have to be the identity even if the initial polygon is closed!) is invariant under the map. Since $P_z$ is only defined as a projective equivalence class, this means that $M_z$ is only defined up to conjugation. Nevertheless, taking conjugation invariant functions (e.g. appropriately normalized eigenvalues) of $M_z$, we obtain, for every $z$, functions that are invariant under the pentagram map. It is shown in [@ovsienko2010pentagram] that the so-obtained functions commute under an appropriately defined Poisson bracket and turn the pentagram map into a discrete completely integrable system. See also [@ovsienko2013liouville] for a mode detailed proof. In our paper we utilize pretty much the same idea, but instead of looking at the eigenvalues of $M_z$ we will consider $M_z$ itself. It is not quite well-defined, but we will show that its $z$ derivative at $z = 1$ is, and that it coincides with Glick’s operator.

Infinitesimal monodromy {#Sec:im}
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In this section we define the infinitesimal monodromy and show that it does not depend on the choices we need to make to formulate the definition, namely on the way we lift the scaling symmetry from projective equivalence classes of polygons to actual polygons.

We start with a closed $n$-gon, $P$, 