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updated link to Cairns-Ghys paper
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There are smooth counter-examples by Cairns and Ghys [Ens. Math. 43, 1997], for instance a smooth non-linearizable action of $SL(2,\mathbb{R})$ on $\mathbb{R}^3$ (fixing the origin) or of $SL(3,\mathbb{R})$ on $\mathbb{R}^8$. By contrast, they show that any $C^k$ action of $SL(n,\mathbb{R})$ on $\mathbb{R}^n$ (same $n$, fixing the origin) is $C^k$-linearizable. Here is a linklink to their paper.

There are smooth counter-examples by Cairns and Ghys [Ens. Math. 43, 1997], for instance a smooth non-linearizable action of $SL(2,\mathbb{R})$ on $\mathbb{R}^3$ (fixing the origin) or of $SL(3,\mathbb{R})$ on $\mathbb{R}^8$. By contrast, they show that any $C^k$ action of $SL(n,\mathbb{R})$ on $\mathbb{R}^n$ (same $n$, fixing the origin) is $C^k$-linearizable. Here is a link to their paper.

There are smooth counter-examples by Cairns and Ghys [Ens. Math. 43, 1997], for instance a smooth non-linearizable action of $SL(2,\mathbb{R})$ on $\mathbb{R}^3$ (fixing the origin) or of $SL(3,\mathbb{R})$ on $\mathbb{R}^8$. By contrast, they show that any $C^k$ action of $SL(n,\mathbb{R})$ on $\mathbb{R}^n$ (same $n$, fixing the origin) is $C^k$-linearizable. Here is a link to their paper.

Source Link
BS.
BS.
  • 9.4k
  • 3
  • 39
  • 49

There are smooth counter-examples by Cairns and Ghys [Ens. Math. 43, 1997], for instance a smooth non-linearizable action of $SL(2,\mathbb{R})$ on $\mathbb{R}^3$ (fixing the origin) or of $SL(3,\mathbb{R})$ on $\mathbb{R}^8$. By contrast, they show that any $C^k$ action of $SL(n,\mathbb{R})$ on $\mathbb{R}^n$ (same $n$, fixing the origin) is $C^k$-linearizable. Here is a link to their paper.