On the mixing properties of piecewise expanding maps under composition with permutations

Details On the mixing properties of piecewise expanding maps under composition with permutations On the mixing properties of piecewise expanding maps under composition with permutations

2012-03-22

arxiv.org/abs/1203.5145v1

Mathematics

Dynamical Systems

Scientific paper

For a mixing and uniformly expanding interval map $f:I\to I$ we pose the following questions. For which permutation transformations $\sigma:I\to I$ is the composition $\sigma\circ f$ again mixing? When $\sigma\circ f$ is mixing, how does the mixing rate of $\sigma\circ f$ typically compare with that of $f$? As a case study, we focus on the family of maps $f(x)=mx\operatorname{mod}1$ for $2\leq m\in\mathbb{N}$. We split $[0,1)$ into $N$ equal subintervals, and take $\sigma$ to be a permutation of these. We analyse those $\sigma \in S_N$ for which $\sigma \circ f$ is mixing, and show that for large $N$, typical permutations will preserve the mixing property. However, when $\sigma\circ f$ is mixing, we will show the existence of permutations in $S_N$ for which the (exponential) mixing rate of $\sigma\circ f$ can be made arbitrarily close to one (as $N\to\infty$).

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