| For a fixed integer d ≥ 2, consider the family of polynomials Pd,lambda( z) = lambda(1 + zd )d, where lambda is a complex parameter. In this work, we study the location of parameters lambda for which Pd,lambda has an attracting cycle of a given length, multiplier of the cycle, 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 Teichmuller 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 Teichmuller space and the tools found in Teichmuller theory prove useful in understanding this Spider Space. By defining a contracting holomorphic mapping on this spider space, we can iterate this mapping to a fixed point in Teichmuller space which in turn determines the parameter we seek.;Finally, we extend the results about these polynomial families to the exponential family Elambda(z) = lambdaez. Here, we are able to constructively prove the existence and location of hyperbolic components in the parameter space of Elambda. |
