The memoir studies glocal holomorphic dynamics, those which are the (local) expression in a germ of a chart of a (global) holomorphic dynamics on a projective complex variety. We establish the existence of germs of a holomorphic foliation in the complex plane none of which are locally conjugate to an algebraic foliation. The proof relies on a Baire argument, replacing unspecified closed sets by proper analytical sets. The analyticity in use (for infinite dimensional spaces) is that of locally convex topologies on the differential algebra of germs of a holomorphic function. We also prove that the generic germ does not satisfy "reasonable" analytic relations. The memoir also discusses the "explicit" depiction of an example of non-glocal system. We present a computable method for the realization of saddle-node foliations, with prescribed Martinet-Ramis invariants. Producing an example thus reduces to characterizing Martinet-Ramis invariants of glocal foliations. A Hermite-Lindemann conjecture is presented in the context of holomorphic foliations. Finally the memoir presents a generalization of Marín-Matte's monodromy construction. This object is a local invariant for singular foliations in the complex plane. Here again we wish to obtain partial characterizations of monodromies attached to glocal foliations. The original hypothesis are weakened and examples are produced, showing the optimality of the new hypothesis.
Analytical properties of the space of convergent power series and glocal holomorphic dynamics