Distributions Of Closed Orbits And Statistical Properties Of Some Hyperbolic Dynamical Systems

In Chapter 1, We consider the billiard flows on the plane with finitely many scatters with smooth boundary satisfying the visibility assumption and describe the asymptotics of the distribution of closed orbits with an even collision number. And we show that the closed orbits of the billiard flow is equidistributed.Chapter 2 deals with some refinements of the central limit theorem for a class of non-uniformly hyperbolic dynamical systems called Young's system, such as local central limit theorem and so-called Berry-Esseen theorem giving the rate of convergence in the central limit theorem.In Chapter 3, we continue to consider the Young's system and show that if the components of a R~d-valued Holder observable f are cohomologously independent f satisfies the multidimensional central limit theorem. Moreover if f is aperiodic, then f satisfies the local multidimensional central limit theorem.In Chapter 4, We consider rational maps of degree d＞2 on the Riemann sphere and prove that they satisfy the large deviation theorem.