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 = (z₁, …, 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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