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E-raamat: Projective Heat Map

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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.
Preface ix
Chapter 1 Introduction
1(12)
1.1 From Geometry to Dynamics
1(2)
1.2 The Projective Heat Map
3(1)
1.3 A Picture of the Julia Set
4(1)
1.4 The Core Results
4(2)
1.5 Deeper Structure
6(2)
1.6 A Few Corollaries
8(1)
1.7 Sketch of the Proofs
8(1)
1.8 Some Comparisons
9(1)
1.9 Outline of the Monograph
10(1)
1.10 Companion Program
11(2)
Part 1 13(48)
Chapter 2 Some Other Polygon Iterations
15(8)
2.1 The Midpoint Theorem
15(1)
2.2 The Midpoint Iteration
15(2)
2.3 Napoleon's Theorem
17(1)
2.4 Napoleon's Iteration
18(1)
2.5 Conformal Averaging
19(4)
Chapter 3 A Primer on Projective Geometry
23(8)
3.1 The Real Projective Plane
23(1)
3.2 Affine Patches
23(1)
3.3 Projective Transformations and Dualities
24(1)
3.4 The Cross Ratio
24(1)
3.5 The Hilbert Metric
25(2)
3.6 Projective Invariants of Polygons
27(2)
3.7 Duality and Relabeling
29(1)
3.8 The Gauss Group
30(1)
Chapter 4 Elementary Algebraic Geometry
31(6)
4.1 Measure Zero Sets
31(1)
4.2 Rational Maps
31(1)
4.3 Homogeneous Polynomials
32(1)
4.4 Bezout's Theorem
32(1)
4.5 The Blow-up Construction
33(4)
Chapter 5 The Pentagram Map
37(10)
5.1 The Pentagram Configuration Theorem
37(1)
5.2 The Pentagram Map in Coordinates
37(2)
5.3 The First Pentagram Invariant
39(1)
5.4 The Poincare Recurrence Theorem
40(1)
5.5 Recurrence of the Pentagram Map
41(1)
5.6 Twisted Polygons
42(1)
5.7 The Pentagram Invariants
42(1)
5.8 Symplectic Manifolds and Torus Motion
43(1)
5.9 Complete Integrability
44(3)
Chapter 6 Some Related Dynamical Systems
47(14)
6.1 Julia Sets of Rational Maps
47(1)
6.2 The One-Sided Shift
48(3)
6.3 The Two-Sided Shift
51(1)
6.4 The Smale Horseshoe
51(1)
6.5 Quasi Horseshoe Maps
52(6)
6.6 The 2-adic Solenoid
58(1)
6.7 The BJK Continuum
59(2)
Part 2 61(60)
Chapter 7 The Projective Heat Map
63(8)
7.1 The Reconstruction Formula
63(1)
7.2 The Dual Map
64(1)
7.3 Formulas for the Projective Heat Map
65(2)
7.4 The Case of Pentagons
67(1)
7.5 Some Speculation
68(3)
Chapter 8 Topological Degree of the Map
71(4)
8.1 Overview
71(1)
8.2 The Lower Bound
71(1)
8.3 The Upper Bound
72(3)
Chapter 9 The Convex Case
75(6)
9.1 Flag Invariants of Convex Pentagons
75(1)
9.2 The Gauss Group Acting on the Unit Square
76(1)
9.3 A Positivity Criterion
76(2)
9.4 The End of the Proof
78(2)
9.5 The Action on the Boundary
80(1)
9.6 Discussion
80(1)
Chapter 10 The Basic Domains
81(8)
10.1 The Space of Pentagons
81(1)
10.2 The Action of the Gauss Group
82(1)
10.3 Changing Coordinates
83(1)
10.4 Convex and Star Convex Classes
84(1)
10.5 The Semigroup
84(2)
10.6 A Global Point of View
86(3)
Chapter 11 The Method of Positive Dominance
89(8)
11.1 The Divide and Conquer Algorithm
