# Cascades in the dynamics of affine interval exchange transformations

## TL;DR

This paper investigates the complex dynamics of a family of affine interval exchange transformations by analyzing their associated affine surface and the Veech group, revealing both trivial and Cantor set accumulation behaviors.

## Contribution

It introduces a novel analysis of affine interval exchange transformations using a modified Rauzy induction and Veech group dynamics on the Disco surface.

## Key findings

- The family is generically dynamically trivial.
- For a Cantor set of parameters, leaves accumulate on a Cantor set.
- The dynamics are linked to the properties of the Veech group.

## Abstract

We describe in this article the dynamics of a $1$-parameter family of affine interval exchange transformations. It amounts to studying the directional foliations of a particular affine surface, the Disco surface. We show that this family displays various dynamical behaviours: it is generically dynamically trivial, but for a Cantor set of parameters the leaves of the foliations accumulate to a (transversely) Cantor set. s study is achieved through the analysis the dynamics of the Veech group of this surface combined a modified version of Rauzy induction in the context of affine interval exchange transformations.