Topologie p-adique sur les mots
Résumé
This is a survey article on the combinatorial aspects of the p-adic metric and p-adic topology on words. We give several equivalent definitions of these notions, illustrated by several examples and properties. After giving a detailed description of the open sets, we prove that the p-adic metric is uniformly equivalent with a metric based on the binomial coefficients defined on words. We also give two examples of converging sequences for the p-adic topology. The first example consists of the sequence of the pn powers of a given word, that converges to the empty word. The second one consists of the sequence of prefixes of the Prouhet-Thue-Morse word: for each prime number p, on can extract from this sequence a subsequence converging to the empty word in the p-adic topology. Most of the proofs are omitted, apart from the very short ones.