89(2)
11.2 Positivity
91(1)
11.3 The Denominator Test
91(2)
11.4 The Area Test
93(1)
11.5 The Expansion Test
93(1)
11.6 The Confinement Test
94(1)
11.7 The Exclusion Test
95(1)
11.8 The Cone Test
95(1)
11.9 The Stretch Test
96(1)
Chapter 12 The Cantor Set
97(12)
12.1 Overview
97(1)
12.2 The Big Disk
98(1)
12.3 The Six Small Disks
99(2)
12.4 The Diffeomorphism Property
101(3)
12.5 The Main Argument
104(1)
12.6 Proof of the Measure Expansion Lemma
105(1)
12.7 Proof of the Metric Expansion Lemma
105(2)
12.8 Discussion
107(2)
Chapter 13 Towards the Quasi Horseshoe
109(6)
13.1 The Target
109(1)
13.2 The Outer Layer
109(2)
13.3 The Inner Layer
111(2)
13.4 The Last Three pieces
113(2)
Chapter 14 The Quasi Horseshoe
115(6)
14.1 Overview
115(1)
14.2 Existence of The Quasi Horseshoe
115(2)
14.3 The Invariant Cantor Band
117(1)
14.4 Covering Property
118(1)
14.5 Subspace Property
118(1)
14.6 Attracting Property
119(2)
Part 3 121(72)
Chapter 15 Sketches for the Remaining Results
123(8)
15.1 The General Setup
123(1)
15.2 The Solenoid Result
124(2)
15.3 Local Structure
126(1)
15.4 The Embedded Graph
127(1)
15.5 Path Connectivity
128(1)
15.6 The Postcritical Set
128(1)
15.7 No Rational Fibration
129(2)
Chapter 16 Towards the Solenoid
131(10)
16.1 The Four Strips
131(1)
16.2 Two Cantor Cones
132(3)
16.3 Using Symmetry
135(3)
16.4 The Limiting Arc
138(3)
Chapter 17 The Solenoid
141(8)
17.1 Recognizing the BJK Continuum
141(1)
17.2 Taking Covers
142(1)
17.3 Connectivity and Unboundedness
143(1)
17.4 The Canonical Loop
144(1)
17.5 Using Symmetry for the Cone Points
144(1)
17.6 The First Cone Point
145(1)
17.7 The Second Cone Point
146(3)
Chapter 18 Local Structure of the Julia Set
149(12)
18.1 Blowing Down the Exceptional Fibers
149(2)
18.2 Everything but One Piece
151(1)
18.3 The Last Piece
151(6)
18.4 The Last Point
157(3)
18.5 Some Definedness Results
160(1)
Chapter 19 The Embedded Graph
161(14)
19.1 Defining the Generator
161(5)
19.2 From Generator to Edge
166(1)
19.3 From Edge to Pentagon
167(1)
19.4 Pre-images of the Pentagon
168(1)
19.5 The First Connector
169(1)
19.6 The Second Connection
170(2)
19.7 The Third Connector
172(1)
19.8 The End of the Proof
173(2)
Chapter 20 Connectedness of the Julia Set
175(12)
20.1 The Region Between the Disks
175(4)
20.2 The Local Diffeomorphism Lemma
179(2)
20.3 A Case by Case Analysis
181(4)
20.4 The Final Picture
185(2)
Chapter 21 Terms, Formulas, and Coordinate Listings
187(6)
21.1 Symbols and Terms
187(2)
21.2 Two Important Numbers
189(1)
21.3 The Maps
189(1)
21.4 Some Special Points
189(1)
21.5 The Cantor Set Pieces
190(1)
21.6 The Horseshoe Pieces
190(2)
21.7 The Refinement
192(1)
21.8 Auxiliary Polygons
192(1)
References 193
Richard Evan Schwartz, Brown University, Providence, RI.
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