Transitivity and invariant measures for the geometric model of the Lorenz equations

Clark Robinson*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

This paper concerns perturbations of the geometric model of the Lorenz equations and their associated one-dimensional Poincaré maps. R. Williams has shown that if a map of the interval of the type arising from the Lorenz equations satisfies f′(x) > 2½ everywhere except at the discontinuity then f is locally eventually onto (l.e.o.) and so topologically transitive [14]. We show roughly that if [formula-omitted] are the end points of the interval, and their iterates [formula-omitted] stay on the same side of the point of discontinuity for 0 ≤ j ≤ k, and f′(x) > 21/(k+1) everywhere, then f is l.e.o. Secondly, we show that the one-dimensional Poincaré map of any Cr perturbation of the geometric model (for large enough r) has an ergodic measure which is equivalent to Lebesgue measure. This result follows by showing it is C1+α and satisfies a theorem of Keller, Wong, Lasota, Li & Yorke.

Original languageEnglish (US)
Pages (from-to)605-611
Number of pages7
JournalErgodic Theory and Dynamical Systems
Volume4
Issue number4
DOIs
StatePublished - Dec 1984

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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