Morphisms preserving the set of words coding three interval exchange

TL;DR

This paper explores the construction of ternary morphisms that preserve words coding three-interval exchanges, based on amicable pairs of Sturmian morphisms, and analyzes their incidence matrices.

Contribution

It introduces a method to construct ternary morphisms from amicable Sturmian pairs and characterizes their incidence matrices within a specific algebraic group.

Findings

Number of amicable pairs with identical incidence matrices determined

Incidence matrices associated with the ternary morphisms characterized

Construction method for ternary morphisms preserving 3-interval exchange words

Abstract

Any amicable pair \phi, \psi{} of Sturmian morphisms enables a construction of a ternary morphism \eta{} which preserves the set of infinite words coding 3-interval exchange. We determine the number of amicable pairs with the same incidence matrix in $SL^\pm(2,N)$ and we study incidence matrices associated with the corresponding ternary morphisms \eta.