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Connected components of partition preserving diffeomorphisms

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Let $f:\mathbb{R}^2\to\mathbb{R}$ be a homogeneous polynomial and $\mathcal{S}(f)$ be the group of diffeomorphisms $h$ of $\mathbb{R}^2$ preserving $f$, i.e. $f \circ h =f$. Denote by $\mathcal{S}_{\mathrm{id}}(f)^{r}$, $(0\leq r \leq \infty)$, the identity component of $\mathcal{S}(f)$ with respect to the weak Whitney $C^{r}_{W}$-topology. We prove that $\mathcal{S}_{\mathrm{id}}(f)^{\infty} = \cdots = \mathcal{S}_{\mathrm{id}}(f)^{1}$ for all $f$ and that $\mathcal{S}_{\mathrm{id}}(f)^{1} ot= \mathcal{S}_{\mathrm{id}}(f)^{0}$ if and only if $f$ is a product of at least two distinct irreducible over $\mathbb{R}$ quadratic forms.

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TitleConnected components of partition preserving diffeomorphisms
SourceMethods Funct. Anal. Topology, Vol. 15 (2009), no. 3, 264-279
MathSciNet MR2567311
CopyrightThe Author(s) 2009 (CC BY-SA)

Authors Information

Sergiy Maksymenko
Topology Department, Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine 

Citation Example

Sergiy Maksymenko, Connected components of partition preserving diffeomorphisms, Methods Funct. Anal. Topology 15 (2009), no. 3, 264-279.


@article {MFAT476,
    AUTHOR = {Maksymenko, Sergiy},
     TITLE = {Connected components of partition preserving diffeomorphisms},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {15},
      YEAR = {2009},
    NUMBER = {3},
     PAGES = {264-279},
      ISSN = {1029-3531},
       URL = {},

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