∞-jets of diffeomorphisms preserving orbits of vector fields

Sergiy Maksymenko

Open Mathematics (2009)

  • Volume: 7, Issue: 2, page 272-298
  • ISSN: 2391-5455

Abstract

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Let F be a C ∞ vector field defined near the origin O ∈ ℝn, F(O) = 0, and (Ft) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝn → ℝn at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney Wr topology. Then contains a subset consisting of maps of the form Fα(x)(x), where α: ℝn → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = . In this paper we present a class of examples of vector fields with degenerate singularities at O for which formally coincides with , i.e. on the level of ∞-jets at O. We also establish parameter rigidity of linear vector fields and “reduced” Hamiltonian vector fields of real homogeneous polynomials in two variables.

How to cite

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Sergiy Maksymenko. "∞-jets of diffeomorphisms preserving orbits of vector fields." Open Mathematics 7.2 (2009): 272-298. <http://eudml.org/doc/269090>.

@article{SergiyMaksymenko2009,
abstract = {Let F be a C ∞ vector field defined near the origin O ∈ ℝn, F(O) = 0, and (Ft) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝn → ℝn at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney Wr topology. Then contains a subset consisting of maps of the form Fα(x)(x), where α: ℝn → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = . In this paper we present a class of examples of vector fields with degenerate singularities at O for which formally coincides with , i.e. on the level of ∞-jets at O. We also establish parameter rigidity of linear vector fields and “reduced” Hamiltonian vector fields of real homogeneous polynomials in two variables.},
author = {Sergiy Maksymenko},
journal = {Open Mathematics},
keywords = {Orbit preserving diffeomorphism; Parameter rigidity; Borel’s theorem; orbit preserving diffeomorphism; parameter rigidity; shift function},
language = {eng},
number = {2},
pages = {272-298},
title = {∞-jets of diffeomorphisms preserving orbits of vector fields},
url = {http://eudml.org/doc/269090},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Sergiy Maksymenko
TI - ∞-jets of diffeomorphisms preserving orbits of vector fields
JO - Open Mathematics
PY - 2009
VL - 7
IS - 2
SP - 272
EP - 298
AB - Let F be a C ∞ vector field defined near the origin O ∈ ℝn, F(O) = 0, and (Ft) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝn → ℝn at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney Wr topology. Then contains a subset consisting of maps of the form Fα(x)(x), where α: ℝn → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = . In this paper we present a class of examples of vector fields with degenerate singularities at O for which formally coincides with , i.e. on the level of ∞-jets at O. We also establish parameter rigidity of linear vector fields and “reduced” Hamiltonian vector fields of real homogeneous polynomials in two variables.
LA - eng
KW - Orbit preserving diffeomorphism; Parameter rigidity; Borel’s theorem; orbit preserving diffeomorphism; parameter rigidity; shift function
UR - http://eudml.org/doc/269090
ER -

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