Palindromes and Sturmian words

An infinite word x over the alphabet A is Sturmian if and only if g x ( n ) = n + 1 for any integer n , where g x ( n ) is the number of distinct words of length n occurring in x . A palindrome is a word that can be read indistinctly from left to right or from right to left. We prove that x is Sturmian if and only if h x ( n ) = 1 + ( n mod 2) for any integer n , where h x ( n ) is the number of palindromes of length n occurring in x . An infinite word x over the alphabet A is generated by a morphism f if there exists a letter c ϵ A such that lim n →∞ f n ( c ) = x . We prove the existence of a morphism that generates the palindromes of any infinite Sturmian word generated by a morphism.