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

For an integer $m\ge 2$, let ${\mathcal{P}}_m$ be the partition of the unit interval $I$ into $m$ equal subintervals, and let ${\mathcal{F}}_m$ be the class of piecewise linear maps on $I$ with constant slope $\pm m$ on each element of ${\mathcal{P}}_m$. We investigate the effect on mixing properties when $f \in $\mathcal{F}$_m$ is composed with the interval exchange map given by a permutation $\sigma \in S_N$ interchanging the $N$ subintervals of ${\mathcal{P}}_N$. This extends the work in a previous paper [N.P. Byott, M. Holland and Y. Zhang, DCDS, {\bf 33}, (2013) 3365–3390], where we considered only the “stretch-and-fold” map $f_{sf}(x)=mx1$.