Rich Stankewitz

Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's

We show that the Julia set of a non-elementary rational semigroup G is uniformly perfect when there is a uniform bound on the Lipschitz constants of the generators of G. This also proves that the limit set of a non-elementary Mobius group is uniformly perfect when there is a uniform bound on the Lipschitz constants of the generators of the group and this implies that the limit set of a finitely generated non-elementary Kleinian group is uniformly perfect.

Completely invariant Julia sets of polynomial semigroups

Let G be a semigroup of rational functions of degree at least two, under composition of functions. Suppose that G contains two polynomials with non-equal Julia sets. We prove that the smallest closed subset of the Riemann sphere which contains at least three points and is completely invariant under each element of G, is the sphere itself.

Completely invariant sets of normality for rational semigroups

Let G be a semigroup of rational functions of degree at least two where the semigroup operation is composition of functions. We prove that the largest open subset of the Riemann sphere on which the semigroup G is normal and is completely invariant under each element of G, can have only 0,1,2, or infinitely many components.

Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups

We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynam- ics, where the semigroup G of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any flnite critical value of any map g 2 G. In general, the Julia set of such a semigroup G may be disconnected, and each Fatou component of such G is either simply connected or doubly connected. In this paper, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of G: Important in this theory is the understanding of various situations which can and cannot occur with respect to how the Julia sets of the maps g 2 G are distributed within the Julia set of the entire semigroup G. We give several results in this direction and show how such results are used to generate (semi) hyperbolic semigroups possessing this postcritically boundedness condition.

Density of repelling fixed points in the Julia set of a rational or entire semigroup

We briefly survey several methods of proof that the Julia set of a rational or entire function is the closure of the repelling cycles, in particular, focusing on those methods which can be extended to the case of semigroups. We then present an elementary proof that the Julia set of either a non-elementary rational or entire semigroup is the closure of the set of repelling fixed points.

Uniformly Perfect Analytic and Conformal Attractor Sets

Conditions are given which imply that analytic iterated function systems (IFSs) in the complex plane have uniformly perfect attractor sets. In particular, it is shown that the attractor set of a finitely generated conformal IFS is uniformly perfect when it contains two or more points. Also, an example of a finitely generated analytic attractor set which is not uniformly perfect is given.

Complex dynamics of Möbius semigroups

We study the dynamics of semigroups of Mobius transformations on the Riemann sphere, especially their Julia sets and attractors. This theory relates to the dynamics of rational functions, rational semigroups, and Mobius groups and we compare and contrast these theories. We particularly examine Caruso’s family of Mobius semigroups, based on a random dynamics variant of the Fibonacci sequence.

Uniformly Perfect Sets, Rational Semigroups, Kleinian Groups and Iterated Functions Systems

These are lecture notes based on the paper of the same title which is to appear in the Proceedings of the AMS. See this paper for more details and for a more extensive bibliography.