New Results on the Computation of Periods of IETs

We introduce a novel technique for computing the periods of (d,k)-IETs based on Rauzy induction R. Specifically, we establish a connection between the set of periods of an interval exchange transformation (IET) T and those of the IET T′ obtained either by applying the Rauzy operator R to T or by considering the Poincaré first return map. Rauzy matrices play a central role in this correspondence whenever T lies in the domain of R (Theorem 4). Furthermore, Theorem 6 addresses the case when T is not in the domain of R, while Theorem 5 deals with IETs having associated reducible permutations. As an application, we characterize the set of periods of oriented 3-IETs (Theorem 8), and we also propose a general framework for studying the periods of (d,k)-IETs. Our approach provides a systematic method for determining the periods of non-transitive IETs. In general, given an IET with d discontinuities, if Rauzy induction allows us to descend to another IET whose periodic components are already known, then the main theorems of this paper can be applied to recover the set of periods of the original IET. This method has been also applied to obtain the set of periods of all (2,k)-IETs and some (3,k)-IETs, k≥1. Several open problems are presented at the end of the paper.