Ergodic Properties of Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature

This note, like the previous one [1], is mainly based on the fact that a geodesic flow on a closed Riemannian manifold of negative curvature satisfies certain conditions (U) formulated below. Therefore it seems appropriate to consider arbitrary dynamical systems satisfying these conditions. A dynamical system is understood throughout this paper to be defined on an m-dimensional connected closed smooth manifold W ^m , to be of class C ² and to have an integral invariant. ^* The dynamical system may have either continuous or discrete time. Since a dynamical system with continuous time is often called a flow, I shall call a system with discrete time a cascade. A flow is determined by giving a vector field f(w) of class C ² on W ^m ; T ^t w, where t runs through all real numbers, denotes the solution of the system of differential equations d ( T t w ) d t = f ( T t w ) , T 0 w = w . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003069515/f3ef54a9-ca7f-48a5-bad6-5d54427f48cf/content/eq3490.tif"/>