Holonomy is a germ, not a globally defined return map

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). The holonomy of a leafwise path is a germ of a local diffeomorphism between local transversals, not a globally defined return map: a representative is defined only on some open neighbourhood of the base point in the transversal, and different representatives of the same germ may differ arbitrarily far from that point. Indeed, the construction of A leafwise path determines a germ of a transverse diffeomorphism composes finitely many chart transports, each of which is produced by the inverse function theorem on a neighbourhood of one point of the path; the resulting map is defined only on a neighbourhood of the base point, and The holonomy germ is independent of the foliation chart chain compares only germs, so nothing in the construction defines values away from the base point, and no global continuation is asserted.

The Poincaré return map of a periodic orbit of a flow is the special case in which the first-return construction defines a map on a fixed section; that is additional structure, not part of the general definition. Statements about "the" return map along a leaf loop must therefore be read as statements about the holonomy germ, as in The holonomy representation and the holonomy group of a leaf, where only the germ ${\rho }_x([a])=h_{a^{-1}}(T,T)$ is used.