The Projective Heat Map

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.

"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 self-contained. The computer program that Schwartz uses is also available on his website, so that the reader can explore the proof in a more hands-on way." - Roland K. W. Roeder, Mathematical Reviews