Non-landing parameter rays of the multicorns

Abstract

It is well known that every rational parameter ray of the Mandelbrot set lands at a single parameter. We study the rational parameter rays of the multicorn ${\mathcal{M}}_d^{\ast }$, the connectedness locus of unicritical antiholomorphic polynomials of degree $d$, and give a complete description of their accumulation properties. One of the principal results is that the parameter rays accumulating on the boundaries of odd period (except period $1$) hyperbolic components of the multicorns do not land, but accumulate on arcs of positive length consisting of parabolic parameters. We also show the existence of undecorated real-analytic arcs on the boundaries of the multicorns, which implies that the centers of hyperbolic components do not accumulate on the entire boundary of ${\mathcal{M}}_d^{\ast }$, and the Misiurewicz parameters are not dense on the boundary of ${\mathcal{M}}_d^{\ast }$.

arXiv admin note: text overlap with arXiv:1209.1753 by other authors

References in corpus (1)

• Orbit Portraits of Unicritical Anti-polynomials

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