Set Of Periodic Points Research Articles

Periodic Orbits of Isometric Flows

Let M be a compact C∞ Riemannian manifold, X a Killing vector field on M, and φt its 1-parameter group of isometries of M. In this, paper, we obtain some basic properties of the set of periodic points of φt. We show that the set of least periods is always finite, and the set P(X, t) of points of M having least period t for the vector field X is a totally geodesic submanifold, with possibly non-empty boundary. Moreover, we show there are at least m geometrically distinct closed geodesic orbits of φt, where m is the number of least periods which are not integral multiples of any other least period.

X.The geometry of discrete vector maps

Summary The geometry of finite and periodic point sets in S1 and of their corresponding vector sets in S 2is discussed. Methods are given for deriving from any S2 set the complete class of S1 sets corresponding. The application of these ideas to crystal analysis is indicated. It is shown that by means of this technique X-ray data may be used to discover crystal structures, not merely to test structures already devised.