Research Projects

Symbolic dynamics is a powerful tool for studying dynamical systems with hyperbolicity, such as Anosov diffeomorphisms and Axiom A maps. It is also used in probability theory, coding, and information theory. Symbolic dynamics refers to a shift space, which is a collection of infinite sequences on a set of symbols, known as an alphabet. A left shift map acts on this sequence, producing dynamics on the space. A subshift of finite type (SFT) is a type of SFT where the alphabet is finite and the sequences are constrained by finite words. A directed graph associated to an SFT is used to study the dynamics of this subshift, with the complexity quantified by topological entropy. Countable Markov shifts are a type of SFT where infinite sequences are given by infinite paths on a directed graph with infinitely many vertices. The main challenge is the loss of compactness of the state space when moving from SET'S to shifts with a countable infinite alphabet. This project aims to study various dynamical aspects of countable Markov shifts from a combinatorial point of view, including defining topological entropy, its relationship with escape rate, and the existence and uniqueness of maximizing invariant measures.