The Research On The Periodic Measures With Deviation Property In The Symbolic Systems And The Non-uniformly Hyperbolic Systems

This thesis studies the deviation properties of periodic measures on symbolicsystems and non-uniformly hyperbolic dynamical systems, including some resultsabout the exponential growth rate of the periodic measures with deviation prop-erty. Presicely丆the systems concerned in this thesis includes: Symbolic systemand non-uniformly hyperbolic system. For this system, given 1 an invariant mea-sure, ' an observable function and亇 an constant, I consider the periodic measure!x with j R 'd!x ! R 'd1j >亇 or j R 'd!x ! R 'd1j丆亇: This thesis concernsthe exponential growth rate (with respect to the period) of the number of suchperiodic measures and hope to bound this exponential growth rate (from topor bottom) by certain characteristics of dynamical systems, such as entropy orgeneralized entropy. In the chapter 2, I get two results on the symbolic systems.The丳rst one, Lemma 2.1.1, states that in such system the exponential growthrate of the periodic measures with deviation property could be controlled fromtop by the supremum of the measure theoretic entropy on a closed subset of in-variant measures. The second one, Theorem 2.1.2, states that the exponentialgrowth rate of the periodic measures with deviation property could be controlled from bottom by the supremum of measure theoretic entropy on an open subsetof invariant measures. In the chapter 3, for non-uniformly hyperbolic systems, Iconcern on the periodic points with a "distance" at least±to a given hyperbolicergodic measure, and prove that the exponential growth rate of the number ofsuch periodic points whose orbits totally contained in aˉxed Pesin block could bebounded from top by the supremum of measure theoretic entropy. This is a gener-alization that deals with non-uniformly expanding situation. Then applying thisresult and using the method of frame bundle developed by Liao, I prove that theexponential growth rate of the number of the periodic measures whose Lyapunovexponents are at least±far from the ones of a given hyperbolic ergodic measureby be bounded from top by the maximum of measure theoretic entropy. More-over, if the given ergodic hyperbolic measure 1 has simple Lyapunov spectrum,then the exponential growth rate of the number of the periodic measures withdeviation property on mean angles between two Oseledec subbundles can be alsobounded from top by the supremum of measure theoretic entropy on some closedset. Notice that the Oseledec splitting is not continuous with the base points(which is just measurable), the Lyapunov exponent of each Oseledec subbundleand the angle between two Oseledec sunbundles are not continuous functions onthe manifold. So these two results are not easy corollaries of Theorem 3.1.1.Fortunately, using the method of frame bundle, the Lyapunov exponents andthe angle function can be presented as integrals of some continuous functions ongeneralized united frame bundle.