Abstract

The purpose of this project is to study thermodynamic formalisms for se- quences of continuous functions on symbolic dynamical systems and problems concerning factor maps between symbolic dynamical systems, and their applica- tions to dimension problems in dynamical systems. Thermodynamic formalism, which was originally developed in statistical mechanics, works well in dimension problems, connecting geometry of sets and symbolic dynamical systems. Ther- modynamic formalisms for sequences of continuous functions have recently de- veloped rapidly in connection with dimension problems for nonconformal maps. Our first goal of this project is to study the properties of equilibrium states for sequences of continuous functions such as uniqueness, the Bernoulli and gen- eralized Gibbs properties. For this purpose, we first consider an almost additive potential on a topologically mixing subshift (X, σX ) on finitely many symbols. We are interested in finding properties of unique equilibrium state for an almost additive potential Φ = {logfn}∞n=1 on X with the property of bounded varia- tion, by using the Ruelle’s transfer operator. We are also interested in finding a class of subadditive potentials with the unique equilibrium states. Secondly, we will investigate the equilibrium states for factor maps between subshifts for sequences of continuous functions. Let π : (X, σX ) → (Y, σY ) be a factor map between subshifts and Φ = {logfn}∞n=1 be a subadditive potential on X. If μ be (unique) generalized Gibbs for Φ, then when is πμ a generalized Gibbs for a subadditive potential Ψ? The question has been studied recently for the case when a subadditive potential is replaced by by a continuous function of summable variation. We would like to generalize the results. Finally, applications could be given to problems in dimension theory for nonconformal maps.