## Symbolic Dynamics

A sequence over a finite alphabet (one- or two-sided, though I'll show only the one-sided case) x1, x2, ... is called recurrent if every word w=xm...xm+l-1 that appears in it once appears in it infinitely often.

That is, for every N there exists k>N for which w=xk...xk+l-1.

Every periodic sequence is recurrent. Every recurrent sequence is strongly recurrent (but the converse is of course false). See that node for examples of (strongly) recurrent sequences that are not periodic.

## Dynamics

To generalize, note that a finite alphabet has only a single metric topology; two sequences are close if they share a long common initial prefix.

So we can use a more interesting space by adding in a metric. A sequence x1, x2, ... over any metric space X (again, one- or two-sided, but I'll only show the one-sided case) is called recurrent if it approaches itself infinitely often, that is if for every n and ε,

d((xn, xn+1, ...), (xn+k, xn+k+1, ...)) ≤ ε
for infinitely many values k.

In other words, if the trajectory of a point x visits any point σnx, then it visits any neighborhood of x infinitely often.

Re*cur"rent (-rent), a. [L. recurrens, -entis, p. pr. of recurrere: cf.F. r'ecurrent. See Recur.]

1.

Returning from time to time; recurring; as, recurrent pains.

2. Anat.

Running back toward its origin; as, a recurrent nerve or artery.

Recurrent fever. Med. See Relapsing fever, under Relapsing. -- Recurrent pulse Physiol., the pulse beat which appears (when the radial artery is compressed at the wrist) on the distal side of the point of pressure through the arteries of the palm of the hand. -- Recurrent sensibility Physiol., the sensibility manifested by the anterior, or motor, roots of the spinal cord (their stimulation causing pain) owing to the presence of sensory fibers from the corresponding sensory or posterior roots.

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