  This paper uses the framework of reverse mathematics to investigate the
strength of two recurrence theorems of topological dynamics. It establishes
that one of these theorems, the existence of an almost periodic point, lies
strictly between WKL and ACA (working over RCA_0). This is the first example of
a theorem with this property. It also shows the existence of an almost periodic
point is conservative over RCA_0 for Pi^1_1 sentences. These results establish
the existence of a new upwards-closed subclass of the PA degrees
