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Let M be a connected, compact, Riemannian manifold. Geodesic flow is a flow on the unit tangent bundle of M . This flow can be studied in dynamics prespective. for example entropy or complexity of the geodesic flow. in this thesis we will follow methods of entropy estimation or computing for geodesic flow. we will follow the method of anthony manning and Ricardo Mañe for proving such result. Maning present two results linking the topological entropy of the geodesic flow on M. we expalin how he find exponential growth rate volume of balls in universal cover as a lower bound for topologycal entropy. another theorem , Mañe represent the equlity between exponential growth rate of avrage of... 

In this thesis, we introduce and prove the inequalities and theorems of Smulian which are about derivatives of convex functions and optimization of linear functionals. Then we show that some old ideas of Smulian can be used to give another proof of theorems of Bourgain. The first theorem of Bourgain associates to any bounded, convex, closed and dentable subset like D of a Banach space X a G-delta subset of the dual space and the second theorem of Bourgain investigates the optimization in Banach spaces. Finally, we characterize subsets of Banach spaces having the Radon-Nikodym property by means of optimization results.