TY - JOUR

T1 - Transitivity and invariant measures for the geometric model of the Lorenz equations

AU - Robinson, Clark

N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 1984/12

Y1 - 1984/12

N2 - 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.

AB - 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.

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U2 - 10.1017/S0143385700002674

DO - 10.1017/S0143385700002674

M3 - Article

AN - SCOPUS:0346513128

VL - 4

SP - 605

EP - 611

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 4

ER -