| Topological Pressure For Geodesic Flows (2007) | |||||||||||||
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| . We give a Riemannian formula for the topological pressure of the geodesic ow of a closed Riemannian manifold. As a consequence we derive an asymptotic formula for the stable norm in cohomology in terms of geodesic arcs for manifolds with topological entropy h top = 0. We introduce a Poincare series also in terms of geodesic arcs and we show that it denes a holomorphic function on the half plane given by those complex numbers with real part > h top . 1. Introduction Let M n be a closed connected C 1 manifold and let g be a Riemannian metric of class C r with r 3. Let t : SM ! SM be the geodesic ow of g acting on the unit sphere bundle SM . Given a continuous function f : SM ! R, let P (f) be the topological pressure of the function f with respect to the geodesic ow . We recall its denition. Given T > 0 and a point (x; v) 2 SM , set f T (x; v) := Z T 0 f( t (x; v)) dt: We say that a set E SM is (T; ")-separated if given (x 1 ; v 1 ) 6= (x 2 ; v 2 ) 2 E, there e... | |||||||||||||
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