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

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) > 2^½ 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) > 2^1/(k+1) everywhere, then f is l.e.o. Secondly, we show that the one-dimensional Poincaré map of any C^r 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 C^1+α and satisfies a theorem of Keller, Wong, Lasota, Li & Yorke.