A polynomial skewproduct with a wandering Fatoudisk
Abstract
Little is known about the existence of wandering Fatou components for rational maps in two complex variables. In 2003 Lilov proved the nonexistence of wandering Fatou components for polynomial skewproducts in the neighborhood of an invariant superattracting fiber. In fact Lilov proved a stronger result, namely that the forward orbit of any vertical disk must intersect a fattened Fatou component of the invariant fiber. Naturally the next class of maps to study are polynomial skewproducts with an invariant attracting (but not superattracting) fiber. Here we show that Lilov's stronger result does not hold in this setting: for some skewproducts there are vertical disks whose orbits accumulate at repelling fixed points in the invariant fiber, and that therefore never intersect the fattened Fatou components. These disks are necessarily Fatou disks, but we also prove that the vertical disks we construct lie entirely in the Julia set. Our results therefore do not answer the existence question of wandering Fatou components in the attracting setting, but show that the question is considerably more complicated than in the superattracting setting.
 Publication:

arXiv eprints
 Pub Date:
 May 2014
 DOI:
 10.48550/arXiv.1405.1340
 arXiv:
 arXiv:1405.1340
 Bibcode:
 2014arXiv1405.1340P
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Complex Variables;
 32H50;
 37F10
 EPrint:
 16 pages, 2 figures