Subword complexity and power avoidance

Abstract

We begin a systematic study of the relations between subword complexity of infinite words and their power avoidance. Among other things, we show that -the Thue-Morse word has the minimum possible subword complexity over all overlapfree binary words and all ( 7 3 )-power-free binary words, but not over all ( 7 3 ) + -power-free binary words; -the twisted Thue-Morse word has the maximum possible subword complexity over all overlap-free binary words, but no word has the maximum subword complexity over all ( 7 3 )-power-free binary words; -if some word attains the minimum possible subword complexity over all square-free ternary words, then one such word is the ternary Thue word; -the recently constructed 1-2-bonacci word has the minimum possible subword complexity over all symmetric square-free ternary words.

Transition Property for $$\alpha $$-Power Free Languages with $$\alpha \ge 2$$ and $$k\ge 3$$ Letters

“…It is well known that the Thue Morse word is a right infinite 2 + -power free word over an alphabet with 2 letters [11]. It follows that the Thue Morse word is α-power free for each α > 2.…”

“…It is well known that there are infinite 2-power free words over an alphabet with 3 letters [11]. Suppose 0, 1, 2 ∈ Σ k .…”

“…Suppose 0, 1, 2 ∈ Σ k . An example is the fixed point of the morphism θ defined by θ(0) = 012, θ(1) = 02, and θ(2) = 1 [11]. If an infinite word t is 2-power free then obviously t is α-power free and α + -power free for each α ≥ 2.…”

On minimal critical exponent of balanced sequences

“…2 is reached by a complementary symmetric Rote sequence. Shallit and Shur [29] proved a number of results connecting factor complexity and critical exponent of sequences. For example, they established that the Thue-Morse sequence and the twisted Thue-Morse sequence have, respectively, the minimum and the maximum factor complexity over all binary overlap-free sequences; the minimal critical exponent of a binary sequence of factor complexity 2n is 5/2; the set of ternary square-free sequences either has no sequence of minimum complexity, or the minimum is reached by the ternary Thue sequence.…”

Transition Property for Cube-Free Words

“…The transition words can be naturally interpreted as transitions in those automata and the transition property forces the automata to be strongly connected. Second, recently it was shown [16,Thm 39] that the transition property implies the existence of infinite α-power-free words of very big subword complexity. Namely, Theorems 1 and 9 imply that for every d ≥ 2 there exists a d-ary cube-free infinite word which contains all two-sided extendable d-ary cube-free finite words as factors.…”