  We study the one-dimensional expanding Lorenz maps and show the existence of
dense subset D of Lorens maps such that each f in D has an uncountable set of
ergodic invariant probabilities with infinite Lyapunov exponent and positive
entropy. Such measures may appear when the singularity has fast recurrence to
itself. Conversely, if the singularity has slow recurrence to itself then the
Lorenz map has an upper bound to the Lyapunov exponent of all invariant
measures.
