Non-existence of absolutely continuous invariant probabilities for exponential maps

TL;DR

This paper proves that certain exponential maps with bounded orbits and transitive points do not admit absolutely continuous invariant probability measures, resolving a long-standing open problem in dynamical systems.

Contribution

It establishes the non-existence of absolutely continuous invariant measures for a class of exponential maps with specific orbit properties, answering a major open question.

Findings

No absolutely continuous invariant probability measure exists for the considered maps.

The result applies to entire maps of the form $z o \lambda ext{exp}(z)$ with bounded zero orbits.

It confirms that Lebesgue almost every point is transitive under these 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.