Complexity of testing morphic primitivity

We analyze the algorithm in [Holub, 2009], which decides whether a given word is a fixed point of a nontrivial morphism. We show that it can be implemented to have complexity in O(mn), where n is the length of the word and m the size of the alphabet

Stable set of self map

The attracting set and the inverse limit set are important objects associated to a self-map on a set. We call \emph{stable set} of the self-map the projection of the inverse limit set. It is included in the attracting set, but is not equal in the general case. Here we determine whether or not the equality holds in several particular cases, among which are the case of a dense range continuous function on an Hilbert space, and the case of a substitution over left infinite words

Avoiding three consecutive blocks of the same size and same sum

We show that there exists an infinite word over the alphabet {0,1,3,4} containing no three consecutive blocks of the same size and the same sum. This answers an open problem of Pirillo and Varricchio from1994

On Two-Sided Infinite Fixed Points of Morphisms

Let \Sigma be a finite alphabet, and let h : \Sigma ! \Sigma be a morphism. Finite and infinite fixed points of morphisms --- i.e., those words w such that h(w) = w --- play an important role in formal language theory. Head characterized the finite fixed points of h, and later, Head and Lando characterized the one-sided infinite fixed points of h. Our paper has two main results. First, we complete the characterization of fixed points of morphisms by describing all two-sided infinite fixed points of h, for both the "pointed" and "unpointed" cases. Second, we completely characterize the solutions to the equation h(xy) = yx in finite words. 1 Introduction and definitions Let \Sigma be a finite alphabet, and let h : \Sigma ! \Sigma be a morphism on the free monoid, i.e., a map satisfying h(xy) = h(x)h(y) for all x; y 2 \Sigma . If a word w (finite or infinite) satisfies the equation h(w) = w, then we call w a fixed point of h. Both finite and infinite fixed points of morph..