An infinite sequence *x*_{1}, *x*_{2}, ... (one- or two-sided, though I'll only show the one-sided case) of symbols is called *strongly recurrent* if the following holds:

- For every word
*w*=*x*_{m}...x_{m+l-1} that appears in the sequence there exists some constant *L* such that *w* appears somewhere in *every* word *x*_{k}...x_{k+L-1} of length *L* that appears in the sequence.

Obviously, every periodic sequence is strongly recurrent. But there also exist aperiodic sequences that are strongly recurrent; the Morse sequence is one example of such a sequence.

In fact, for words over the 2-symbol alphabet {0,1}, there exist only countably many periodic sequences, but uncountably many strongly recurrent sequences.

In terms of the trajectory of the infinite sequence (under the shift operation **S***x*=*x*_{2}*x*_{3}...), we say that a strongly recurrent sequence *returns to every neighbourhood it visits at bounded times*.