related to so-called cross-ratio dynamics, see [@arnold2018cross Section 6.2.1].

[**Acknowledgments.**]{} The authors are grateful to Boris Khesin, Valentin Ovsienko, Richard Schwartz, and Sergei Tabachnikov for comments and discussions. A.I. was supported by NSF grant DMS-2008021.

Background: twisted polygons, corner invariants, and scaling {#Sec:back}
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In this section we briefly recall standard notions related to the pentagram map, concentrating on what will be used in the sequel.

A [*twisted $n$-gon*]{} is a bi-infinite sequence of points $v_i \in {\mathbb P}^2$ such that $v_{i+n}=M(v_i)$ for all $i\in \mathbb{Z}$ and a certain projective transformation $M\in {{\mathbb P}\mathrm{GL}}({\mathbb P}^2)$ called the [*monodromy*]{}. A twisted $n$-gon generalizes the notion of a closed $n$-gon as we recover a closed $n$-gon when the monodromy is equal to the identity. We denote the space of twisted $n$-gons by ${\mathcal{P}}_n$.

The pentagram map takes a twisted $n$-gon to a twisted $n$-gon (preserving the monodromy) so it can be regarded as a densely defined map from the space ${\mathcal{P}}_n$ of twisted $n$-gons to itself. From now on, we will assume that polygons are in sufficiently general position so as to allow for all constructions to go through unhindered.

We say that two twisted $n$-gons $\{v_i\}$ and $\{v_i'\}$ are projectively equivalent 