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

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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 are the end points of the interval, and their iterates 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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