Repelling periodic points and landing of rays for post-singularly bounded exponential maps

TL;DR

This paper proves that for certain exponential maps with bounded singular orbits, repelling periodic points and hyperbolic set points are accessible via rays, extending classical polynomial results to exponential dynamics.

Contribution

It establishes the landing of periodic rays at repelling points for exponential maps with bounded singular orbits, providing a new proof for polynomial cases and extending accessibility results.

Findings

Repelling periodic points are landing points of rays in exponential maps.

Points in hyperbolic sets are accessible by finitely many rays.

The singular value itself is accessible in these exponential maps.

Abstract

We show that repelling periodic points are landing points of periodic rays for exponential maps whose singular value has bounded orbit. For polynomials with connected Julia sets, this is a celebrated theorem by Douady, for which we present a new proof. In both cases we also show that points in hyperbolic sets are accessible by at least one and at most finitely many rays. For exponentials this allows us to conclude that the singular value itself is accessible.