Smooth Generalized Interval Exchange Transformations with Wandering Intervals, from explicit Derived from pseudo-Anosov maps

TL;DR

This paper constructs explicit derived pseudo-Anosov maps on surfaces, introduces a measure with mixing properties, and develops a generalized interval exchange transformation with a wandering interval linked to self-similar IETs.

Contribution

It provides a new explicit construction of derived pseudo-Anosov maps with wandering intervals and establishes their ergodic properties and associated flows.

Findings

Existence of a measure with mixing properties supported on the stable foliation

Construction of a uniquely ergodic generalized interval exchange transformation with a wandering interval

The flow and G.I.E.T. are ^1 when the map is ^2

Abstract

Starting from any pseudo-Anosov map $\varphi$ on a surface of genus $g \geqslant 2$, we construct explicitly a family of Derived from pseudo-Anosov maps $f$ by adapting the construction of Smale's Derived from Anosov maps on the two-torus. This is done by perturbing $\varphi$ at some fixed points. We first consider perturbations at every conical fixed point and then at regular fixed points. We establish the existence of a measure $\mu$, supported by the non-trivial unique minimal component of the stable foliation of $f$, with respect to which $f$ is mixing. In the process, we construct a uniquely ergodic Generalized Interval Exchange Transformation with a wandering interval that is semi-conjugated to a self-similar Interval Exchange Transformation. This Generalized Interval Exchange Transformation is obtained as the Poincar\'e map of a flow renormalized by $f$ which parametrizes stable foliation. When $f$ is $\mathcal{C}^2$, the flow and the Generalized Interval Exchange Transformation are~$\mathcal{C}^1$.