Projective heat map - Schwartz, Richard Evan (Providence : American Mathematical Society, 2017)
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.
Introduction -- Some other polygon iterations -- A primer on projective geometry -- Elementary algebraic geometry -- The pentagram map -- Some related dynamical systems -- 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 -- 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.
Includes bibliographical references.
Keywords: Mappings (Mathematics); Projective spaces; Projective geometry