Germes de difféomorphismes et de champs de vecteurs en classe de différentiabilité finie

Dumortier, Freddy; Roussarie, Robert

doi:10.5802/aif.910

Cited by 7 publications

(5 citation statements)

References 6 publications

(38 reference statements)

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“…Let us indicate how to derive, in a standard way [DR1], the similar result for vector fields thanks to the uniqueness of the solutions of the equations in the proof of Theorem 2. Suppose that X is a vector field near 0 ∈ R c × R such that its flow…”

Section: Generalisationsmentioning

confidence: 99%

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Partially hyperbolic fixed points with constraints

Bonckaert

1996

Trans. Amer. Math. Soc.

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Abstract. We investigate the local conjugacy, at a partially hyperbolic fixed point, of a diffeomorphism (vector field) to its normally linear part in the presence of constraints, where the change of variables also must satisfy the constraints. The main result is applied to vector fields respecting a singular foliation, encountered, by F. Dumortier and R. Roussarie, in the desingularization of families of vector fields.

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Section: Generalisationsmentioning

confidence: 99%

“…We claim in fact that Ψ t := X −t •H −1 •NX t •H equals the identity. A straightforward calculation [DR1] shows that NX 1 • Ψ t = Ψ t • NX 1 (index 1 means time one). Now we are in the situation of the 'path part' (C) of the proof of Theorem 2, this time with f = f t = NX 1 .…”

Section: Generalisationsmentioning

confidence: 99%

Partially hyperbolic fixed points with constraints

Bonckaert

1996

Trans. Amer. Math. Soc.

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“…Lebesgue measure in this C ∞ setting. For C κ+2 diffeomorphisms f , there seems to be no general result that f is the time-1 map of a C κ+1 vector field ( [DR83] gives results depending on the smoothness of the vector field). Therefore, we first perturb f to a C ∞ diffeomorphism, next use the asymptotics obtained from the vector field, and finally interpret the results for f using an extra approximation argument, see Lemma 2.3.…”

Section: Set-upmentioning

confidence: 99%

Regular variation and rates of mixing for infinite measure preserving almost Anosov diffeomorphisms

Bruin¹,

Terhesiu²

2018

Ergod. Th. Dynam. Sys.

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“…It should be mentioned that Dumortier and Roussarie [2] have dealt with the finite determining of germs of vector fields on R2 having the following normal form: X(r, 6) = (4>i(0)rk+1 + 0(rk+2))^ + (a(6) + rkcj>2(0) + O^'))f or certain functions 9\,02 and a. We emphasize that the techniques and methods used there are quite different from ours.…”

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confidence: 97%

Vector fields in the vicinity of a circle of critical points

Mattéi¹,

Teixeira²

1986

Trans. Amer. Math. Soc.

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In this paper the Cs-conjugacy between vector fields on R2 having a circle of critical points is studied. 0. Statement of the problem. We are interested in studying vector fields on R2 having a circle of critical points as well as diffeomorphisms on R2 having a circle of fixed points. We can produce examples of such vector fields (resp. diffeomorphisms) by blowing up in polar coordinates germs of C°° vector fields (resp. diffeomorphisms) on R2 at 0 having the form X(x,y) = (x2 + y2)k with b ,¿ 0 and k being a positive integer (resp. ip -Id +X).A natural question in the study of singularities is: "Under which conditions is a germ of a singularity Cs-determined by a finite jet and which finite jets are determining?" On R2 there is the following result due to F. Dumortier [1]:If a germ satisfies an inequality of Lojawiewicz and has a characteristic orbit, then it is C°-determined. (By "characteristic orbit" we mean an orbit which tends to the singularity or leaves the singularity with a well-defined direction. Observe that the field given above does not have a characteristic orbit.)Consider germs of vector fields in R2 at 0 having the form -Z(x,y) = rk-X(x,y), where r2 = x2 + y2, k is a positive integer and X is a germ of C°° vector fields at 0 with X(0) = 0 and the eigenvalues of X (0) are A = a±ib with a/0 and b ^ 0.Consider the germs of diffeomorphisms in the plane of the form y5 = Id +Z. Denote by Z and ip the respective blowing ups (in polar coordinates) of Z and ip.Call Xk and Dk the spaces of such Z and k).We recall that:(i) The number a -ab~x is a topological invariant for the structural stability in D2 (see for instance [6]).(ii) The diffeomorphism Tp is formally imbedded in a flow represented by a vector field H. Moreover the (k + l)-jet of H at 0 is Hk = (x2 + y2)Xo(x,y), where X0 is the linearized vector field of X (see [4]).

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Germes de difféomorphismes et de champs de vecteurs en classe de différentiabilité finie

Cited by 7 publications

References 6 publications

Partially hyperbolic fixed points with constraints

Partially hyperbolic fixed points with constraints

Regular variation and rates of mixing for infinite measure preserving almost Anosov diffeomorphisms

Vector fields in the vicinity of a circle of critical points

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