Negative slope algorithm and multiplicative Rauzy induction of 3-interval exchange transformations

Abstract

The induced transformation on a properly chosen interval of a 2-interval exchange transformation is a 2-interval exchange transformation again. This induction is explicitly integrated by using the simple continued fraction algorithm. Therefore, we can say that the simple continued fraction algorithm acts well as a ‘‘multiplicative Rauzy induction’’ on the family of 2-interval exchange transformations. Now, we have the following question; On what kind of family of 3-interval exchange transformations does the negative slope algorithm act well as a multiplicative Rauzy induction? The purpose of this paper is to give the answer to this question.