# Moscow Mathematical Journal

Volume 15, Issue 3, July–September 2015 pp. 527–592.

Analyticity in Spaces of Convergent Power Series and Applications

**Authors**:
Loïc Teyssier

**Author institution:**Laboratoire I.R.M.A., Université de Strasbourg

**Summary: **

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

2010 Math. Subj. Class. 46G20, 58B12, 34M99, 37F75.

**Keywords:**Infinite-dimensional holomorphy, complex dynamical systems, holomorphic solutions of differential equations, Liouvillian integrability of foliations.

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