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Topological and Measurable Dynamics of Lorenz Maps

In: Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems

Author

Listed:
  • Gerhard Keller

    (Universität Erlangen-Nürnberg, Mathematisches Institut)

  • Matthias St. Pierre

    (Universität Erlangen-Nürnberg, Mathematisches Institut)

Abstract

We investigate the dynamics of Lorenz maps, in particular the asymptotical behaviour of the trajectory of typical points. For Lorenz maps f with negative Schwarzian derivative we give a classification of the possible metric attractors and show that either f has an ergodic absolutely continuous invariant probability measure of positive entropy or the iterates of typical points spend most of their time shadowing the trajectory of one of the two critical values. Our main tool therefore is the construction of Markov extensions for Lorenz maps which provide a unified framework to approach both the topological and the measurable aspects of the dynamics. We study the bifurcation diagram of a smooth two parameter family of Lorenz maps which describes the parameter dependence of the kneading invariant and show that essentially every admissible kneading invariant actually occurs if the family is sufficiently rich. Finally, we adress the problem whether the kneading invariant depends monotonously on the parameters.

Suggested Citation

  • Gerhard Keller & Matthias St. Pierre, 2001. "Topological and Measurable Dynamics of Lorenz Maps," Springer Books, in: Bernold Fiedler (ed.), Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pages 333-361, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-56589-2_15
    DOI: 10.1007/978-3-642-56589-2_15
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