Hardcover ISBN:  9781470435141 
Product Code:  SURV/219 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470440503 
Product Code:  SURV/219.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470435141 
eBook: ISBN:  9781470440503 
Product Code:  SURV/219.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 
Hardcover ISBN:  9781470435141 
Product Code:  SURV/219 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470440503 
Product Code:  SURV/219.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470435141 
eBook ISBN:  9781470440503 
Product Code:  SURV/219.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 219; 2017; 196 ppMSC: Primary 26; 37; 51
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.
ReadershipUndergraduate and graduate students and researchers interested in analysis, geometry, and topology.

Table of Contents

Chapters

Introduction

Part 1

Some other polygon iterations

A primer on projective geometry

Elementary algebraic geometry

The pentagram map

Some related dynamical systems

Part 2

The projective heat map

Topological degree of the map

The convex case

The basic domains

The method of positive dominance

The Cantor set

Towards the quasi horseshoe

The quasi horseshoe

Part 3

Sketches for the remaining results

Towards the solenoid

The solenoid

Local structure of the Julia set

The embedded graph

Connectedness of the Julia set

Terms, formulas, and coordinate listings


Additional Material

Reviews

This monograph contains considerable background on the techniques used, including basics of projective geometry, algebraic geometry, and dynamical systems, in order to make it accessible for graduate students and relatively selfcontained. The computer program that Schwartz uses is also available on his website, so that the reader can explore the proof in a more handson way.
Roland K. W. Roeder, Mathematical Reviews


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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.
Undergraduate and graduate students and researchers interested in analysis, geometry, and topology.

Chapters

Introduction

Part 1

Some other polygon iterations

A primer on projective geometry

Elementary algebraic geometry

The pentagram map

Some related dynamical systems

Part 2

The projective heat map

Topological degree of the map

The convex case

The basic domains

The method of positive dominance

The Cantor set

Towards the quasi horseshoe

The quasi horseshoe

Part 3

Sketches for the remaining results

Towards the solenoid

The solenoid

Local structure of the Julia set

The embedded graph

Connectedness of the Julia set

Terms, formulas, and coordinate listings

This monograph contains considerable background on the techniques used, including basics of projective geometry, algebraic geometry, and dynamical systems, in order to make it accessible for graduate students and relatively selfcontained. The computer program that Schwartz uses is also available on his website, so that the reader can explore the proof in a more handson way.
Roland K. W. Roeder, Mathematical Reviews