Let f be a continuous map of an interval I/R. The periods of the periodic points of f are described by a theorem of Sarkovskii [6,12], which defines an ordering in the set N* of all positive integers in such a way that if n # N* is a period of x # I, every integer following n in Sarkovskii's ordering is the period of some point of I. Quite different situations may arise when the dynamical system generated by the iterations of a single map is replaced by a continuous semiflow. As will be shown in this article, if , : R + _I ร I is a continuous semiflow on I, the existence of a periodic orbit implies that , is the identity; or equivalently, the only compact orbits are reduced to the fixed points of ,. If I is replaced by a circle, the existence of a periodic point implies that , is a periodic flow.
Of course, these and other results involving also |-stable points and non-wandering points depend in an essential way on the fact that , is a one-dimensional dynamical system. To reach similar conclusions for flows and semiflows in higher dimensions, stronger hypotheses on the single maps of , have to be introduced. Thus, if , is a continuous semiflow of holomorphic self-maps of the open unit ball of a complex Hilbert space, the existence of a periodic orbit spanning a dense linear subspace of the entire space implies that , is a periodic flow. In the case of complex dimension one, i.e., when , acts holomorphically on the open unit disc of the complex plane, the periodicity of , follows from weaker conditions: namely from the hypothesis whereby the set of non-wandering points of , is not empty.
- Let M be a metric space with the topology defined by a distance d. For a continuous map f : M ร M, we will denote the iterates of f, writing f 0 =identity map, f 1 = f and f n = f n&1 b f for n=1, 2, ... . If f is a homeomorphism, we will extend this notation to all n # Z writing f n =( f &1 ) &n when n is a negative integer. The dynamical system generated iterating f is a discrete dynamical system.