Date of Submission

Spring 2011

Academic Program

Mathematics

Advisor

Jim Belk

Abstract/Artist's Statement

A topological quadratic is a two-sheeted branched covering map on the complex plane with one branch point. Such a map is called postcritically nite if the orbit of the branch point under iteration is nite. Two such maps have the same dynamics if there exists a self-homeomorphism of the complex plane conjugating the rst to the second. The study of postcritically nite branched covers was initiated by Thurston, who characterized when such a map is homotopic to a conjugate of a polynomial map. The problem of which
polynomial this would be, however, was left unsolved.

There are exactly three quadratic polynomials for which the branch point has period 3: the rabbit, the corabbit, and the airplane. In 2006, V. Nekrashevych and L. Bartholdi solved the twisted rabbit problem, which asked \given a topological quadratic whose branch point is periodic with period three, to which quadratic polynomial is it Thurston equivalent?"

Using braids and the mapping class group of a complex plane with punctures, we provide a new solution to the twisted rabbit problem. In addition, we solve the \twisted three-eared
rabbit" problem, which is the analogous question for period-four quadratic polynomials.

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Dissertation/Thesis

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