Uniform recurrence
= Uniform recurrence
= Uniformly recurrent
{synonym}
A two-sided <sequence> over a finite <alphabet> is <uniformly recurrent> if every finite <word over an alphabet> occurring in it occurs with bounded gaps. More precisely, for each such word some $M$ ensures that every length-$M$ interval contains a complete occurrence. This is equivalent to being a <minimal point> of the <full shift>: a finite cover of the <orbit closure> by preimages of a word's <cylinder set> bounds its return gaps; conversely, bounded gaps pass to all points of the <orbit closure> and make every forward <orbit> dense there. The property concerns finite words, not infinite <integer intervals>.