Bounded type interval exchange maps

TL;DR

This paper explores the generalization of bounded type conditions from circle rotations to interval exchange maps, linking arithmetic properties with dynamical behavior.

Contribution

It introduces a new framework for understanding bounded type conditions in the context of interval exchange maps, extending classical results from circle rotations.

Findings

Bounded partial quotients characterize bounded type irrationals.

Recurrence times in circle rotations are bounded below for bounded type irrationals.

The paper extends these concepts to interval exchange maps.

Abstract

Irrational numbers of bounded type have several equivalent characterizations. They have bounded partial quotients in terms of arithmetic characterization and in the dynamics of the circle rotation, the rescaled recurrence time to $r$-ball of the initial point is bounded below. In this paper, we consider how the bounded type condition of irrational is generalized into interval exchange maps.