A family of projectively natural polygon iterations

TL;DR

This paper introduces a one-parameter family of polygon iteration maps that generalize the projective heat map, revealing diverse dynamical behaviors in projective geometry.

Contribution

It extends Schwartz's work by constructing a new family of maps that exhibit similar dynamics to the projective heat map, broadening understanding of projective polygon iterations.

Findings

The family of maps shows diverse dynamical behaviors.

The maps generalize the projective heat map.

Potential applications in projective geometry dynamics.

Abstract

The pentagram map was invented by Richard Schwartz in his search for a projective-geometric analogue of the midpoint map. It turns out that the dynamical behavior of the pentagram map is totally different from that of the midpoint map. Recently, Schwartz has constructed a related map, the projective heat map, which empirically exhibits similar dynamics as the midpoint map. In this paper, we will demonstrate that there is a one-parameter family of maps which behaves a lot like Schwartz's projective heat map.