A note on the pentagram map and tropical geometry

TL;DR

This paper explores the pentagram map's behavior within tropical geometry, revealing quasi-recursive dynamics and connecting geometric transformations to tropical frameworks.

Contribution

It introduces a novel application of tropical geometry to analyze the pentagram map's action on twisted polygons' moduli space.

Findings

Pentagram map exhibits quasi-recursive behavior in certain domains.

Tropical geometry provides a new framework for understanding the map's dynamics.

Connection established between the pentagram automaton and tropical geometric structures.

Abstract

In this note we study the action of the pentagram map on the moduli space of twisted polygons. The action with respect to the canonical coordinate turns out to be applicable to the framework of tropical geometry. As an application, we induce quasi-recursiveness of the pentagram map on some domain, which is induced from dynamical property of the pentagram automaton.