Full families of generalized interval exchange transformations

TL;DR

This paper introduces full families of generalized interval exchange transformations (GIETs), extending classical results of Poincaré by showing how any Rauzy renormalization path can be realized within these finite-dimensional parameter families.

Contribution

It defines and constructs full families of GIETs that realize all possible Rauzy renormalization paths, extending classical results to a broader class of transformations.

Findings

Full families of GIETs can realize any prescribed Rauzy renormalization path.

GIETs and IETs with the same renormalization path are semi-conjugated.

Extension of Poincaré's classical relation to GIETs.

Abstract

We consider generalized interval exchange transformations, or briefly GIETs, that is bijections of the interval which are piecewise increasing homeomorphisms with finite branches. When all continuous branches are translations, such maps are classical interval exchange transformations, or briefly IETs. The well-known Rauzy renormalization procedure extends to a given GIET and a Rauzy renormalization path is defined, provided that the map is infinitely renormalizable. We define full families of GIETs, that is optimal finite dimensional parameter families of GIETs such that any prescribed Rauzy renormalization path is realized by some map in the family. In particular, a GIET and a IET with the same Rauzy renormalization path are semi-conjugated. This extends a classical result of Poincar\'e relating circle homeomorphisms and irrational rotations.