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Rich words

Glen, A. (2008) Rich words. In: LaCIM seminar series, 12 September, University of Québec, Montréal, Canada.

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In 2001, Droubay, Justin, and Pirillo showed that any finite word w of length │w│ contains at most │w│ + 1 distinct palindromes (including the empty word). Inspired by this result, Justin and I recently initiated a unified study of finite and infinite words that are characterized by containing the maximal number of distinct palindromes, which we call ”rich words” (in view of their palindromic richness). In this talk I will explore various properties of this intriguing class of words. In particular, I will show that a characteristic property of rich words is that all “complete returns” to any palindromic factor are palindromes. These words encompass the well-known family of episturmian words, originally introduced by Droubay et al. in 2001. Another special class of rich words consists of sequences with ”abundant palindromic prefixes”, which were introduced and studied by Fischler in 2006 in relation to Diophantine approximation. Other examples of rich words have appeared in many different contexts; they include complementation-symmetric Rote sequences, symbolic codings of trajectories of symmetric interval exchange transformations, and a certain class of words associated with β-expansions where β is a simple Parry number.

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