Measurable dynamics of escaping exponential maps

Dynamics of exponential maps has been studied extensively since 1980s. There are, however, still many aspects for which we do not have a clear understanding. Pioneering works of Lyubich and Rees tell that the map $e^z$ is not recurrent and also not ergodic with respect to the Lebesgue measure. As one of the typical examples of exponential maps with escaping singular behaviours (escaping exponential maps, for abbreviation), their results suggest the question whether this kind of measurable behaviour could be true in general for escaping exponential maps. 

In recent works, joint with Jun Wang and also Jiaxing Huang we construct various examples of escaping exponential with different measurable properties. This shows that exponential maps may have complicated dynamical behaviours from a measurable point of view. These examples were constructed using perturbations in the parameter space, a technique used jointly with Magnus Aspenberg to study the Lebesgue measure of the set of escaping exponential maps, and hyperbolic approximations of these maps.