Adam L. Epstein

Bounded hyperbolic components of quadratic rational maps

Let

Quasiconformal variation of slit domains

\mathcal{H}

Geography of the cubic connectedness locus: Intertwining surgery

be a hyperbolic component of quadratic rational maps possessing two distinct attracting cycles. We show that

Integrality and rigidity for postcritically finite polynomials

\mathcal{H}

From local to global analytic conjugacies

has compact closure in moduli space if and only if neither attractor is a fixed point.

Remarks on the period three cycles of quadratic rational maps

We use quasiconformal variations to study Riemann mappings onto variable single slit domains when the slit is the tail of an appropriately smooth Jordan arc. In the real analytic case our results answer a question of Dieter Gaier and show that the function κ in Lowners differential equation is real analytic.

ON INVARIANCE OF ORDER AND THE AREA PROPERTY FOR FINITE-TYPE ENTIRE FUNCTIONS

Abstract We exhibit products of Mandelbrot sets in the two-dimensional complex parameter space of cubic polynomials. Cubic polynomials in such a product may be renormalized to produce a pair of quadratic maps. The inverse construction intertwining two quadratics is realized by means of quasiconformal surgery. The associated asymptotic geography of the cubic connectedness locus is discussed in the Appendix.

Infinitesimal Thurston Rigidity and the Fatou-Shishikura Inequality

We give an arithmetic proof of rigidity for postcritically finite polynomials of prime power degree.

A parabolic Pommerenke-Levin-Yoccoz inequality

Let

On Thurston's pullback map

f_1