Abstract
We give the full description of the \({C^r_\delta}\) embeddings of a given diffeomorphism \({F \colon \mathbb{R}^2 \supset U \to \mathbb{R}^2}\) of class C r such that F(0) = 0 and \({d^{(r)}F(x) = d^{(r)}F(0) + O(\|x\|^{\delta}), \ \|x\|\to 0}\) with a hyperbolic fixed point. That is we determine all families of \({C^r_\delta}\) diffeomorphisms of the plane defined in a neighbourhood of the origin such that \({F^t\circ F^s=F^{t+s}}\), t,s ≥ 0, F 1 = F and the mapping \({t \mapsto F^t(x)}\) is continuous. To describe these semigroups we determine the real logarithms and all continuous groups of the real non-singular matrices.
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Dedicated to Professor János Aczél on his 90th birthday
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Zdun, M.C., Solarz, P. Embeddings of diffeomorphisms of the plane in regular iteration semigroups. Aequat. Math. 89, 149–160 (2015). https://doi.org/10.1007/s00010-014-0273-7
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DOI: https://doi.org/10.1007/s00010-014-0273-7