Atlas: Makienko's Conjecture and indecomposable continua by Clinton Curry

Atlas home || Conferences | Abstracts | about Atlas

Spring Topology and Dynamical Systems Conference 2008
March 13-15, 2008
University of Wisconsin Milwaukee and Marquette University
Milwaukee, WI, USA

Organizers
Ric Ancel, Karen Brucks, Craig Guilbault, Chris Hruska, Suzanne Hruska, Boris Okun (UWM); Paul Bankston (Marquette); Lois Kailhofer (Alverno College).

View Abstracts
Conference Homepage

Makienko's Conjecture and indecomposable continua
by
Clinton Curry
University of Alabama at Birmingham
Coauthors: John C. Mayer, Jonathan Meddaugh, Jim Rogers

The residual Julia set of a rational function is defined as its Julia set minus the boundaries of its Fatou components. It is a well-known fact that, when a component of the Fatou set is fully invariant under some power of the map, the residual Julia set is empty. Based on Sullivan's dictionary, Peter M.~Makienko conjectured that the converse is true: when the residual Julia set of a rational map is empty, there is a Fatou component which is fully invariant under a power of the map. Until now, this has been confirmed only for Julia sets which are not connected and for Julia sets which are locally connected. We prove that any counterexample to Makienko's conjecture is an indecomposable continuum. It is not known if the Julia set of a rational function can be an indecomposable continuum.

Date received: February 27, 2008