Decimations and sturmian words

Recent Results on Extensions of Sturmian Words

“…The k-decimation of x is the word obtained in withdrawing all letters whose occurrence has a number that is not a multiple of k. It appears that the Fibonacci word is invariant under all decimation. This is a general result, already mentioned by Rauzy and proved by Justin and Pirillo [34].…”

How to build billiard words using decimations

“…Image by decimations of standard billiard words have been already studied, on a two-letter alphabet (k = 2). These words are invariant by standard decimations, see [10] or [3] for a Christoffel version of this result. Roughly speaking, they are the only invariant words if we use two standard decimation with coprime values of n [10].…”

“…Dv is the word consisting of the remaining letters. For a two-letters alphabet A = {a, b}, this notion of n-decimation has been introduced by Rauzy [13], and used in [7] for k = 2, and then independently by Justin and Pirillo [10] (when k = 2 and for cutting sequences) and the author [3] (for k = 2 and for Christoffel words), see also [4] and [12] for some generalizations. For k = 2 and finite words, we have C = aV b, where C is the Christoffel word and V the cutting sequence, and the Christoffel approach is more convenient in this case.…”

Characterizations of finite and infinite episturmian words via lexicographic orderings

“…Nowadays, for most authors, only the aperiodic Sturmian words are considered to be 'Sturmian'. In several of our previous papers (see [9,12,15,19,21] for instance), we have referred to aperiodic Sturmian words as 'proper Sturmian' to highlight the fact that such Sturmian words correspond to the most common sense of 'Sturmian' now. In the present paper, the term 'Sturmian' will refer to both aperiodic and periodic Sturmian words.…”