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.
