The Variational Principle On Non-compact Invariant Set

In this thesis, we study sets of generic points for shift-invariant measures in the countable symbolic system. For a shift-invariant Borel probability measure μ, we denote by Gμ the set of all generic points of μ. We obtain a formula of Billingsley dimension of Gμ with respect to the metric defined by a Gibbs mea-sure. This dimension formula is given by a variational principle. An application is given to the continued fraction system. We stress that Billingsley dimension with respect to many Gibbs measures (including Gauss measure) is obtained.This thesis consists of five chapters. In Chapter1, we give an introduction to the relationship between the dimension theory and the theory in dynamical systems and present the origin and the development of the main problem we consider in this thesis. We also present the main results of this thesis.In Chapter2, we recall some basic facts about dynamical systems, the con-tinued fraction, entropy and Hausdorff dimension. Especially, some important properties about countable symbolic system are also included.Chapter3is devoted to the construction of a generic point with a given growth. We use this generic point to "sprout" a Cantor subset of Gμ.After the preparations we prove the dimension formula given by a variational principle in Chapter4.In Chapter5, an application is given to a class of expanding interval maps with infinitely many branches. We determine the Billingsley dimension of generic points set in continued fraction with respect to Gibbs measures. Actually, we also determine the Hausdorff dimension of generic points set with respect to Euclidean metric. In addition, we are concerned with the quasi-generic points set in finite symbolic dynamical systems. The Billingsley dimension of this set is given, which extends Bowen’s result.