A note on hyperbolic leaves and wild laminations of rational functions

Abstract

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.

Keywords:

• laminations
• regular leaf space
• Julia sets
• orbifold

Acknowledgements

We thank Carlos Cabrera and Juan Rivera-Letelier for useful discussions. The second author has been partially supported by the NSF and NSERC. The third author has been partially supported by a postdoctoral fellowship of the German Academic Exchange Service (DAAD), and later by the EPSRC fellowship EP/E052851/1. He would also like to thank the Institute for Mathematical Sciences at Stony Brook and the Simons endowment for their continued support and hospitality.

Notes

1. [email protected]

2. [email protected]

3. We should note that there is a different construction of laminations for rational maps, due to Su [13]. These laminations are never locally compact.

4. In [6], Proposition 7.6, it is stated (incorrectly) that such isolated leaves can only occur for Lattès and Chebyshev polynomials. Proposition 4.5 provides a corrected version of this assertion.

Additional information

Notes on contributors

Mikhail Lyubich

1

Lasse Rempe

2