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Softcover ISBN:  9781470441760 
Product Code:  MEMO/265/1288 
List Price:  $85.00 
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Softcover ISBN:  9781470441760 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 265; 2020; 105 ppMSC: Primary 37
The socalled “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.

Table of Contents

Chapters

1. Introduction

2. Invariant laminations: general properties

3. Special types of invariant laminations

4. Applications: Spaces of topological polynomials


Additional Material

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The socalled “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.

Chapters

1. Introduction

2. Invariant laminations: general properties

3. Special types of invariant laminations

4. Applications: Spaces of topological polynomials