A note on hyperbolic leaves and wild laminations of rational functions

TL;DR

This paper investigates the local compactness of affine orbifold laminations associated with rational functions, providing a counterexample that reveals complex interactions between hyperbolic leaves and Julia sets.

Contribution

It demonstrates that affine orbifold laminations are not always locally compact by constructing a counterexample involving hyperbolic leaves intersecting the Julia set.

Findings

Counterexample shows non-local compactness of certain laminations

Hyperbolic leaves can intersect Julia sets in these laminations

Raises questions about the structure of leaves in rational dynamics

Abstract

We study the affine orbifold laminations that were constructed by Lyubich and Minsky. An important question left open in their construction is whether these laminations are always locally compact. We show that this is not the case. The counterexample we construct has the property that the regular leaf space contains (many) hyperbolic leaves that intersect the Julia set; whether this can happen is itself a question raised by Lyubich and Minsky.