = {Decay of Correlations, GMY-towers}, author = {Marks Ruziboev} }