monodromies are related by $\tilde M_z=\Phi_z M_z\Phi_z^{-1}.$ Differentiating this and using that $\Phi_1 = { {I}}$, we get $$\left.\frac{d}{dz}\right\vert_{z=1}\tilde M_z = 
\left.\frac{d}{dz}\right\vert_{z=1} M_z + \left[\left.\frac{d}{dz}\right\vert_{z=1} \Phi_z, M_1 \right].$$ This identity in particular shows that the infinitesimal monodromy of a *twisted* polygon is in general not well-defined, due to the extra commutator term in the right-hand side. But for a closed polygon we have $M_1 = { {I}}$, so the extra term vanishes and we get the desired identity.

Before we proceed to the proof of the main theorem, let us mention one property of the infinitesimal monodromy:

The infinitesimal monodromy of a closed polygon is preserved by the pentagram map.

The pentagram map preserves the monodromy and commutes with the scaling. The infinitesimal monodromy is defined using monodromy and scaling and is thus preserved as well.

This result in fact follows from our main theorem, because Glick shows in [@glick2020limit Theorem 3.1] that his operator has this property. However, the proof based on Glick’s definition is quite non-trivial, while in our approach it is immediate. The observation that the infinitesimal monodromy is preserved by the pentagram map was in fact our motivation to conjecture that it should coincide with Glick’s 