What is ergodic theory

Ergodic theory involves the study of transformations on measure spaces. Interchanging the words “measurable function” and “probability density function” translates many results from real analysis to results in probability theory. Ergodic theory is no exception. Ergodic theory has fundamental applications in probability theory, starting from areas that are very well understood, such as finite state Markov chains. In this paper we define the main objects of interest in the study of ergodic theory, and in particular focus on some applications on so-called interval exchange transformations. Studying this particular class of transformations illuminates the general theory, for as we show, these transformations illustrate the independence of some notions that a priori might see connected.