Mananifold

Recurrence in phenomena, physical or otherwise, has piqued the curiosity of humans for a long time. Without going into a historical detour on the study of periodic phenomena (of which the author is painfully ignorant), we cut to the chase and introduce one of the foundational results of ergodic theory: the Poincaré Recurrence Theorem. The theorem essentially states that under certain transformations of a space (to be made precise shortly), the system will almost return to its initial state under repeated iterations of the transformation. Preliminaries In order to state the theorem, we recall the definition of a probability space and a measure-preserving transformation here. Definition. [Probability Space] A probability space $(X,\mathcal{B},\mu)$ is a nonempty set $X$, a $\sigma$-algebra $\mathcal{B}$ on $X$, and a measure $\mu$ on $(X,\mathcal{B})$ with $\mu(X)=1$. Assume further that the space $(X,\mathcal{B},\mu)$ is complete, in the sense that all subsets of measurable subs...