A note on hyperbolic leaves and wild laminations of rational functions

We study the affine orbifold laminations that were constructed by Lyubich and Minsky (J. Differential Geom. 47(1) (1997), pp. 17-94). An important question left open in the original 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 was itself a question raised by Lyubich and Minsky.