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Dynamical systems on an infinite-dimensional torus: Fundamentals of hyperbolic theory
S. D. Glyzin,
A. Yu. Kolesov Centre of Integrable Systems, P.G. Demidov Yaroslavl State University
Abstract:
On the infinite-dimensional torus
$\mathbb{T}^{\infty}=E/2\pi\mathbb{Z}^{\infty}$, where
$E$ is an infinite-dimensional real Banach space and
$\mathbb{Z}^{\infty}$ is an abstract integer lattice, a special class of diffeomorphisms
$\mathrm{Diff}(\mathbb{T}^{\infty})$ is considered. This class consists of mappings $G\colon\mathbb{T}^{\infty}\to\mathbb{T}^{\infty}$ for which the differentials
$DG$ and
$D(G^{-1})$ are uniformly bounded and uniformly continuous on
$\mathbb{T}^{\infty}$. A systematic exposition of elements of hyperbolic theory is given for diffeomorphisms in
$\mathrm{Diff}(\mathbb{T}^{\infty})$. First, known results are presented (the hyperbolicity criterion, the theorem on
$C^1$-robustness of the hyperbolicity property, the Hadamard–Perron Theorem, and the existence of stable and unstable invariant foliations), and then new statements are established. The latter include a theorem on topological conjugacy (under certain additional conditions) of a hyperbolic diffeomorphism from
$\mathrm{Diff}(\mathbb{T}^{\infty})$ with a linear hyperbolic automorphism, as well as results related to this theorem on topological mixing and structural stability.
UDC:
517.926+
517.938
MSC: 37D20,
37C05,
37C15,
46T20,
37E30,
58B20 Received: 16.03.2023
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