Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A characterization of balanced episturmian sequences. | A characterization of balanced episturmian sequences Genevieve Paquin y Laurent Vuillonz Submitted Nov 21 2006 Accepted Apr 24 2007 Published May 9 2007 Mathematics Subject Classification 68R15 Abstract It is well-known that Sturmian sequences are the non ultimately periodic sequences that are balanced over a 2-letter alphabet. They are also characterized by their complexity they have exactly n 1 distinct factors of length n. A natural generalization of Sturmian sequences is the set of infinite episturmian sequences. These sequences are not necessarily balanced over a k-letter alphabet nor are they necessarily aperiodic. In this paper we characterize balanced epistur-mian sequences periodic or not and prove Fraenkel s conjecture for the special case of episturmian sequences. It appears that balanced episturmian sequences are all ultimately periodic and they can be classified in 3 families. 1 Introduction Sturmian sequences are exactly the non ultimately periodic balanced sequences over a 2-letter alphabet 6 18 . A sequence s is balanced if for every letter a the number of a s in any two n-length factors differs by at most 1 for any n. Sturmian sequences are also characterized by their number of n-length factors they always have n 1 factors of length n for every n. For Sturmian sequences the two conditions are equivalent. There are two different generalizations of Sturmian sequences for alphabets of cardinality k 3. A natural generalization of Sturmian sequences is the set of infinite episturmian sequences. It was first obtained by a construction due to de Luca 17 which uses the palindromic closure. The class of strict episturmian sequences over a 3-letter alphabet also appears in 19 and is studied in 3 . The set of episturmian sequences have been extensively studied by Droubay Justin and Pirillo 9 15 16 more recently. The second generalization of Sturmian sequences is the set of balanced sequences studied in 5 22 23 . with the support of NSERC Canada yLaboratoire .
