Perfect subspaces of quadratic laminations
| Autor: | Lex Oversteegen, Vladlen Timorin, Alexander Blokh |
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| Rok vydání: | 2018 |
| Předmět: |
Pure mathematics
Mathematics::Dynamical Systems 37F20 (Primary) 37F10 37F50 (Secondary) General Mathematics 010102 general mathematics Dynamical Systems (math.DS) Mandelbrot set Mathematics::Geometric Topology 01 natural sciences Linear subspace 010101 applied mathematics Hausdorff distance Unit circle Quadratic equation FOS: Mathematics Equivalence relation Mathematics - Dynamical Systems 0101 mathematics Invariant (mathematics) Quotient Mathematics |
| Zdroj: | Science China Mathematics. 61:2121-2138 |
| ISSN: | 1869-1862 1674-7283 |
| DOI: | 10.1007/s11425-017-9305-3 |
| Popis: | The combinatorial Mandelbrot set is a continuum in the plane, whose boundary can be defined, up to a homeomorphism, as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by Douady and, in different terms, by Thurston. Thurston used quadratic invariant laminations as a major tool. As has been previously shown by the authors, the combinatorial Mandelbrot set can be interpreted as a quotient of the space of all limit quadratic invariant laminations. The topology in the space of laminations is defined by the Hausdorff distance. In this paper, we describe two similar quotients. In the first case, the identifications are the same but the space is smaller than that taken for the Mandelbrot set. The result (the quotient space) is obtained from the Mandelbrot set by "unpinching" the transitions between adjacent hyperbolic components. In the second case, we do not identify non-renormalizable laminations while identifying renormalizable laminations according to which hyperbolic lamination they can be "unrenormalised" to. Comment: 29 pages, 4 figures |
| Databáze: | OpenAIRE |
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