Abstract

This proposal focuses on the study of equilibrium states for sequences of continuous functions on symbolic dynamical systems as well as their applications to dimension problems in dynamical systems, continuing our previous line of research. The thermodynamic formalism for sequences of continuous functions is a generalization of that for continuous functions and it has recently evolved rapidly in connection with dimension problems for non-conformal repellers. Developing such a theory will enable us understand equilibrium states for continuous functions from a more general point of view and it helps us study the dimensions of non-uniform hyperbolic sets. The theory of equilibrium states for sequences of continuous functions has been applied to the further study of various areas in symbolic dynamical systems, for example, theory of relative pressure, theory of factor maps between subshifts, and theory of factors of Gibbs measures. The first aim of this project is to study the relations between the generalized Gibbs measures for sequences of continuous functions and the (weak) Gibbs measures for continuous functions. It is known that a generalized Gibbs measure for a sequence of continuous functions can be a weak Gibbs measure for a continuous function. We want to know under which conditions such a phenomenon occurs. The special case of this study is closely related to the study of factor maps between subshifts, in particular, existence of a continuous saturated compensation function. The results that we plan to obtain in our project will also help us study under which conditions a factor of the generalized Gibbs measure for a sequence of continuous functions is a (weak) Gibbs measure for a continuous function. To approach the problems, we will study (relative) pressure theory for continuous functions and sequences of continuous functions as well as the theory of factor map between subshifts. Secondly, we wish to study properties of equilibrium states for sequences of continuous functions such as uniqueness and the Bernoulli properties. To this end, our main goal is to generalize Ruelle’s Perron-Frobenius Theorem and the theory of g-measures for continuous functions to those for sequences of continuous functions. Furthermore, we will study the equilibrium states of sequences of continuous functions for β-transformations. For this purpose, we will study β-shifts. Finally, we plan to apply pressure theory for sequences of continuous functions to problems in dimension theory.