Breadcrumb
Mixing properties of piecewise expanding maps under composition with permutations
Thu 14 March 2013, 15:00
Yiwei Zhang
Pontificia Universidad Cat�³lica de Chile
Ergodic Theory and Dynamical Systems
Organisers: Andrew Ferguson, Thomas Jordan
ABSTRACT
We consider the effect on the mixing properties of piecewise smooth interval map $f$, when its domain is divided into $N$ equal subintervals and $f$ is composed with a permutation of these. In particular, we study in details the case where $f$ is the stretch-and-fold map $f_{sf}(x)=mx\mod1$ for an integer $m\geq2$(orientation-preserving) together with the case where $f$ is the m-fold Zigzag map(alternately orientation-preserving and orientation-reversing). In the case of $m=2,$ the map $f_{sf}$ is the doubling map and $f_{zz}$ is the tent map.
We give a combinatorial description of these permutations $\sigma$ for which $\sigma\circ f$ is still (topologically) mixing and investigate their mixing rates (as measured by the modulus of the second largest eigenvalue of the transfer operator). In contrast to the situation for continuous time diffusive systems, we show that the composition with a permutation can't improve the mixing rate of $f$, but typically makes it worse. Moreover, we take comparisons on the worst mixing rates between the two systems by using standard circulant matrices together with a variant of circulant matrices (which we called mirror-circulant matrices). Finally, we discuss/illustrate various distributions (e.g. geometric and probability distributions) of the second largest eigenvalues.