Non-existence of absolutely continuous invariant probabilities for exponential maps

Abstract

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.

4 pages. Similar to the version published in Fundamenta in February 2008

References in corpus (1)

• Ergodic properties of sub-hyperbolic functions with polynomial Schwarzian derivative