Infinite words without palindrome

We show that there exists an uniformly recurrent infinite word whose set of factors is closed under reversal and which has only finitely many palindromic factors. 1 Notations For a finite word w = w1 · · ·wn, the reversal of w is the word w̃ = wn · · ·w1. This notation is extended to sets by setting F = {w̃ | w ∈ F} for any set F of finite words. A word w is a palindrome if w̃ = w. A set F of finite words is closed under reversal if F = F . For an infinite word x, we denote respectively by Fac(x) and Pal(x), the set of factors of x and the set of factors of x which are palindrome. An infinite word is uniformly recurrent if each of its factors occurs infinitely many times with bounded gap. Equivalently, x is uniformly recurrent if for any integer m, there is an integer n such that any factor of x of length n contains all factors of x of length m. If x uniformly recurrent and if Pal(x) is infinite, the set of factors of x is closed under reversal, that is Fac(x) = Fac(x). In the following examples, we show that the converse does not hold. 2 Over a 4-letter alphabet Let A be the alphabet A = {0, 1, 2, 3}. Define by induction the sequence (xn)n≥0 of words over A by x0 = 01 and xn+1 = xn23x̃n. The first values are x1 = 012310, x2 = 01231023013210 and x3 = 012310230132102301231032013210. We denote by x the limit of the sequence (xn)n≥0. We claim that the word x has the following properties • x is uniformly recurrent, • Fac(x) is closed under reversal : Fac(x) = Fac(x), • Pal(x) is finite : Pal(x) = A.