On the mixing properties of piecewise expanding maps under composition with permutations, II: Maps of non-constant orientation

Abstract

For an integer m ≥ 2, let P m be the partition of the unit interval I into m equal subintervals, and let F m be the class of piecewise linear maps on I with constant slope ±m on each element of P m . We investigate the effect on mixing properties when f ∈ F m is composed with the interval exchange map given by a permutation σ ∈ S N interchanging the N subintervals of P N . This extends the work in a previous paper [N.P. Byott, M. Holland and Y. Zhang, DCDS, 33, (2013) 3365-3390], where we considered only the "stretch-and-fold" map f sf (x) = mx mod 1.

Deceleration of one-dimensional mixing by discontinuous mappings

“…The present study could be extended to a larger collection of interval exchange transformations with finite order, however the similarities of the mixing rates for small diffusivity to the rates τ when κ = 0 implies that the conclusions of [28] predict well the asymptotic mixing rates and it is unlikely that adding diffusion would not highlight anything of further interest than already discussed herein. The model is highly idealised in relation to real fluid mixing problems.…”

“…IETs with diffusion have been used as toy models to investigate mixing [26,27] but there has been no investigation of the mixing rates of this larger parameter space. Bounds have been found on the mixing rates for permutations composed with expanding maps on the unit interval, with the conclusion that permutations do not improve the mixing rate and typically make it worse [28]. It is similarly reported for shears composed with a slip deformation that the combined mechanisms can slow the rate of mixing of material segregation [29].…”

“…However, for κ = 0 the transfer matrix for the Fourier coefficients can not be feasibly truncated, thus it can not be found from the computational method already presented. The rate of mixing for the map when κ = 0 can be calculated from matrices defining the probability transition between Markov partitions for the map, the derivation of which closely follows methods developed for permutations composed with expanding maps and we briefly outline the results of [28].…”

“…The construction and results of [28] have a direct comparison with the current presented model. Consider the graphical representation of the system studied here; the halving map composed with a permutation, shown in Figure 6.…”

“…It can easily be shown that an eigenvalue τ of the transformation σ −1 • f , is equivalent to the eigenvalue τ of (σ • M B ) −1 , where σ −1 denotes the inverse permutation of σ. This arises from the construction of the Markov transition matrices in the proof of the bounds from [28] (see the Appendix for details).…”