On the number of factors in codings of three interval exchange

Abstract

We consider exchange of three intervals with permutation $(3,2,1)$. The aim of this paper is to count the cardinality of the set $3\iet(N)$ of all words of length $N$ which appear as factors in infinite words coding such transformations. We use the strong relation of 3iet words and words coding exchange of two intervals, i.e., Sturmian words. The known asymptotic formula $# 2\iet(N)/N^3\sim\frac1{\pi^2}$ for the number of Sturmian factors allows us to find bounds $\frac1{3\pi^2} + o(1) \leq # 3\iet(N)/N^4 \leq \frac2{\pi^2} + o(1)$.