Almost all interval exchange transformations with flips are nonergodic

Abstract

Here we prove that almost all interval exchange transformations which reverse orientation, in at least one interval, have a periodic point where the derivative is − 1. Therefore they are periodic in an open neighborhood of the periodic point.

On the Hausdorff dimension of minimal interval exchange transformations with flips

“…The proof is based on the argument presented in : first, we show that the exponential tail of the roof function implies that the corresponding Markov map is fast decaying in a sense of and then using check that this property implies the estimation we are interested in. Section 8 completes the proof of Theorem : we show the lower bound applying a construction related to the one described in Nogueira in . Using the same idea, we prove Theorem and Proposition .…”

“…Sometimes, since we also work with the signed permutation , we can specify that or was the winner or the loser. Remark We will iterate the Rauzy induction map, however Rauzy induction is not defined everywhere, thus the iteration stops if we arrive to a point outside its domain of definition (see for an example when it stops). In case of an IET (without flips) which satisfies Keane's condition this never happens.…”

Affine interval exchange transformations with flips and wandering intervals

“…We shall need the Rauzy induction [16,15,9] to obtain a minimal, self-similar IET whose associated matrix satisfies all the hypotheses of Theorem 1.…”

Stability of periodic points in piecewise isometries of Euclidean spaces

“…This is the basic argument that drives the proof that almost every interval exchange map with flips is non-ergodic in [6]. Namely, the fact that the map acts locally as an isometry implies that, if we take d := min k=1,...,p {dist(T k · x 0 , D)}, where p is the period of x 0 and D is the discontinuity set, then the ball B d (x 0 ) follows the orbit of x 0 at all iterations and, therefore, the Lebesgue measure supported on the union of the iterates of B d (x 0 ) is an invariant measure, hence preventing any invariant non-atomic Borel measure from being ergodic.…”