Expansivity properties and rigidity for non-recurrent exponential maps

TL;DR

This paper establishes that certain exponential maps are uniquely determined by their combinatorial structure, proves hyperbolicity of the postsingular set under non-recurrence, and links bounded postsingular sets to combinatorial non-recurrence.

Contribution

It introduces a combinatorial framework for exponential maps, constructs puzzles to analyze their dynamics, and connects geometric properties with combinatorial non-recurrence.

Findings

Exponential maps with non-recurrent, non-escaping singular values are uniquely determined by combinatorics.

Hyperbolicity of the postsingular set is proven for non-recurrent singular values.

Bounded postsingular sets imply combinatorial non-recurrence when the singular value is in the Julia set.

Abstract

We show that an exponential map $f_c(z)=e^z+c$ whose singular value $c$ is combinatorially non-recurrent and non-escaping is uniquely determined by its combinatorics, i.e. the pattern in which its dynamic rays land together. We do this by constructing puzzles and parapuzzles in the exponential family. We also prove a theorem about hyperbolicity of the postsingular set in the case that the singular value is non-recurrent. Finally, we show that boundedness of the postsingular set implies combinatorial non-recurrence if $c$ is in the Julia set.