zbMATH — the first resource for mathematics

Expanding attractors. (English) Zbl 1164.37317
Summary: The article is a survey on local and global structures (including classification results) of expanding attractors of diffeomorphisms \(f: M\to M\) of a closed smooth manifold \(M\). Beginning with the most familiar expanding attractors (Smale solenoid; DA-attractor; Plykin attractor; Robinson–Williams attractors), one reviews the Williams theory, Bothe’s classification of one-dimensional solenoids in 3-manifolds, Grines–Plykin–Zhirov’s classification of one-dimensional expanding attractors on surfaces, and Grines–Zhuzhoma’s classification of codimension one expanding attractors of structurally stable diffeomorphisms. The main theorems are endowed with ideas of proof.

37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
Full Text: DOI