Diophantine type of interval exchange maps

TL;DR

This paper explores Diophantine conditions for interval exchange maps, generalizing Roth-type conditions from circle rotations, and establishes relationships among various conditions for different numbers of intervals.

Contribution

It introduces and compares multiple Roth-like Diophantine conditions for interval exchange maps, extending classical results from circle rotations.

Findings

For three-interval maps, (D) and (A) are equivalent and imply (Z), (U), (R).

For four or more intervals, several relations are established, but some implications remain open.

The paper generalizes Roth-type conditions to higher-dimensional interval exchange transformations.

Abstract

Roth type irrational rotation numbers have several equivalent arithmetical characterizations as well as several equivalent characterizations in terms of the dynamics of the corresponding circle rotations. In this paper we investigate how to generalize Roth-like Diophantine conditions to interval exchange maps. If one considers the dynamics in parameter space one can introduce two nonequivalent Roth-type conditions, the first (condition (Z)) by means of the Zorich cocyle, the second (condition (A)) by means of a further acceleration of the continued fraction algorithm introduced in [10]. A third very natural condition (condition (D)) arises by considering the distance between the discontinuity points of the iterates of the map. If one considers the dynamics of an interval exchange map in phase space then one can introduce the notion of Diophantine type by considering the asymptotic scaling of return times pointwise or w.r.t. uniform convergence (resp. condition (R) and (U)). In the case of circle rotations all the above conditions are equivalent. For interval exchange maps of three intervals we show that (D) and (A) are equivalent and imply (Z), (U) and (R) which are equivalent among them. For maps of four intervals or more we prove several results, the only relation which we cannot decide is whether (Z) implies (R) or not.