A study of the dynamics of a family of antipolynomials

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We call the connectedness locus of the family f_c (z) =z^^<-d>+c of antipolynomials the multicorn.We have shown that Julia sets depend continuously with respect to Hausdorff metric throughout the closure of hyperbolic components of odd periods, hence that immediate basins of attracting cycles converge to those of parabolic cycles. We have also shown that parabolic arcs do not intersect themselves, that closures of distinct parabolic arcs intersect only at cusp points, that the 0-Ecalle height point on the arc is a land point of an internal ray of angle 0 and its converse. Using these facts, we have shown that critical value maps are branched coverings of degree d+1 over the open unit disk.We have shown that the multicorn is not locally connected near the main hyperbolic component and that it is not locally pathwise connected near the principal parabolic arcs of maximally tuned hyperbolic components of odd periods not on the arcs of symmetry.We have shown that, on the boundary of hyperbolic components of odd periods, the holomorphic indices of parabolic cycles are real and diverge to +* as the parameter approaches a cusp point and antiholomorphic bifurcation occurs outside hyperbolic components if the index is greater than 1.We have calculated the Grotzsch defects of fixed points and 2-periodic points of polynomials P_c (z) =z^d+c and have shown their continuity.