On the mixing properties of piecewise expanding maps under composition with permutations. II: Maps of non-constant orientation (Q2797930)

The first author et al. [Discrete Contin. Dyn. Syst. 33, No. 8, 3365--3390 (2013; Zbl 1308.37004)] studied quantitative mixing properties of maps on the interval obtained by composing a piecewise smooth interval map with a permutation of the subintervals obtained by dividing the interval into finitely many subintervals of equal length. For the map \(f:x\mapsto mx\pmod{1}\) for an integer \(m\geq2\) composed with a permutation of \(N\) equal subintervals the permutations for which the resulting map is topologically mixing are described in combinatorial terms, and it is shown that the proportion of such permutations tends to \(1\) as \(N\to\infty\). In contrast to the case of continuous-time diffusive systems, they showed that composition with a permutation typically reduces and never improves the rate of mixing. Here a similar analysis is done for the more general class of maps with constant slope \(\pm m\) on each of the subintervals \([\frac{j}{m},\frac{j+1}{m})\). The main technique is to exploit spectral properties of an associated transfer operator to control rates of mixing.