A characterization of overlap-free morphisms

Algebraic Irrational Binary Numbers Cannot Be Fixed Points of Non-trivial Constant Length or Primitive Morphisms

“…A simpler proof of this theorem has been given by Berstel and Se e bold in [1]. Define a morphism to be overlap-free if and only if the image of any overlap-free word is itself overlap-free.…”

“…Thue [11] proved in 1912 that the only binary overlap-free morphisms are + k and E b + k , with k 0, where + is the morphism defined by +(0)=01 and +(1)=10, and E the morphism defined by E(0)=1 and E(1)=0. The proof of Berstel and Se e bold in [1] consists in first showing that if a morphism h has the property that the word h(01101001) is overlap-free, then the morphism h is either equal to + k or to E b + k for some k 0. Thue's result can easily be deduced from this theorem.…”

“…Theorem 1. Let x a positive real number whose binary expansion is a fixed point of a morphism on the alphabet [0,1]. If the morphism is either of constant length 2 or primitive, then the number x is either rational or transcendental.…”

“…The first is an extension of a theorem due to Ferenczi and Mauduit [4], which in turn is a combinatorial reformulation of Ridout's theorem (see [8] for example), and is used in case the real number has an overlap in its binary expansion. An overlap is a block of the form WWa, where a is the first letter of the word W. The second, due to Se e bold [10,1] gives a simple and nice characterization of the fixed points of binary morphisms that are overlap-free. The third, proved originally by Mahler [6] and later by Dekking [3], involves the transcendence of the Thue Morse number, defined as the number whose binary expansion is the fixed point beginning in 0 of the morphism 0 Ä 01, 1 Ä 10.…”

Combinatorics on Words – A Tutorial

“…Thus w is a test word if {w} is a test set. It has been shown [BS93] that abbabaab is a test word for overlap-free morphisms. We refer to [RS99] and [RW02b] for a detailed study of test words in various cases.…”