Abstract

The purpose of this project is to study dimension problems in dynamical systems using techniques in symbolic dynamics. Let T be a continuous expanding map of a Riemannian manifold and K be a compact invariant set. Our research aims to study the Hausdorff dimension of K, measures of full dimension and their properties, and to determine uniqueness of such measure for a nonconformal expanding map T . In particular, we will consider an expanding map of the n-torus given by an integer-valued diagonal matrix for n ≥ 2 and a compact set K whose symbolic representation is a topologically mixing shift of finite type. Our main approach will be to use the results of Gatzouras-Peres and Shin. We will also study how our strategy can be applied to more general maps such as an expanding map given by a real-valued diagonal matrix. Throughout the project we will study the theory of equilibrium states, relative pressure, compen- sation functions and Markov partitions. New results in these areas will be applied to the dimension problems in dynamics.