The Nielsen-Thurston classification of the mapping classes proved that every

orientation preserving homeomorphism of a closed surface, up to isotopy, is either

periodic, reducible, or pseudo-Anosov. Pseudo-Anosov maps have particularly nice

structure because they expand along one foliation by a factor of λ > 1 and contract

along a transversal foliation by a factor of λ. The number λ is called the dilatation

of the pseudo-Anosov. Thurston showed that every dilatation λ of a pseudo-Anosov

map is an algebraic unit, and conjectured that every algebraic unit λ whose Galois

conjugates lie in the annulus A = {z : 1 λ < |z| < λ} is a dilatation of some pseudo-Anosov on some surface S.

Pseudo-Anosovs have a huge role in Teichmuller theory and geometric topology.

The relation between these and complex dynamics has been well studied, inspired

by Thurston.

In this project, I develop a new connection between the dynamics of quadratic

polynomials on the complex plane and the dynamics of homeomorphisms of surfaces.

In particular, given a quadratic polynomial, we show that one can construct

an extension of it which is generalized pseudo-Anosov homeomorphism. Generalized

pseudo-Anosov means the foliations have infinite singularities that accumulate

on finitely many points. We determine for which quadratic polynomials such an extension

exists. My construction is related to the dynamics on the Hubbard tree

which is a forward invariant subset of the filled Julia set that contains the critical orbit.