Dynamical Systems IX

1. Front Matter

   Pages I-9

   PDF

2. Hyperbolic Sets

   • D. V. Anosov, V. V. Solodov

   Pages 10-92

3. Strange Attractors

   • R. V. Plykin, E. A. Sataev, S. V. Shlyachkov

   Pages 93-139

4. Cascades on Surfaces

   • S. Kh. Aranson, V. Z. Grines

   Pages 141-175

5. Dynamical Systems with Transitive Symmetry Group. Geometric and Statistical Properties

   • A. V. Safonov, A. N. Starkov, A. M. Stepin

   Pages 177-230

6. Back Matter

   Pages 231-238

   PDF

This volume is devoted to the "hyperbolic theory" of dynamical systems (DS), that is, the theory of smooth DS's with hyperbolic behaviour of the tra­ jectories (generally speaking, not the individual trajectories, but trajectories filling out more or less "significant" subsets in the phase space. Hyperbolicity the property that under a small displacement of any of a trajectory consists in point of it to one side of the trajectory, the change with time of the relative positions of the original and displaced points resulting from the action of the DS is reminiscent of the mot ion next to a saddle. If there are "sufficiently many" such trajectories and the phase space is compact, then although they "tend to diverge from one another" as it were, they "have nowhere to go" and their behaviour acquires a complicated intricate character. (In the physical literature one often talks about "chaos" in such situations. ) This type of be­ haviour would appear to be the opposite of the more customary and simple type of behaviour characterized by its own kind of stability and regularity of the motions (these words are for the moment not being used as a strict ter­ 1 minology but rather as descriptive informal terms). The ergodic properties of DS's with hyperbolic behaviour of trajectories (Bunimovich et al. 1985) have already been considered in Volume 2 of this series. In this volume we therefore consider mainly the properties of a topological character (see below 2 for further details).

• Ergodic flow
• Ergodischer Fluß
• Isotopieklasse
• Symmetry group
• diffeomorphism
• homogener Fluß
• homogenous flow
• hyperbolic set
• hyperbolische Menge
• isotopy class
• seltsame Attraktor
• strange attractor

• Steklov Mathematical Institute, Moscow GSP-1, Russia

  D. V. Anosov

Policies and ethics