Note on powers in three interval exchange transformations

TL;DR

This paper investigates the repetition structure of 3-interval exchange transformation words, linking their finiteness properties to continued fraction coefficients of a parameter, and provides bounds on their index.

Contribution

It establishes a connection between the finiteness of the index of 3iet words and the boundedness of continued fraction coefficients of a key parameter, offering bounds on the index.

Findings

Finiteness of the index is equivalent to bounded continued fraction coefficients.

Provides upper and lower bounds on the index of 3iet words.

Links combinatorial properties of 3iet words to number theoretic conditions.

Abstract

We study repetitions in infinite words coding exchange of three intervals with permutation (3,2,1), called 3iet words. The language of such words is determined by two parameters $\varepsilon,\ell$. We show that finiteness of the index of 3iet words is equivalent to boundedness of the coefficients of the continued fraction of $\varepsilon$. In this case we also give an upper and lower estimate on the index of the corresponding 3iet word.