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Journal of Dynamical and Control Systems
Published in: Journal of Dynamical and Control Systems 1/2017

29-03-2016

Real-Formal Orbital Rigidity for Germs of Real Analytic Vector Fields on the Real Plane

Author: Jessica Angélica Jaurez-Rosas

Published in: Journal of Dynamical and Control Systems | Issue 1/2017

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Abstract

For \(n \geqslant 2\), we consider \(\mathcal {V}^{\mathbb {R}}_{n}\) the class of germs of real analytic vector fields on \(\left (\mathbb {R}^{2}, \widehat {0}\right )\) with zero (n−1)-jet and nonzero n-jet. We prove, for generic germs of \(\mathcal {V}^{\mathbb {R}}_{n}\), that the real-formal orbital equivalence implies the real-analytic orbital equivalence, that is, the real-formal orbital rigidity takes place. This is the real analytic version of Voronin’s formal orbital rigidity theorem.
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Footnotes
1
We shall say that an analytic (formal) orbital equivalence is strict if the linear part of the analytic (formal) change of coordinates is the identity matrix and the constant term of the analytic scalar function (the real formal series) is 1. In a similar way, we define a strict real-analytic (real-formal) orbital equivalence.
 
2
A cross-section to \(\mathbb {D}_{\mathbb {C}}\) at a point \(\mathfrak {d} \in \mathbb {D}_{\mathbb {C}}\) is the germ of a complex curve which is contained in \(\mathbb {M}_{\mathbb {C}}\) and intersects \(\mathbb {D}_{\mathbb {C}}\) transversally at \(\mathfrak {d}\).
 
3
This map is defined modulo conjugacy, the latter given by the changes of cross-sections.
 
4
This group modulo a simultaneous conjugacy will be referred to as vanishing holonomy group of \(\widetilde {\mathcal {F}}^{\mathbb {C}}_{\upsilon }\), being independent of a cross-section or even a base point
 
5
It is important to notice that when \(\mathfrak {p}_{i} \in \mathbb {D}_{\mathbb {R}}\), S i is the complexification of a real analytic curve, that is, the Taylor series of ϕ i has real coefficients. On the other hand, if \(\mathfrak {p}_{i} \notin \mathbb {D}_{\mathbb {R}}\) then there exists another singular point \(\mathfrak {p}_{j} \notin \mathbb {D}_{\mathbb {R}}\) which is the conjugate point of \(\mathfrak {p}_{i}\). Furthermore, S j is the conjugate curve of S i , that is, if \(\phi _{i} (x) = {\sum }_{k \geqslant 1} a_{k} x^{k}\) then \(\phi _{j} (x) = {\sum }_{k \geqslant 1} \overline {a}_{k} x^{k}\).
 
6
Given f a germ of holomorphic map at \(\widehat {0} \in \mathbb {C}^{n}\), the conjugated function of f, denoted by \(\overline {f(\bar {z})}\), is the germ of holomorphic function at \(\widehat {0} \in \mathbb {C}^{n}\) satisfying the following condition: if \(f(z) = {\sum }_{\mathbf {k} \in \mathbb {N}^{n}} a_{\mathbf {k}} z^{\mathbf {k}}\) then \(\overline {f(\bar {z})} = {\sum }_{\mathbf {k} \in \mathbb {N}^{n}} \overline {a}_{\mathbf {k}} z^{\mathbf {k}}\).
 
7
A formal transformation \(\mathbf {x} (u,x) \in \mathbb {C}[[u-u_{0},x-x_{0}]]\) is a solution of the differential equation (3) intersecting \(\widehat {\Gamma }\) if the following formal equalities are satisfied: \(\tfrac {\text {d} \, \mathbf {x}}{\text {d} \, u} = Q (u, \mathbf {x}(u,x))\) and x 0+Φ(x) = x(u 0+Ψ(x),x).
 
8
From the uniqueness of solutions of nonautonomous formal differential equations (depending on a parameter) intersecting a fixed formal cross-section (see [15]), the formal transformation \(\widetilde {H}_{1} \left (\mathbf {x}^{(1)}_{j} \left (u, u_{j}; {\Phi }^{(1)}_{j} (x)\right ), u \right )\) is equal to \( \mathbf {x}^{(2)}_{j} \left (\widetilde {H}_{2} \left (\mathbf {x}^{(1)}_{j}\left (u, u_{j}; {\Phi }^{(1)}_{j} (x)\right ), u \right ), \widetilde {H}_{2}\left ({\Phi }^{(1)}_{j} (x), u_{j}\right ); \widetilde {H}_{1} \left ({\Phi }^{(1)}_{j} (x), u_{j}\right )\right ) \, . \) The expression (6) is obtained by taking u = u j+1 in the previous equivalent transformations.
 
9
We can prove the real-formal orbital rigidity theorem without this condition, but in this case the proofs of the following results would be a little more complicated.
 
10
That is, given k+1≤i≤m there exists k+1≤j≤m such that \(\mathcal {L}_{j}\) is the conjugate of \(\mathcal {L}_{i}\), that is, if \(\psi _{i} (x) = {\sum }_{r \geqslant 1} a_{r} x^{r}\) then \(\psi _{j} (x) = {\sum }_{r \geqslant 1} \bar {a}_{r} x^{r}\), where \(\bar {a}\) is the complex conjugate of \(a \in \mathbb {C}\).
 
11
Since F i (x)=(f i (x),ψ i ∘f i (x)) where f i is a germ of biholomorphism at \(0 \in \mathbb {C}\), the coefficients of the Taylor series of f j are the complex conjugates of the respective coefficients of the Taylor series of f i . As a consequence, F l is the complexification of a germ of real analytic diffeomorphism whenever 1≤l≤k.
 
12
Considering the nonautonomous real analytic equation \(\tfrac {\text {d} \, x}{\text {d} \, u} = \vartheta _{j}(x,u)\) whose extended fase portrait coincides with the foliation \(H_{0} (\widetilde {\mathcal {F}}_{\vartheta })\), \(\widetilde {H}_{0}\) will be the inverse of \( (x,u) \mapsto (\phi _{\vartheta _{j}} (u,x),u)\), where \(\phi _{\vartheta _{j}}(u,x)\) is the flow of the nonautonomous equation satisfying \(\phi _{\vartheta _{j}}(q_{j},x) = x\).
 
13
Since \(\mathcal {J}_{j,x} (0, q_{j}) \neq 0\), \(\mathcal {J}_{j,u} (0, q_{j}) = 0\) (Lemma 6) and the curve T j is transversal to the divisor \(\mathbb {D}_{\mathbb {R}}\), \(\mathcal {J}_{j} \vert _{T_{j}}\) is invertible in a neighborhood of (0,q j )∈T j .
 
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Metadata
Title
Real-Formal Orbital Rigidity for Germs of Real Analytic Vector Fields on the Real Plane
Author
Jessica Angélica Jaurez-Rosas
Publication date
29-03-2016
Publisher
Springer US
Published in
Journal of Dynamical and Control Systems / Issue 1/2017
Print ISSN: 1079-2724
Electronic ISSN: 1573-8698
DOI
https://doi.org/10.1007/s10883-016-9314-y

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