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Beyond Hyperbolicity: Expansion Properties of One-Dimensional Mappings

In: From Topology to Computation: Proceedings of the Smalefest

Author

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  • John Guckenheimer
  • Stewart Johnson

Abstract

Smale’s definition of Axiom A [26] was a brilliant insight into the structure of dynamical systems that led to the geometric characterization of structurally stable diffeomorphisms. “Real-world” dynamical systems are often found with attractors that are structurally unstable, and these have been described as belonging to the “dark realm” [22]. The iterations of one-dimensional mappings provide one route to illuminating a small corner of the dark realm, and much has been discovered about these iterations during the past 15 years. The keys to understanding the dynamics of these transformations lies in 1. rigid topological structure described by kneading theory 2. distortion properties of iterates on intervals of monotonicity 3. properties of expansion that are determined by growth rates of the derivative along trajectories of critical points.

Suggested Citation

  • John Guckenheimer & Stewart Johnson, 1993. "Beyond Hyperbolicity: Expansion Properties of One-Dimensional Mappings," Springer Books, in: Morris W. Hirsch & Jerrold E. Marsden & Michael Shub (ed.), From Topology to Computation: Proceedings of the Smalefest, chapter 22, pages 227-236, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-2740-3_22
    DOI: 10.1007/978-1-4612-2740-3_22
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