Analyticity in spaces of convergent power series and applications

We study the analytic structure of the space of germs of an analytic function at the origin of $\mathbb C^m$, namely the space $\mathbb C\{\mathbf z\}$, where $\mathbf z=(z_1,\dots,z_m)$, equipped with a convenient locally convex topology. We are particularly interested in studying the properties of analytic sets of $\mathbb C\{\mathbf z\}$ as defined by the vanishing loci of analytic maps. While we notice that $\mathbb C\{\mathbf z\}$ is not Baire we also prove it enjoys the analytic Baire property: the countable union of proper analytic sets of $\mathbb C\{\mathbf z\}$ has empty interior. This property underlies a quite natural notion of a generic property of $\mathbb C\{\mathbf z\}$, for which we prove some dynamics-related theorems. We also initiate a program to tackle the task of characterizing glocal objects in some situations.