Spatially quasi-periodic solutions of the Euler equation

We develop a framework for studying quasi-periodic maps and diffeomorphisms on ${\mathbb{R}}^n$. As an application, we prove that the Euler equation is locally well posed in a space of quasi-periodic vector fields on ${\mathbb{R}}^n$. In particular, the equation preserves the spatial quasi-periodicity of the initial data. Several results on the analytic dependence of solutions on the time and the initial data are proved.