On the accumulation sets of exponential rays

We show that there exist non-landing exponential rays with bounded accumulation sets. By introducing folding models of certain rays, we prove that each of the corresponding accumulation sets is an indecomposable continuum containing part of the ray, an indecomposable continuum disjoint from the ray or a Jordan arc.

Any counterexample to Makienko’s conjecture is an indecomposable continuum

Makienko’s conjecture, a proposed addition to Sullivan’s dictionary, can be stated as follows: the Julia set of a rational function R:ℂ∞→ℂ∞ has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko’s conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational function whose Julia set is an indecomposable continuum.

INDECOMPOSABLE CONTINUA AND MISIUREWICZ POINTS IN EXPONENTIAL DYNAMICS

In this paper we describe several new types of invariant sets that appear in the Julia sets of the complex exponential functions Eλ(z) = λez where λ ∈ ℂ in the special case when λ is a Misiurewicz parameter, so that the Julia set of these maps is the entire complex plane. These invariant sets consist of points that share the same itinerary under iteration of Eλ. Previously, the only known types of such invariant sets were either simple hairs that extend from a definite endpoint to ∞ in the right half plane or else indecomposable continua for which a single hair accumulates everywhere upon itself. One new type of invariant set that we construct in this paper is an indecomposable continuum in which a pair of hairs accumulate upon each other, rather than a single hair having this property. The second type consists of an indecomposable continuum together with a completely separate hair that accumulates on this continuum.

Singularities in the boundaries of local Siegel disks

Bounded irreducible local Siegel disks include classical Siegel disks of polynomials, bounded irreducible Siegel disks of rational and entire functions, and the examples of Herman and Moeckel. We show there are only two possibilities for the structure of the boundary of such a disk: either the boundary admits a nice decomposition onto a circle or it is an indecomposable continuum.

Rotation and periodicity in plane separating continua

We prove that ifFis an orientation-preserving homeomorphism of the plane that leaves invariant a continuum Λ which irreducibly separates the plane into exactly two domains, then the convex hull of the rotation set ofFrestricted to Λ is a closed interval and each reduced rational in this interval is the rotation number of a periodic orbit in Λ. We also show that the interior and exterior rotation numbers ofFassociated with Λ are contained in the convex hull of the rotation set ofFrestricted to Λ and that if this set is nondegenerate then Λ is an indecomposable continuum.