Palindromes in infinite ternary words

¹ Doppler Institute & Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, Praha 2 120 00, Czech Republic; l.balkova@centrum.cz
² Doppler Institute & Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, Praha 2 120 00, Czech Republic; edita.pelantova@fjfi.cvut.cz
³ Institut de Mathématiques de Luminy, Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France.
⁴ Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, Praha 2 120 00, Czech Republic; starosta@iml.univ-mrs.fr

Received: 3 February 2009
Accepted: 9 June 2009

Abstract

We study infinite words u over an alphabet satisfying the property , where denotes the number of palindromic factors of length n occurring in the language of u. We study also infinite words satisfying a stronger property : every palindrome of u has exactly one palindromic extension in u. For binary words, the properties and coincide and these properties characterize Sturmian words, i.e., words with the complexity C(n) = n + 1 for any . In this paper, we focus on ternary infinite words with the language closed under reversal. For such words u, we prove that if C(n) = 2n + 1 for any , then u satisfies the property and moreover u is rich in palindromes. Also a sufficient condition for the property is given. We construct a word demonstrating that on a ternary alphabet does not imply .