Resumen de Transcendental entire maps with two singular values

Asli Deniz

• This dissertation concerns holomorphic dynamics. It can be classified into three topics, namely, studying a specific family, landing properties of rays and introducing a new notion termed holomorphic explosion.

  In the first part, we investigate the characteristic dynamical properties of a specific one parameter family of transcendental entire maps with two singular values. We mainly study the interplay between the dynamical plane and the parameter plane. This family has a rich parameter space, and we illustrate the dissertation's more general results with this model family.

  The second part concerns the ray structure/landing properties of rays in both the dynamical plane and the parameter plane. Under some mild hypothesis, we prove that in hyperbolic components, all rational rays land. We illustrate the proof by our model family, by considering its main hyperbolic component. Furthermore, we give a proof of a landing theorem for periodic rays for transcendental entire maps which have bounded postsingular set. The main tool in the proof is hyperbolic geometry.

  As a last piece of work regarding rays, under very general conditions, for a given parabolic parameter, we give a transparent proof for landing of (a) parameter ray(s) at the parabolic parameter in question. The proof is based on parabolic implosion.

  In the third part, we introduce a new concept, called holomorphic explosion, which is produced by continuous extension of a holomorphic motion to an isolated boundary point of its parameter set. We study the fundamental properties and apply the concept to two concrete examples.¿¿¿¿