Mating Non-Renormalizable Quadratic Polynomials

Abstract

In this paper we prove the existence and uniqueness of matings of the basilica with any quadratic polynomial which lies outside of the 1/2-limb of M, is non-renormalizable, and does not have any non-repelling periodic orbits. Basic properties for R a and f • •

One hundred years of complex dynamics

“…In the domain of complex dynamics, early extensions of Yoccoz' work were achieved by Roesch to the family of Newton cubics, and she also extended the use to other specific families [139][140][141][142][143][144][145]. Aspenberg & Yampolsky [146] (see also [147]) used an analogue of the Yoccoz puzzle to study the family of quadratic polynomials with a critical point of period two-where all hyperbolic maps are known to be either matings or Wittner captures. The basic idea appeared, quite non-rigorously, in the 1995 Cornell thesis of Jiaqi Luo.…”

Counting hyperbolic components

“…We can then obtain the result for B n by the usual birational equivalence. In fact it suffices to compute the degree in ζ of H h n,1 − H h n,2 where H n,1 H n,2 = h n ζ,ρ (1) and H h n,j is the homogenised version of H n,j . The zero set of H h n,1 − H h n,2 is the union of sets A d for d dividing n. We claim that the degree is 2 n − 1.…”

“…Instead of using R as in the case of V n for n ≥ 3, we use a parametrisation which extends that used in [1] and [17]:…”

Sierpinski curve Julia sets for quadratic rational maps

“…The second slice, Per 2 (0), consists of all quadratic rational maps having a period two critical orbit. Such quadratic rational maps has been investigated by M. Aspenberg and M. Yampolsky [1], where the authors consider the mating between the Basilica with other suitable quadratic polynomials. Among other results they proved that the two Fatou components containing the period two critical orbit have non-empty intersection.…”