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Perfect subspaces of quadratic laminations

Authors
  • Blokh, Alexander1
  • Oversteegen, Lex1
  • Timorin, Vladlen2
  • 1 University of Alabama at Birmingham, Department of Mathematics, Birmingham, AL, 35294, USA , Birmingham (United States)
  • 2 National Research University Higher School of Economics, Faculty of Mathematics, Moscow, 119048, Russia , Moscow (Russia)
Type
Published Article
Journal
Science China Mathematics
Publisher
Science China Press
Publication Date
Nov 22, 2018
Volume
61
Issue
12
Pages
2121–2138
Identifiers
DOI: 10.1007/s11425-017-9305-3
Source
Springer Nature
Keywords
License
Yellow

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.

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