Variational Principles And Multifractal Analysis In Topological Dynamical Systems

The thesis consists of two parts. The first part consisting of Chapter1and Chapter2investigates the dynamical systems acted by groups and establishes the variational principles for local sofic pressure and sofic topological pressure for groupoids. The second part studies the historic set by adopting the idea of multifractal anlysis. In Chapter3, a finer description about divergence points is presented in Olsen’s frame-work and the topological pressures of the level sets are computed. In Chapter4, we compute the packing entropy of the level sets in historic set. In Chapter5, applying the construction of Moran fractal, we investigate the topological pressure of the strong scrambled set. In Chapter6, the chaotic phenomenon in multifractal structure is in-vestigated and the strong scrambled set in multifractal structure is also computed. In Chapter7, we compute the packing dimensions of the divergence points of self-similar measures with OSC. This solves the conjecture about packing dimension posed by Olsen and Winter (J. London Math. Soc.(2)67(2003)).