Palindromes in infinite ternary words

TL;DR

This paper investigates the properties of infinite ternary words related to palindromes, establishing conditions under which they satisfy certain palindrome-related properties and constructing examples to illustrate their relationships.

Contribution

It characterizes ternary infinite words with specific complexity and palindrome properties, extending known results from binary to ternary alphabets and providing new constructions.

Findings

Ternary words with complexity 2n+1 satisfy property P and are rich in palindromes.

Property PE is not implied by property P in ternary words, demonstrated by a constructed example.

For words with language closed under reversal, specific conditions ensure palindrome properties.

Abstract

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.