4 research outputs found
On Theta-palindromic Richness
In this paper we study generalization of the reversal mapping realized by an
arbitrary involutory antimorphism . It generalizes the notion of a
palindrome into a -palindrome -- a word invariant under . For
languages closed under we give the relation between
-palindromic complexity and factor complexity. We generalize the notion
of richness to -richness and we prove analogous characterizations of
words that are -rich, especially in the case of set of factors
invariant under . A criterion for -richness of
-episturmian words is given together with other examples of
-rich words.Comment: 14 page
Palindromes in infinite ternary words
We study infinite words u over an alphabet A satisfying the property
P : P(n)+ P(n+1) = 1+ #A for any n in N, where P(n) denotes the number of
palindromic factors of length n occurring in the language of u. We study also
infinite words satisfying a stronger property PE: every palindrome of u has
exactly one palindromic extension in u. For binary words, the properties P and
PE coincide and these properties characterize Sturmian words, i.e., words with
the complexity C(n)=n+1 for any n in N. 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 n in N, then u satisfies the property P and
moreover u is rich in palindromes. Also a sufficient condition for the property
PE is given. We construct a word demonstrating that P on a ternary alphabet
does not imply PE.Comment: 12 page
On Brlek-Reutenauer conjecture
Brlek and Reutenauer conjectured that any infinite word u with language
closed under reversal satisfies the equality 2D(u)=\sum_{n=0}^{\infty} T(n) in
which D(u) denotes the defect of u and T(n) denotes C(n+1)-C(n)+2-P(n+1)-P(n),
where C and P are the factor and palindromic complexity of u, respectively.
Brlek and Reutenauer verified their conjecture for periodic infinite words. We
prove the conjecture for uniformly recurrent words. Moreover, we summarize
results and some open problems related to defect, which may be useful for the
proof of Brlek-Reutenauer Conjecture in full generality