Analyticity in spaces of convergent power series and applications

TL;DR

This paper investigates the analytic structure of spaces of germs of analytic functions at the origin in complex m-space, focusing on their properties, analytic sets, and generic behaviors within a locally convex topology.

Contribution

It introduces the analytic Baire property for these spaces, studies properties of analytic sets, and explores dynamics and glocal objects in this context.

Findings

The space of germs is not Baire but has the analytic Baire property.

Proper analytic sets in the space have empty interior.

Established dynamics-related theorems and initiated a program on glocal objects.

Abstract

We study the analytic structure of the space of germs of an analytic function at the origin of \ww C^{\times m} , namely the space \germ{\mathbf{z}} where \mathbf{z}=\left(z\_{1},\cdots,z\_{m}\right) , equipped with a convenient locally convex topology. We are particularly interested in studying the properties of analytic sets of \germ{\mathbf{z}} as defined by the vanishing locus of analytic maps. While we notice that \germ{\mathbf{z}} is not Baire we also prove it enjoys the analytic Baire property: the countable union of proper analytic sets of \germ{\mathbf{z}} has empty interior. This property underlies a quite natural notion of a generic property of \germ{\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.