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Science China Mathematics

, Volume 61, Issue 12, pp 2121–2138 | Cite as

Perfect subspaces of quadratic laminations

  • Alexander Blokh
  • Lex Oversteegen
  • Vladlen TimorinEmail author
Articles
  • 18 Downloads

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.

Keywords

complex dynamics laminations Mandelbrot set Julia set 

MSC(2010)

37F20 37F10 37F50 

Notes

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.

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Copyright information

© Science in China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Alexander Blokh
    • 1
  • Lex Oversteegen
    • 1
  • Vladlen Timorin
    • 2
    Email author
  1. 1.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA
  2. 2.Faculty of MathematicsNational Research University Higher School of EconomicsMoscowRussia

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