| On Two Noteworthy Deformations Of Negatively Curved Riemannian Metrics (1999) | |||||||||||||||
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| . Let M be a closed connected C 1 Riemannian manifold whose geodesic ow is Anosov. Let be a smooth 1-form on M . Given 2 R small, let hEL () be the topological entropy of the Euler-Lagrange ow of the Lagrangian L (x; v) = 1 2 jvj 2 x x (v); and let hF () be the topological entropy of the geodesic ow of the Finsler metric, F (x; v) = jvj x x (v): We show that h 00 EL (0) + h 00 F (0) = h 2 Var(); where Var() is the variance of with respect to the measure of maximal entropy of and h is the topological entropy of . We derive various consequences from this formula. 1. Introduction Let M be a closed connected C 1 Riemannian manifold whose geodesic ow is Anosov. This happens for example, when all the sectional curvatures are negative. Let be a smooth 1-form on M . We think of as a function : TM ! R such that for each x 2 M , x is a linear functional of T x M . For 2 R consider the 1-parameter family of convex superlinear Lagrangians ... | |||||||||||||||
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