Recurrent word
In mathematics, a recurrent word or sequence is an infinite word over a finite alphabet in which every factor occurs infinitely many times. An infinite word is recurrent if and only if it is a sesquipower.
A uniformly recurrent word is a recurrent word in which for any given factor X in the sequence, there is some length nX such that X appears in every block of length nX. The terms minimal sequence and almost periodic sequence are also used.
Examples
- The easiest way to make a recurrent sequence is to form a periodic sequence, one where the sequence repeats entirely after a given number m of steps. Such a sequence is then uniformly recurrent and nX can be set to any multiple of m that is larger than twice the length of X. A recurrent sequence that is ultimately periodic is purely periodic.
- The Thue–Morse sequence is uniformly recurrent without being periodic, nor even eventually periodic.
- All Sturmian words are uniformly recurrent.