Abstract
The combinatorial Mandelbrot set is a continuum in the plane, whose boundary is defined as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by Douady (1984) and, separately, by Thurston (1985) who used quadratic invariant geolaminations as a major tool. We showed earlier that the combinatorial Mandelbrot set can be interpreted as a quotient of the space of all limit quadratic invariant geolaminations with the Hausdorff distance topology. In this paper, we describe two similar quotients. In the first case, the identifications are the same but the space is smaller than that used for the Mandelbrot set. The resulting quotient space is obtained from the Mandelbrot set by ıpinching" the transitions between adjacent hyperbolic components. In the second case we identify renormalizable geolaminations that can be ırenormalized" to the same hyperbolic geolamination while no two non-renormalizable geolaminations are identified.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Blokh A, Levin G. Growing trees, laminations and the dynamics on the Julia set. Ergodic Theory Dynam Systems, 2002, 22: 63–97
Blokh A, Mimbs D, Oversteegen L, et al. Laminations in the language of leaves. Trans Amer Math Soc, 2013, 365: 5367–5391
Blokh A, Oversteegen L, Ptacek R, et al. The combinatorial Mandelbrot set as the quotient of the space of geolami-nations. Contemp Math, 2016, 669: 37–62
Blokh A, Oversteegen L, Ptacek R, et al. Laminations from the main cubioid. Discrete Contin Dyn Syst, 2016, 36: 4665–4702
Blokh A, Oversteegen L, Ptacek R, et al. Laminational models for some spaces of polynomials of arbitrary degree. Mem Amer Math Soc, 2017, in press
Douady A, Hubbard J H. Étude dynamique des polynômes complexes. Orsay: Publications Mathématiques d'Orsay, 1984
Kiwi J. Wandering orbit portraits. Trans Amer Math Soc, 2002, 354: 1473–1485
Kiwi J. Real laminations and the topological dynamics of complex polynomials. Adv Math, 2004, 184: 207–267
Levin G. On backward stability of holomorphic dynamical systems. Fund Math, 1998, 158: 97–107
Milnor J. Dynamics in One Complex Variable. Annals of Mathematical Studies, vol. 160. Princeton: Princeton University Press, 2006
Schleicher D. Appendix: Laminations, Julia sets, and the Mandelbrot set. In: Complex Dynamics Families and Friends. Wellesley: A K Peters, 2009, 111–130
Thurston W. The combinatorics of iterated rational maps. In: Complex Dynamics Families and Friends. Wellesley: A K Peters, 2008, 1–108
Acknowledgements
The first named author was partially supported by National Science Foundation of USA (Grant No. DMS-1201450). The second named author was partially supported by National Science Foundation of USA (Grant No. DMS-1807558). The third named author was partially supported by the Russian Academic Excellence Project ‘5-100’. The authors thank the referees for valuable and helpful remarks.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the Memory of Lei Tan
Rights and permissions
About this article
Cite this article
Blokh, A., Oversteegen, L. & Timorin, V. Perfect subspaces of quadratic laminations. Sci. China Math. 61, 2121–2138 (2018). https://doi.org/10.1007/s11425-017-9305-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-017-9305-3