Topological and Measurable Dynamics of Lorenz Maps

Gerhard Keller and Matthias St. Pierre ()
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Gerhard Keller: Universität Erlangen-Nürnberg, Mathematisches Institut
Matthias St. Pierre: Universität Erlangen-Nürnberg, Mathematisches Institut

A chapter in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, 2001, pp 333-361 from Springer

Abstract: 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.

Keywords: Periodic Orbit; Bifurcation Diagram; Global Attractor; Schwarzian Derivative; Periodic Attractor (search for similar items in EconPapers)
Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-56589-2_15

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DOI: 10.1007/978-3-642-56589-2_15

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