Searcharxiv⌕ Search

arXiv subjects

Daniel Panazzolo

Publications and source records attributed to Daniel Panazzolo.

10 recordsLinked to original sources
arXiv2026

Bruno ideal and the variety of centers for singular germs of vector fields

Given a logarithmic analytic vector field $\partial$, we consider the formal ideal $B(\partial)$ defined by the collinearity locus of the semi-simple and nilpotent components of~$\partial$. Assuming that the eigenvalues of the linear part of $\partial$ satisfy the so-called Bruno arithmetic condition, we prove that $B(\partial)$ is in fact an analytic ideal. Moreover, $\partial$ is analytically normalizable when restricted to this ideal. As a consequence, the vanishing locus $V$ of $B(\partial)$ is an analytic variety, and the foliation defined by $\partial|_{V}$ is analytically linearizable.

math.DS↗
arXiv2026

Piecewise Smooth Dynamical Systems Regularized by Convolution

We present a general regularization procedure for piecewise smooth vector fields whose discontinuity locus is a variety of normal crossings type. We show that such regularization can be smoothed through a finite sequence of blowings-up, thereby reducing the problem to study of the dynamics of a smooth vector field in a manifold with corners. The procedure will be illustrated in the cases of piecewise smooth vector fields on $\mathbb{R}^2$ with discontinuity locus $x=0$ or $xy=0$, and on $\mathbb{R}^3$ with discontinuity locus $xyz=0$. We will see that some unexpected dynamical phenomena may arise even in the case of piecewise constant vector fields.

math.DS↗
arXiv2024

Automorphisms and derivations on algebras endowed with formal infinite sums

We establish a correspondence between automorphisms and derivations on certain algebras of generalised power series. In particular, we describe a Lie algebra of derivations on a field $k(\!(G)\!)$ of generalised power series, exploiting our knowledge of its group of valuation preserving automorphisms. The correspondence is given by the formal Taylor expansion of the exponential. In order to define the exponential map, we develop an appropriate notion of summability of infinite families in algebras. We show that there is a large class of algebras in which the exponential induces the above correspondence.

math.RA↗
arXiv2021

Rigidity of saddle loops

A saddle loop is a germ of a holomorphic foliation near a homoclinic saddle connection. We prove that they are classied by their Poincar{\'e} rst-return map. We also prove that they are formally rigid when the Poincar{\'e} map is multivalued. Finally, we provide a list of all analytic classes of Liouville-integrable saddle loops.

math.DS↗
arXiv2018

Generalized Flow-Box property for singular foliations

We introduce a notion of generalized Flow-Box property valid for general singular distributions and sub-varieties (based on a dynamical interpretation). Just as in the usual Flow-Box Theorem, we characterize geometrical and algebraic conditions of (quasi) transversality in order for an analytic sub-variety $X$ (not necessarily regular) to be a section of a line foliation. We also discuss the case of more general foliations. This study is originally motivated by a question of Jean-Francois Mattei (concerning the strengthening of a Theorem of Mattei) about the existence of local slices for a (non-compact) Lie group action.

math.CA↗
arXiv2017

Regularization of Discontinuous Foliations: Blowing up and Sliding Conditions via Fenichel Theory

We study the regularization of an oriented 1-foliation $\mathcal{F}$ on $M \setminus Σ$ where $M$ is a smooth manifold and $Σ\subset M$ is a closed subset, which can be interpreted as the discontinuity locus of $\mathcal{F}$. In the spirit of Filippov's work, we define a sliding and sewing dynamics on the discontinuity locus $Σ$ as some sort of limit of the dynamics of a nearby smooth 1-foliation and obtain conditions to identify whether a point belongs to the sliding or sewing regions.

math.DS↗
arXiv2017

PSL(2,C), the exponential and some new free groups

We prove a normal form result for the groupoid of germs generated by PSL(2,C) and the exponential map. As consequences, we generalize a result of Cohen about the group of translations and powers, and prove that the subgroup of Homeo(R,+infinity) generated by the positive affine maps and the exponential map is isomorphic to a HNN-extension.

math.DS↗
arXiv2016

(1 + d/dz)^(-1)

We investigate the structure of fully non-linear P.D.E.'s in holomorphic functions, with emphasis on the functorial generalisation of so called "irregular" O.D.E.'s. Highlights are an implicit function theorem removing the perturbation conditions of Nash-Moser type, best possible existence results when the singularity of the linearised P.D.E. is at worst bi-dimensional, and various, again optimal, corollaries on existence of centre manifolds and conjugation to normal form of 3-dimensional vector fields.

math.AG↗
arXiv2007

Resolution of Singularities of Vector Fields in Dimension Three

Let X be an analytic vector field defined in a real analytic manifold of dimension three. We prove that all the singularities of X can be made elementary by a finite number of blowing-ups in the ambient space. New version: Some misprints have been corrected and the text have been slightly reorganized. The final version appears in Acta Mathematica -Volume 197, Number 2, 167-289.

math.AG↗