Abstract not found

The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.

There are circular square-free words of length n on three symbols for n 18. This proves a conjecture of R. J. Simpson.

We characterize the squares occurring in infinite overlap-free binary words and construct various α power-free binary words containing infinitely many overlaps. 1

Introduction In his classical 1912 paper [15] (see also [3]), A. Thue gave the first example of an overlap-free infinite word, i. e., of a word which contains no subword of the form axaxa for any symbol a and word x. Thue's example is known now as the Thue-Morse word w TM = 01101001100101101001011001101001 : : :: It was rediscovered several times, can be constructed in many alternative ways and occurs in various fields of mathematics (see the survey [1]). The set of all overlap-free words was studied e. g. by Fife [8] who described all binary overlap-free infinite words and Seebold [13] who proved that the Thue-Morse word is essentially the only binary overlap-free word which is a fixed point of a morphism. Nowadays the theory of overlap-free words is a part of a more general theory of pattern avoidance [5]. J.-P. Allouche and J. Shallit [2] asked if the initial Thue's construction of an overlap-free wo

It is shown that 2+-repetition, i.e. a word of the form uvuvu where u is a letter and v is a word, is the smallest repetition which can be avoided in infinite words over binary alphabet. Such binary words avoiding pattern uvuvu, finite or infinite, are called as 2+-free words and those words are the main topic of this work. It is shown here that 2+-free words over binary alphabet can be presented as words built from special kind of blocks, called Morse-blocks, with some rules. In particular, the given presentation by these blocks is unique for 2+-free words long enough. Moreover, it is also shown that the language generated by this presentation can be described by some automaton. In fact, the corresponding presentation in blocks for finite 2

We show that there exist binary circular 5=2 power free words of every length.