Recurrent geodesics on a surface of negative curvature

The results necessary for the development of this paper are contained in a paper by G. D. Birkhoff,t in a paper by J. Hadamard.J and in an earlier paper by the present writer. § In this earlier paper, as in the present paper, only those geodesies on the given surfaces of negative curvature are considered which, if continued indefinitely in either sense, lie wholly in a finite portion of space. A class of curves is introduced, each of which consists of an unending succession of the curve segments by which the given surface, when rendered simply connected, is bounded. It is shown how a curve of this class can be chosen so as to uniquely characterize some geodesic lying wholly in a finite portion of space. Conversely, it is shown that every geodesic lying wholly in a finite portion of space, is uniquely characterized by some curve of the above class. The results of the earlier paper on geodesies, and the representation obtained there, will be used in the present paper to establish various theorems concerning sets of geodesies and their limit geodesies. In particular, the existence of a class of geodesies called recurrent geodesies of the discontinuous type,\\ will be established. This class of geodesies offers the first proof that has been given in the general theory of dynamical systems, of the existence of recurrent motions of the discontinuous type. For a more complete treatment of the questions of the existence of surfaces of negative curvature, the reader is referred to the paper by Hadamard, already cited.