Basic ergodic theory

Consider a ''random'' process where the next state depends on the previous one but in a chaotic manner - say we consider a particle moving in a box. One would expect that as time passes, the trajectory of the particle fills the entire cube, and does so in a uniform way, so that different subsets are visited with frequency comparable to their volume. The path of the particle would then be called ergodic. Ergodic theory offers tools for analyzing dynamical systems, in particular as the time parameter of the system goes to infinity. Central questions are whether any subset of positive measure in our measure space is visited infinitely often, and whether the process is equidistributed, meaning that the measure of the subset tells the probability that the process lies in the subset at a given time.

In this post, we present the basics of ergodic theory, and use them to prove a strong version of Weyl's equidistribution result for the numbers $\gamma,2\gamma,3\gamma...$ modulo $1$ with $\gamma$ irrational. Another application of ergodic theory we present is the fact that almost all real numbers are normal in the sense that each finite sequence of digits occurs with the expected frequency, in each base. We also sketch a proof of the law of large numbers, assuming merely that the iid. random variables have an expectation. We require only the very basics of measure theory.