---
author:
- 'Quinton Aboud[^1] and Anton Izosimov[^2]'
bibliography:
- 'ref.bib'
title: The limit point of the pentagram map and infinitesimal monodromy
---

Introduction
============

The pentagram map, introduced by R.Schwartz in [@schwartz1992pentagram], is a discrete dynamical system on the space of planar polygons. The definition of this map is illustrated in Figure \[Fig1\]: the image of the polygon $P$ under the pentagram map is the polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$ (i.e., diagonals connecting second-nearest vertices).

(VK7) at (0,0); (VK6) at (1.5,-0.5); (VK5) at (3,1); (VK4) at (3,2); (VK3) at (1,3); (VK2) at (-0.5,2.5); (VK1) at (-1,1.5);

(VK7) – (VK6) – (VK5) – (VK4) – (VK3) – (VK2) – (VK1) – cycle; (VK7) – (VK5); (VK6) – (VK4); (VK5) – (VK3); (VK4) – (VK2); (VK3) – (VK1); (VK2) – (VK7); (VK1) – (VK6);

; ; ; ; ; ; ;

(Ap) – (Bp) – (Cp) – (Dp) – (Ep) – (Fp) – (Gp) – cycle;

at (-0.9,2.3) () [$P$]{}; at (2,1.5) () [$P'$]{};

The pentagram map has been an especially popular topic in the last decade, mainly due to its connections with integrability [@ovsienko2010pentagram; @soloviev2013integrability] and the theory of cluster algebras 