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JOURNALS // Regular and Chaotic Dynamics // Archive

Regul. Chaotic Dyn., 2006 Volume 11, Issue 2, Pages 225–246 (Mi rcd670)

This article is cited in 4 papers

On the 70th birthday of L.P. Shilnikov

Expanding attractors

V. Z. Grinesa, E. V. Zhuzhomab

a Department of Mathematics, Agriculture Academy of Nizhny Novgorod, 97 Gagarin str., 603107 Nizhny Novgorod, Russia
b Department of Applied Mathematics, Nizhny Novgorod State Technical University, 24 Minin str., 603600 Nizhny Novgorod, Russia

Abstract: 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.

Keywords: Axiom A diffeomorphisms, (codimension one) expanding attractors, structurally stable diffeomorphisms, hyperbolic automorphisms.

MSC: 37D20, 37C70, 37C15

Received: 28.05.2005
Accepted: 15.11.2005

Language: English

DOI: 10.1070/RD2006v011n02ABEH000347



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