Non-existence of absolutely continuous invariant probabilities for exponential maps
Volume 198 / 2008
Fundamenta Mathematicae 198 (2008), 283-287
MSC: Primary 37F10.
DOI: 10.4064/fm198-3-6
Abstract
We show that for entire maps of the form $z \mapsto \lambda \exp(z)$ such that the orbit of zero is bounded and Lebesgue almost every point is transitive, no absolutely continuous invariant probability measure can exist. This answers a long-standing open problem.
