A Class Of Graph Directed Recurrent Sets And The Ruelle Operator

The Ruelle-Perron-Frobenius theorem is well known in fractal geometry, moreover to be a standard tool in dynamical systems, thermodynamic formalism and multifractal formalism.Early, D.Ruelle introduced a convergence theorem to study the equilibrium state of the infinite one-dimensional lattice gas.After a few years, Bowen set up the theorem as the convergence of the iterations of a certain operator on the space of continuous functions on a symbolic space. Recently, Fan and Lau continued to study the operator by adopting the iterated function system. Now, There is a vast literature on the Ruelle operator and the related eigenproblem and the convergence property.In this paper we will extend the theorem to a class of graph directed recurrent sets by the skill of the paper [6].There are three chapters in this paper.First chapter is the introduction.In our second chapter, we first present the definitions about the Ruelle operator and the theorem in [5] and [6], and graph directed recurrent sets.Chapter 3 first gives the definition of a class of graph directed recurrent sets in this paper, then give proof of the related Ruelle operator theorem.