An exchange of \(k\geq2\) intervals with permutation \(\pi\) is a bijection \(T:J\rightarrow J\), where \(J\) is a semi-closed interval, there is a partition \(J=J_{0}\cup\cdots\cup J_{k-1}\) of \(J\) into a disjoint union of semi-closed subintervals satisfying \(J_{0}<J_{1}<\cdots<J_{k-1}\) (i.e., each value in \(J_{i-1}\) is smaller than each value in \(J_{i}\)), and there exist numbers \(c_{0},\ldots,c_{k-1}\) such that \(T( x) =x+c_{i}\) for \(x\in J_{i}\); \(\pi\) is a permutation of \(\{0,1,\ldots,k-1\}\) such that \(T(J_{i})<T(J_{i^{\prime}})\) for \(\pi( i) <\pi( i^{\prime }) \) (determined by the order of the intervals \(T( J_{i}) \)). \(T\) is a symmetric interval exchange if \(\pi\) is the permutation \(i\mapsto k-i-1\). The orbit \(\rho,T( \rho) ,T^{2}( \rho) ,\ldots\) of a point \(\rho\in J\) can be coded by an infinite word \(u_{\rho }=u_{0}u_{1}u_{2}\cdots\) over the alphabet \(\{0,1,\dots,k-1\}\) where \(u_{n}=i\) if \(T^{n}( \rho) \in J_{i}\). If for every \(\rho\in J\) and for every \(n\in N\) the word \(u_{\rho}\) contains the maximum possible number \((k-1)n+1\) of distinct factors of length \(n,\) then the transformation \(T\) and the word \(u_{\rho}\) are said to be non-degenerate.
The paper deals with the so-called class-\(P\) conjecture with respect to purely morphic words coding symmetric non-degenerate 3-interval exchange transformations. Such words are known to contain infinitely many palindromes. The class \(P\) consists of all morphisms \(\varphi\) such that there exists a palindrome \(p\) such that for every \(a\in A\) one has \(\varphi( a) =pp_{a}\) for some palindrome \(p_{a}\). The main result states that if an infinite word \(u\) coding a non-degenerate 3-interval exchange transformation \(T\) with permutation \((321)\) is a fixed point of a primitive morphism \(\xi\) then \(\xi\) or \(\xi^{2}\) is a conjugate of a morphism from the class \(P\).
A morphism \(\varphi\) is primitive if, for some \(k\geq1\), for any pair of symbols \(a\), \(b\), \(\varphi^{k}( a) \) contains \(b\). The morphisms \(\varphi\), \(\psi\) are conjugates if there exists a word \(w\) such that, for all letters \(a\), \(w\varphi( a) =\psi( a) w\) or, for all \(a\), \(\varphi( a) w=w\psi( a) \).