Abstract
Automata, Logic and Semantics For certain generalized Thue-Morse words t, we compute the critical exponent, i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.
Highlights
It is a well-known fact that the Norwegian mathematician Axel Thue (1863–1922) was the first to explicitly construct and study the combinatorial properties of an infinite overlap-free word over a 2-letter alphabet, obtained as the fixpoint of the morphism μ : {a, b}∗ → {a, b}∗ defined by μ(a) = ab; μ(b) = ba: μ(m) = m = abbabaabbaababba · · ·
That survey mentions some generalizations of the ThueMorse word, and recently other ones were considered in [2, 8]
Let us mention that the notions of ‘fractional power’ and ‘critical exponent’ have received growing attention in recent times, especially in relation to Sturmian and episturmian words; see for instance [4, 5, 7, 10, 11, 12, 13, 15, 18]
Summary
Morse in 1921 in the study of symbolic dynamics, this overlap-free word is called the Thue-Morse word. 1365–8050 c 2007 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France It was shown in [2] that the word tb,m contains arbitrarily long squares, which extends a result previously established by Brlek [6] for m. Let us mention that the notions of ‘fractional power’ and ‘critical exponent’ have received growing attention in recent times, especially in relation to Sturmian and episturmian words; see for instance [4, 5, 7, 10, 11, 12, 13, 15, 18]
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