Laminational Models for Some Spaces of Polynomials of Any Degree

Vladlen Timorin, Lex Oversteegen, Ross Ptacek, Alexander Blokh

ISBN 10: 1470441764 / ISBN 13: 9781470441760

Published by American Mathematical Society, US, 2021

About this Item

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. The authors 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.

Seller Inventory # LU-9781470441760

Bibliographic Details

• Title: Laminational Models for Some Spaces of Polynomials of Any Degree
• Publisher: American Mathematical Society, US
• Publication Date: 2021
• Language: English
• ISBN 10: 1470441764
• ISBN 13: 9781470441760
• Binding: Paperback
• Condition: New
• Weight: 333 grams

About this title

Synopsis

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. The authors 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.

About the Author

Alexander Blokh, University of Alabama at Birmingham, AL, USA.

Lex Oversteegen, University of Alabama at Birmingham, AL, USA.

Ross Ptacek, National Research University Higher School of Economics, Moscow, Russia.

Vladlen Timorin, National Research University Higher School of Economics, Moscow, Russia.