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To each triangulation of any surface with marked points on the boundary andorbifold points of order three, we associate a quiver (with loops) withpotential whose Jacobian algebra is finite dimensional and gentle. We study thestability scattering diagrams of such gentle algebras and use them to provethat the Caldero--Chapoton map defines a bijection between reachable$\tau$-rigid pairs and cluster monomials of the generalized cluster algebraassociated to the surface by Chekhov and Shapiro.

Gentle algebras arising from surfaces with orbifold points of order 3, Part I: scattering diagrams