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Laminational Models for Some Spaces of Polynomials of Any Degree

About this Title

Alexander Blokh, Department of Mathematics University of Alabama at Birmingham, Birmingham, AL 35294, Lex Oversteegen, Department of Mathematics University of Alabama at Birmingham, Birmingham, AL 35294, Ross Ptacek, Faculty of Mathematics, Laboratory of Algebraic Geometry and its Applications, National Research University Higher School of Economics, 6 Usacheva str., Moscow, Russia, 119048 and Vladlen Timorin, Faculty of Mathematics, Laboratory of Algebraic Geometry and its Applications, National Research University Higher School of Economics, 6 Usacheva str., Moscow, Russia, 119048

Publication: Memoirs of the American Mathematical Society
Publication Year: 2020; Volume 265, Number 1288
ISBNs: 978-1-4704-4176-0 (print); 978-1-4704-6144-7 (online)
DOI: https://doi.org/10.1090/memo/1288
Published electronically: March 31, 2020
Keywords: Complex dynamics, laminations, Mandelbrot set, Julia set
MSC: Primary 37F20; Secondary 37F10, 37F50

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Invariant laminations: general properties
  • 3. Special types of invariant laminations
  • 4. Applications: Spaces of topological polynomials

Abstract

The so-called “pinched disk” model of the Mandelbrot set is due to A. Douady, J. H. Hubbard and W. P. Thurston. It can be described in the language of geodesic laminations. The combinatorial model is the quotient space of the unit disk under an equivalence relation that, loosely speaking, “pinches” the disk in the plane (whence the name of the model). The significance of the model lies in particular in the fact that this quotient is planar and therefore can be easily visualized. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating that the Mandelbrot set is locally connected.

For parameter spaces of higher degree polynomials no combinatorial model is known. One possible reason may be that the higher degree analog of the MLC conjecture is known to be false. We investigate to which extent a geodesic lamination is determined by the location of its critical sets and when different choices of critical sets lead to essentially the same lamination. This yields models of various parameter spaces of laminations similar to the “pinched disk” model of the Mandelbrot set.

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