Non-existence of absolutely continuous invariant probabilities for exponential maps

Neil Dobbs, Bartlomiej Skorulski

Published 2007-12-30, updated 2009-02-18Version 4

We show that for entire maps of the form $z \mapsto \lambda \exp(z)$ such that the orbit of zero is bounded and such that Lebesgue almost every point is transitive, no absolutely continuous invariant probability measure can exist. This answers a long-standing open problem.