On the mixing properties of piecewise expanding maps under composition with permutations
Byott, NP; Holland, M; Zhang, Y
Date: 1 January 2013
Journal
Discrete and Continuous Dynamical Systems - Series A
Publisher
American Institute of Mathematical Sciences
Publisher DOI
Abstract
We consider the effect on the mixing properties of a piecewise smooth interval map f when its domain is divided into N equal subintervals and f is composed with a permutation of these. The case of the stretch-and-fold map f(x)=mxmod1 for integers m≥2 is examined in detail. We give a combinatorial description of those permutations σ for ...
We consider the effect on the mixing properties of a piecewise smooth interval map f when its domain is divided into N equal subintervals and f is composed with a permutation of these. The case of the stretch-and-fold map f(x)=mxmod1 for integers m≥2 is examined in detail. We give a combinatorial description of those permutations σ for which σ∘f is still (topologically) mixing, and show that the proportion of such permutations tends to 1 as N→∞. We then investigate the mixing rate of σ∘f (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 composition with a permutation cannot improve the mixing rate of f, but typically makes it worse. Under some mild assumptions on m and N, we obtain a precise value for the worst mixing rate as σ ranges through all permutations; this can be made arbitrarily close to 1 as N→∞ (with m fixed). We illustrate the geometric distribution of the second largest eigenvalues in the complex plane for small m and N, and propose a conjecture concerning their location in general. Finally, we give examples of other interval maps f for which composition with permutations produces different behaviour than that obtained from the stretch-and-fold map.
Mathematics and Statistics
Faculty of Environment, Science and Economy
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