  Lie-Poisson structure of the Lorenz'63 system gives a physical insight on its
dynamical and statistical behavior considering the evolution of the associated
Casimir functions. We study the invariant density and other recurrence features
of a Markov expanding Lorenz-like map of the interval arising in the analysis
of the predictability of the extreme values reached by particular physical
observables evolving in time under the Lorenz'63 dynamics with the classical
set of parameters. Moreover, we prove the statistical stability of such an
invariant measure. This will allow us to further characterize the SRB measure
of the system.
