Solenoidal attractors of diffeomorphisms of annular sets

An arbitrary diffeomorphism $\Pi$ of an annular set of the form $K=B\times \mathbb{T}$ is considered, where $B$ is a ball in a Banach space and $\mathbb{T}$ is a (finite- or infinite-dimensional) torus. A system of effective sufficient conditions is proposed which ensure that $P$ has a global attractor $A=\bigcap_{n\geqslant 0}\Pi^n(K)$ that can be represented as a generalized solenoid, that is, the inverse limit $\mathbb{T}\xleftarrow{G}\mathbb{T}\xleftarrow{G}\cdots\xleftarrow{G}\mathbb{T}\xleftarrow{G}\cdots$, where $G$ is an expanding linear endomorphism of the torus $\mathbb{T}$. Furthermore, the restriction $\Pi|_{A}$ is topologically conjugate to a shift map of the solenoid.Bibliography: 25 titles.

annular set, diffeomorphism, attractor, generalized solenoid, shift map, hyperbolicity