Quasi-periodic functions, diffeomorphisms, and their applications

I will start with a review of the main properties of quasi-periodic (and almost periodic) functions on $\mathbb{R}^n$. Almost periodic functions were introduced by H. Bohr and studied by Bochner, von Neumann, and others. Quasi-periodic functions appear naturally in applications as a generalization of periodic functions. I will introduce the quasi-periodic diffeomorphisms on $\mathbb{R}^n$ and will show that they form a topological group. As an application, I will construct spatially quasi-periodic solutions of a class of partial differential equations appearing in fluid dynamics.