Embeddings of diffeomorphisms of the plane in regular iteration semigroups

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.