Weak mixing for interval exchange transformations and translation flows.

*(English)*Zbl 1136.37003Let \(d\geq 2\) be a natural number and let \(\pi\) be an irreducible permutation of \(\{1,\dots, d\}\), i.e., \(\pi(\{1,\dots, k\})\neq \{1,\dots, k\}\) for \(1\leq k\leq d\). For a given \(\lambda\in \mathbb R^d_+\), take an interval \(I\) of length \(\sum \lambda_i\) and cut it into \(d\) subintervals of lengths \(\lambda_1,\dots, \lambda_d\). Next, glue the subintervals together in another order, according to the permutation \(\pi\) and preserving the orientation. We obtain an interval \(I\) of the same length and hence we defined a map \(f: I\to I\), which is denoted by \(f= f_{\lambda,\pi}\) and called the interval exchange transformation (i.e.t.). The authors investigate the ergodic properties of i.e.t.’s. If \(\pi\) is the rotation of \(\{1,\dots, d\}\) that is, if \(\pi(i+ 1)\equiv\pi(i)+ 1\text{\,mod\,}d\) for all \(i\in\{1,\dots, d\}\), then \(f: I\to I\) is conjugate to a rotation of the circle, and so it is not weakly mixing for every\(\lambda\in\mathbb R^d_+\).

As one of the main results of this paper, the authors prove the following theorem: Let \(\pi\) be an irreducible permutation, which is not a rotation. Then, for Lebesgue almost every \(\lambda\in\mathbb R^d_+\), \(f_{\lambda,\pi}: I\to I\) is weakly mixing. After noticing the relations between the i.e.t.’s and the translation flows on the translation surfaces, the authors obtain similar results as the above theorem for translation flows on translation surfaces.

As one of the main results of this paper, the authors prove the following theorem: Let \(\pi\) be an irreducible permutation, which is not a rotation. Then, for Lebesgue almost every \(\lambda\in\mathbb R^d_+\), \(f_{\lambda,\pi}: I\to I\) is weakly mixing. After noticing the relations between the i.e.t.’s and the translation flows on the translation surfaces, the authors obtain similar results as the above theorem for translation flows on translation surfaces.

Reviewer: Akihiko Morimoto (Nagoya)