Critical and Singular Dynamics in the Lorenz Equations

Dynamical systems "trvith complicated orbit structures are best described by suitable invariant measures. Sinai, Ruelle and Bowen showed, in the 70's: that a special class of invariant measures (now called SBR measures) which provide substantial information on the dynamical and statistical properties, can be constructed for uniformly hyperbolic systems. The question arises as to what extent \veaker hyperbolicity conditions still guarantee the existence of SBR measures. vVe introduce a class of flows in R^3 , inspired by a system of differential equation proposed by Lorenz, in which the presence of a singularity and of criticalities constitute obstructions to uniform hyperbolicity. \Ve prove that a weaker form of hyperbolicity exists and is present in a (measure-theoretically) persistent way in one-parameter families. It is expected that such non-uniform hyperbolicity implies the existence of an SBR measure.