On a~class of Anosov diffeomorphisms on the infinite-dimensional torus

S. D. Glyzin ; A. Yu. Kolesov ; N. Kh. Rozov

Izvestiya. Mathematics , Tome 85 (2021) no. 2, pp. 177-227

Résumé

We study a quite natural class of diffeomorphisms $G$ on $\mathbb{T}^{\infty}$, where $\mathbb{T}^{\infty}$ is the infinite-dimensional torus (the direct product of countably many circles endowed with the topology of uniform coordinatewise convergence). The diffeomorphisms under consideration can be represented as the sums of a linear hyperbolic map and a periodic additional term. We find some constructive sufficient conditions, which imply that any $G$ in our class is hyperbolic, that is, an Anosov diffeomorphism on $\mathbb{T}^{\infty}$. Moreover, under these conditions we prove the following properties standard in the hyperbolic theory: the existence of stable and unstable invariant foliations, the topological conjugacy to a linear hyperbolic automorphism of the torus and the structural stability of $G$.

Détail

Keywords: diffeomorphism, hyperbolicity, infinite-dimensional torus, topological conjugacy, structural stability.
Mots-clés : invariant foliations