David Brown

Ph.D. (2001) Cornell University

First Position

Assistant Professor at Ithaca College

Dissertation

Using Spider Theory to Explore Parameter Spaces

Research Area

Dynamical Systems, Teichmüller Theory

Abstract

For a fixed integer $d\ge2$, consider the family of polynomials $P_{d,\lambda}(z)=\lambda(1+z/d)^d$, where $\lambda$ is a complex parameter. In this work, we study the location of parameters $\lambda$ for which $P_{d,\lambda}$ has an attracting cycle of a given length, multiplier, and combinatorial type.

Two main tools are used in determining an algorithm for finding these parameters: the well-established theories of external rays in the dynamical and parameter planes and Teichmüller theory. External rays are used to specify hyperbolic components in parameter space of the polynomials and study the combinatorics of the attracting cycle. A properly normalized space of univalent mappings is then employed to determine a linearizing neighborhood of the attracting cycle.

Since the image of a univalent mapping completely determines the mapping, we visualize these maps concretely on the Riemann sphere; with discs for feet and curves as legs connected at infinity, these maps conjure a picture of fat-footed spiders. Isotopy classes of these spiders form a Teichmüller space, and the tools found in Teichmüller theory prove useful in understanding the Spider Space. By defining a contracting holomorphic mapping on this spider space, we can iterate this mapping to a fixed point in Teichmüller space which in turn determines the parameter we seek.

Finally, we extend the results about these polynomial families to the exponential family $E_\lambda(z)=\lambda e^z$. Here, we are able to constructively prove the existence and location of hyperbolic components in the parameter space of $E_\lambda$.