No Finite Invariant Density for Misiurewicz Exponential Maps

TL;DR

This paper proves that for certain exponential maps with bounded singular value orbits, there is no integrable invariant density on the complex plane, highlighting a fundamental limitation in their statistical behavior.

Contribution

It establishes the non-existence of finite invariant densities for Misiurewicz exponential maps with bounded singular orbits.

Findings

No integrable invariant density exists for these maps.

The result applies to exponential maps with bounded singular value orbits.

This contributes to understanding the ergodic properties of complex exponential dynamics.

Abstract

For exponential mappings such that the orbit of the only singular value 0 is bounded, it is shown that no integrable density invariant under the dynamics exists on the complex plane.