Inventiones mathematicae

, Volume 204, Issue 3, pp 869–893 | Cite as

Non-landing parameter rays of the multicorns

  • Hiroyuki Inou
  • Sabyasachi MukherjeeEmail author


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 \(\mathscr {M}_d^*\), 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 \(\mathscr {M}_d^*\), and the Misiurewicz parameters are not dense on the boundary of \(\mathscr {M}_d^*\).

Mathematics Subject Classification

37F45 37F10 37F20 30D05 



We would like to thank Adam Epstein for many helpful discussions. Special thanks go to Dierk Schleicher for his useful suggestions to improve the original manuscript, and for allowing us to reproduce one of the figures from [8]. The second author gratefully acknowledges the support of Deutsche Forschungsgemeinschaft DFG during this work.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsKyoto UniversityKyotoJapan
  2. 2.Jacobs University BremenBremenGermany
  3. 3.Institute for Mathematical SciencesStony Brook UniversityStony BrookUSA

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