The Projective Heat Map ( Mathematical Surveys and Monographs )

Publication series ：Mathematical Surveys and Monographs

Author： Richard Evan Schwartz  

Publisher： American Mathematical Society‎

Publication year： 2017

E-ISBN: 9781470435141

P-ISBN(Paperback): 9781470440503

Subject： O185 projective (projection) Geometry, geometry

Keyword： 暂无分类

Language： ENG

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The Projective Heat Map

Description

This book introduces a simple dynamical model for a planar heat map that is invariant under projective transformations. The map is defined by iterating a polygon map, where one starts with a finite planar $N$-gon and produces a new $N$-gon by a prescribed geometric construction. One of the appeals of the topic of this book is the simplicity of the construction that yet leads to deep and far reaching mathematics. To construct the projective heat map, the author modifies the classical affine invariant midpoint map, which takes a polygon to a new polygon whose vertices are the midpoints of the original. The author provides useful background which makes this book accessible to a beginning graduate student or advanced undergraduate as well as researchers approaching this subject from other fields of specialty. The book includes many illustrations, and there is also a companion computer program.

Chapter

Cover

Title page

Contents

Preface

Chapter 1. Introduction

1.1. From Geometry to Dynamics

1.2. The Projective Heat Map

1.3. A Picture of the Julia Set

1.4. The Core Results

1.5. Deeper Structure

1.6. A Few Corollaries

1.7. Sketch of the Proofs

1.8. Some Comparisons

1.9. Outline of the Monograph

1.10. Companion Program

Part 1 .

Chapter 2. Some Other Polygon Iterations

2.1. The Midpoint Theorem

2.2. The Midpoint Iteration

2.3. Napoleon’s Theorem

2.4. Napoleon’s Iteration

2.5. Conformal Averaging

Chapter 3. A Primer on Projective Geometry

3.1. The Real Projective Plane

3.2. Affine Patches

3.3. Projective Transformations and Dualities

3.4. The Cross Ratio

3.5. The Hilbert Metric

3.6. Projective Invariants of Polygons

3.7. Duality and Relabeling

3.8. The Gauss Group

Chapter 4. Elementary Algebraic Geometry

4.1. Measure Zero Sets

4.2. Rational Maps

4.3. Homogeneous Polynomials

4.4. Bezout’s Theorem

4.5. The Blow-up Construction

Chapter 5. The Pentagram Map

5.1. The Pentagram Configuration Theorem

5.2. The Pentagram Map in Coordinates

5.3. The First Pentagram Invariant

5.4. The Poincare Recurrence Theorem

5.5. Recurrence of the Pentagram Map

5.6. Twisted Polygons

5.7. The Pentagram Invariants

5.8. Symplectic Manifolds and Torus Motion

5.9. Complete Integrability

Chapter 6. Some Related Dynamical Systems

6.1. Julia Sets of Rational Maps

6.2. The One-Sided Shift

6.3. The Two-Sided Shift

6.4. The Smale Horseshoe

6.5. Quasi Horseshoe Maps

6.6. The 2-adic Solenoid

6.7. The BJK Continuum

Part 2 .

Chapter 7. The Projective Heat Map

7.1. The Reconstruction Formula

7.2. The Dual Map

7.3. Formulas for the Projective Heat Map

7.4. The Case of Pentagons

7.5. Some Speculation

Chapter 8. Topological Degree of the Map

8.1. Overview

8.2. The Lower Bound

8.3. The Upper Bound

Chapter 9. The Convex Case

9.1. Flag Invariants of Convex Pentagons

9.2. The Gauss Group Acting on the Unit Square

9.3. A Positivity Criterion

9.4. The End of the Proof

9.5. The Action on the Boundary

9.6. Discussion

Chapter 10. The Basic Domains

10.1. The Space of Pentagons

10.2. The Action of the Gauss Group

10.3. Changing Coordinates

10.4. Convex and Star Convex Classes

10.5. The Semigroup

10.6. A Global Point of View

Chapter 11. The Method of Positive Dominance

11.1. The Divide and Conquer Algorithm

11.2. Positivity

11.3. The Denominator Test

11.4. The Area Test

11.5. The Expansion Test

11.6. The Confinement Test

11.7. The Exclusion Test

11.8. The Cone Test

11.9. The Stretch Test

Chapter 12. The Cantor Set

12.1. Overview

12.2. The Big Disk

12.3. The Six Small Disks

12.4. The Diffeomorphism Property

12.5. The Main Argument

12.6. Proof of the Measure Expansion Lemma

12.7. Proof of the Metric Expansion Lemma

12.8. Discussion

Chapter 13. Towards the Quasi Horseshoe

13.1. The Target

13.2. The Outer Layer

13.3. The Inner Layer

13.4. The Last Three pieces

Chapter 14. The Quasi Horseshoe

14.1. Overview

14.2. Existence of The Quasi Horseshoe

14.3. The Invariant Cantor Band

14.4. Covering Property

14.5. Subspace Property

14.6. Attracting Property

Part 3 .

Chapter 15. Sketches for the Remaining Results

15.1. The General Setup

15.2. The Solenoid Result

15.3. Local Structure

15.4. The Embedded Graph

15.5. Path Connectivity

15.6. The Postcritical Set

15.7. No Rational Fibration

Chapter 16. Towards the Solenoid

16.1. The Four Strips

16.2. Two Cantor Cones

16.3. Using Symmetry

16.4. The Limiting Arc

Chapter 17. The Solenoid

17.1. Recognizing the BJK Continuum

17.2. Taking Covers

17.3. Connectivity and Unboundedness

17.4. The Canonical Loop

17.5. Using Symmetry for the Cone Points

17.6. The First Cone Point

17.7. The Second Cone Point

Chapter 18. Local Structure of the Julia Set

18.1. Blowing Down the Exceptional Fibers

18.2. Everything but One Piece

18.3. The Last Piece

18.4. The Last Point

18.5. Some Definedness Results

Chapter 19. The Embedded Graph

19.1. Defining the Generator

19.2. From Generator to Edge

19.3. From Edge to Pentagon

19.4. Pre-images of the Pentagon

19.5. The First Connector

19.6. The Second Connection

19.7. The Third Connector

19.8. The End of the Proof

Chapter 20. Connectedness of the Julia Set

20.1. The Region Between the Disks

20.2. The Local Diffeomorphism Lemma

20.3. A Case by Case Analysis

20.4. The Final Picture

Chapter 21. Terms, Formulas, and Coordinate Listings

21.1. Symbols and Terms

21.2. Two Important Numbers

21.3. The Maps

21.4. Some Special Points

21.5. The Cantor Set Pieces

21.6. The Horseshoe Pieces

21.7. The Refinement

21.8. Auxiliary Polygons

References

Back Cover

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