Cone criterion on an infinite-dimensional torus

On the infinite-dimensional torus $\mathbb{T}^{\infty} = E/2\pi\mathbb{Z}^{\infty}$, where $E$ is an infinite-dimensional real Banach space, $\mathbb{Z}^{\infty}$is an abstract integer lattice, a special class of diffeomorphisms $\mathrm{Diff}(\mathbb{T}^{\infty})$ is considered. This class consists of the mappings $G\colon \mathbb{T}^{\infty} \to \mathbb{T}^{\infty}$ such that the differentials $DG$ and $D(G^{-1})$ are uniformly bounded and uniformly continuous on $\mathbb{T}^{\infty}$. For diffeomorphisms from $\mathrm{Diff}(\mathbb{T}^{\infty})$, we establish the validity of the so-called cone criterion, which is a classical result of finite-dimensional hyperbolic theory (that is, the hyperbolicity criterion formulated in terms of fields of invariant horizontal and vertical cones).