Multifractal Analysis And Large Deviation Property In Some Nonuniformly Hyperbolic Systems

We say (X, d,f) is a topological dynamical system means that (X, d) is a com-pact metric space together with a continuous self-map f:Xâ†'X. In topological dynamics one studies the asymptotic properties of continuous maps, such as topolog-ical entropy, topological pressure, chaos and Lyapunov exponents. The well-known notions of topological and measure entropy constitute essential invariants in the char-acterization of the complexity of a dynamical system in classical ergodic theory. The relationship between these two quantities is the so-called variational principle. We focus on the dimension-like entropy(pressure) and establish conditional variational principles for certain non-compact sets.The thesis is devoted to multifractal analysis and large deviation property, which are related to Birkhoff ergodic theorem, via orbit-gluing properties in dynamical sys-tems. Our results can be applied to some nonuniformly hyperbolic systems. The subject of chapter 1 and chapter 2 is the multifractal analysis in some nonuniform-ly hyperbolic systems satisfying nonuniform specification property. In chapter 3, we introduce Pesin theory and define weak shadowing property which is applicable for nonuniformly hyperbolic systems satisfying Pesin theory. In chapter 4 and chapter 5, we give conditional variational principles for level sets and historic sets of Birkhoff averages separately under weak shadowing property. Meanwhile, similar results can be obtained for maps with limit shadowing property. The main purpose of chapter 6 is to study the large deviation property via Pesin theory.