Totally Rank One Interval Exchange Transformations

TL;DR

This paper investigates the properties of irreducible interval exchange transformations, establishing conditions for their powers, and proves that almost all such transformations are totally rank one, with implications for their density in the weak closure.

Contribution

It introduces a condition relating powers of induced maps, sets up a skew production of Rauzy induction, and proves the prevalence of totally rank one transformations.

Findings

Almost all irreducible interval exchange transformations are totally rank one.

The skew production of Rauzy induction map is ergodic.

Rank one transformations are dense in the weak closure for almost all transformations.

Abstract

For irreducible interval exchange transformations, we study the relation between the powers of induced map and the induced maps of powers and raise a condition of equivalence between them. And skew production of Rauzy induction map is set up and verified to be ergodic regard to a product measure. Then we prove that almost all the interval exchange transformations are totally rank one (rank one for all powers of positive integers) by interval. As a corollary, for almost all interval exchange transformations, rank one transformations are dense $G_{\delta}$ in the weak closure.