Stacky geometry and logarithmic topology of transversely affine foliations
We study the stacky and logarithmic-topological structures associated with transversely affine affine foliations. To such a foliation we attach its holonomy group and the corresponding quotient stack, which provides the natural geometric base for the multiplicative developing coordinate. Holomorphic and meromorphic reparametrisations are then identified with endomorphisms of this stack and of its compactification, yielding a geometric Singer-type theorem. We then classify these reparametrisations according to the geometry of the holonomy group. On the logarithmic-topological side, we pass to the Kato-Nakayama space, where the residues define canonical boundary characters, govern the induced linear dynamics on the boundary tori, and give rise to a canonical logarithmic lift of the developing map. In this way, the quotient stack controls the linear part of the theory, while the Kato-Nakayama space captures its logarithmic-topological and dynamical content.