Ergodic exponential maps with escaping singular behaviours

TL;DR

This paper constructs specific ergodic exponential maps where the singular value escapes to infinity, contrasting with previous results showing the non-ergodicity of the standard exponential function.

Contribution

It introduces new ergodic exponential maps with escaping singular values, expanding understanding of dynamics in complex exponential functions.

Findings

Constructed ergodic exponential maps with escaping singular values.

Contrasts with Lyubich's 1987 result that $e^z$ is not ergodic.

Shows existence of ergodic maps with divergent singular orbits.

Abstract

We construct exponential maps for which the singular value tends to infinity under iterates while the maps are ergodic. This is in contrast with a result of Lyubich from 1987 which tells that $e^z$ is not ergodic.