Continued Fraction Algorithms for Interval Exchange Maps: an Introduction

Abstract

Rotations on the circle T = R/Z are the prototype of quasiperiodic dynamics. They also constitute the starting point in the study of smooth dynamics on the circle, as attested by the concept of rotation number and the celebrated Denjoy theorem. In these two cases, it is important to distinguish the case of rational and irrational rotation number. But, if one is interested in the deeper question of the smoothness of the linearizing map, one has to solve a small divisors problem where the diophantine approximation properties of the irrational rotation number are essential. The classical continuous fraction algorithm generated by the Gauss map G(x) = {x} (where x ∈ (0, 1) and {y} is the fractional part of a real number y) is the natural way to analyze or even define these approximation properties. The modular group GL(2,Z) is here of fundamental importance, viewed as the group of isotopy classes of diffeomorphisms of T, where act the linear flows obtained by suspension from rotations.