Ergodic Properties of Full Branch Maps

Abstract

Dynamical System is the study of a point in state space and its behavior over the time. This system contain functions which describe time dependence of a point in a topological space. There are some examples of dynamical system such as celestial bodies motion, flow of water through a pipe, the movement of clock etc. Markov chain in dynamical system is a stochastic process that describes sequence of possible events whose probabilities relies on the states achieved in the previous event. In simple terms, outcomes of future state is dependent solely on current state. Markov chain has many applications e.g. lines of cus- tomers arriving at the airport or the ques in front of cash counter of shop, Exchange change rates of currency etc. Ergodic theory play a huge role in dynamical systems. Ergodic theory explains the statistical properties of dynamical system which are deterministic. By deter- ministic dynamical system, we mean the equation which determines the dynamics which do not have any random noise, perturbations. In this thesis, we will discuss ergodic prop- erties of full branch map. We will describe more general result about Markov maps which indirectly implies to full branch map. We will discuss Markov Chains, Markov Measure and Conjugacy of Markov Shifts and Interval Maps to reach out to our main result.