On the number of factors in codings of three interval exchange

TL;DR

This paper investigates the complexity of coding three-interval exchange transformations, establishing bounds on the number of factors in such codings and relating them to Sturmian words.

Contribution

It provides new bounds on the number of factors in three-interval exchange codings, connecting their growth to known results for Sturmian words.

Findings

Derived bounds for the number of factors in 3-interval exchange codings.

Connected the factor count of 3-interval exchanges to Sturmian words.

Established asymptotic relationships involving nd  constants.

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 $\frac1{3\pi^2} + o(1) \leq # 3\iet(N)/N^4 \leq \frac2{\pi^2} + o(1)$.