Elements of hyperbolic theory on an infinite-dimensional torus

Abstract

On the infinite-dimensional torus $\mathbb{T}^{\infty}=E/2\pi\mathbb{Z}^{\infty}$, where $E$ is an infinite-dimensional Banach torus and $\mathbb{Z}^{\infty}$ is an abstract integer lattice, a special class of diffeomorphisms $\operatorname{Diff}(\mathbb{T}^{\infty})$ is considered. It consists of the maps $G\colon\mathbb{T}^{\infty}\to\mathbb{T}^{\infty}$ whose differentials $DG$ and $D(G^{-1})$ are uniformly bounded and uniformly continuous on $\mathbb{T}^{\infty}$. For diffeomorphisms in $\operatorname{Diff}(\mathbb{T}^{\infty})$ elements of hyperbolic theory are presented systematically, starting with definitions and some auxiliary facts and ending by more advanced results. The latter include a criterion for hyperbolicity, a theorem on the $C^1$-roughness of hyperbolicity for diffeomorphisms in $\operatorname{Diff}(\mathbb{T}^{\infty})$, the Hadamard-Perron theorem, as well as a fundamental result of hyperbolic theory, the fact that each Anosov diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ has a stable and an unstable invariant foliation. Bibliography: 34 titles.