Perturbations of exponential maps: Non-recurrent dynamics

TL;DR

This paper investigates the measure and approximation properties of non-recurrent parameters in exponential maps, showing they form a measure-zero set and can be approximated by hyperbolic parameters, extending previous results.

Contribution

It proves that non-recurrent parameters in exponential maps have Lebesgue measure zero and can be approximated by hyperbolic parameters, advancing understanding of parameter space structure.

Findings

Non-recurrent parameters have Lebesgue measure zero.

The set of escaping parameters also has Lebesgue measure zero.

Non-recurrent parameters can be approximated by hyperbolic parameters.

Abstract

We study perturbations of non-recurrent parameters in the exponential family. It is shown that the set of such parameters has Lebesgue measure zero. This particularly implies that the set of escaping parameters has Lebesgue measure zero, which complements a result of Qiu from 1994. Moreover, we show that non-recurrent parameters can be approximated by hyperbolic ones.