Extremal words in morphic subshifts

Extremal words in morphic subshifts

Given an infinite word x over an alphabet A, a letter b occurring in x, and a total order σ on A, we call the smallest word with respect to σ starting with b in the shift orbit closure of x an extremal word of x. In this paper we consider the extremal words of morphic words. If x = g(fω(a)) for some morphisms f and g, we give two simple conditions on f and g that guarantees that all extremal wo...

Extremal Words in the Shift Orbit Closure of a Morphic Sequence

Suffix Conjugates for a Class of Morphic Subshifts - (Extended Abstract)

Let A be a finite alphabet and f : A → A be a morphism with an iterative fixed point f(α), where α ∈ A. Consider the subshift (X , T ), where X is the shift orbit closure of f(α) and T : X → X is the shift map. Let S be a finite alphabet that is in bijective correspondence via a mapping c with the set of nonempty suffixes of the images f(a) for a ∈ A. Let S ⊂ S be the set of infinite words s = ...

Maximal Blocks in Morphic and Automatic Words

Let w be a morphic word over a finite alphabet Σ, and let ∆ be a subalphabet. We study the behavior of maximal blocks of ∆ letters in w, and prove the following: let (ik, jk) denote the starting and ending positions, respectively, of the k’th maximal ∆-block in w. Then lim supk→∞(jk/ik) is algebraic if w is morphic, and rational if w is automatic. As a results, we show that the same holds if (i...

Almost Rich Words as Morphic Images of Rich Words

We focus on Θ-rich and almost Θ-rich words over a finite alphabet A, where Θ is an involutive antimorphism over A∗. We show that any recurrent almost Θ-rich word u is an image of a recurrent Θ′-rich word under a suitable morphism, where Θ′ is also an involutive antimorphism. Moreover, if the word u is uniformly recurrent, we show that Θ′ can be set to the reversal mapping. We also treat one spe...