Sabyasachi Mukherjee

• The goal of this thesis is to study the dynamics of unicritical antiholomorphic polynomials, and to explore the combinatorial and topological properties of the multicorns, which are the connectedness loci of the maps under consideration. We, on one hand, prove many topological differences between the multicorns and their holomorphic counterparts, the multibrot sets, and on the other hand, study the self-similarity property of the multicorns. We show that the parameter rays accumulating on the boundaries of odd period (except period one) hyperbolic components of the multicorns do not land; they non-trivially accumulate on intervals of parabolic parameters. We also study the behavior of the ‘straightening maps’ from the ‘baby multicorns’ to the original multicorns. More precisely, we prove discontinuity of the straightening map (for even degree multicorns) at infinitely many explicit parameters. Both of these results are in stark contrast with the corresponding situation for the multibrot sets, where all rational parameter rays are known to land, and all baby multibrot sets are known to be homeomorphic to the original ones.