We consider words coding exchange of three intervals with
permutation (3,2,1), here called 3iet words. Recently, a
characterization of substitution invariant 3iet words was
provided. We study the opposite question: what are the morphisms
fixing a 3iet word? We reveal a narrow connection of such
morphisms and morphisms fixing Sturmian words using the new notion
of amicability.

A simple Parry number is a real number $\beta \>1$ such that the Rényi expansion of $1$ is finite, of the form ${d}_{\beta}\left(1\right)={t}_{1}\cdots {t}_{m}$. We study the palindromic structure of infinite aperiodic words ${u}_{\beta}$ that are the fixed point of a substitution associated with a simple Parry number $\beta $. It is shown that the word ${u}_{\beta}$ contains infinitely many palindromes if and only if ${t}_{1}={t}_{2}=\cdots ={t}_{m-1}\ge {t}_{m}$. Numbers $\beta $ satisfying this condition are the so-called Pisot numbers. If ${t}_{m}=1$ then ${u}_{\beta}$ is an Arnoux-Rauzy word. We show that if $\beta $ is a confluent Pisot number then $\mathcal{P}(n+1)+\mathcal{P}\left(n\right)=\mathcal{C}(n+1)-\mathcal{C}\left(n\right)+2$, where...

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