Simple proofs and extensions of a result of L. D. Pustylnikov on the nonautonomous Siegel theorem
\textit{L. D. Pustyl'nikov} [Math. USSR, Sb. 23, 382--404 (1975; Zbl 0324.34051)] extended to the nonautonomous case the Siegel theorem of linearization of analytic mappings. Five proofs of a generalization of this result are presented in this paper. More precisely, if a sequence \(f_n\) of analytic mappings of \({\mathbb{C}}^d\) has a common fixed point \(f_n(0)=0\), and the maps \(f_n\) converge to a linear mapping \(A_\infty=\text{diag}(e^{2\pi i\omega_1}, \ldots , e^{2\pi i\omega_d})\), \(\omega = (\omega_1, \ldots, \omega_d)\in{\mathbb{R}}^d\), so fast that \[ \sum_n \|f_n-A_\infty\|_{L^\infty(B)}<\infty\,, \] then \(f_n\) is nonautonomously conjugate to the linearization. That is, there exists a sequence \(h_n\) of analytic mappings fixing the origin satisfying \(h_{n+1}\circ f_n=A_\infty\,h_n\). The key point is that the functions \(h_n\) are defined in a large domain and they are bounded. It is shown that \(\sum_n \|h_n- \text{Id}\|_{L^\infty(B)}<\infty\). Some results are also provided in the case in which \(f_n\) converges to a nonlinearizable map \(f_\infty\) or to a nonelliptic linear mapping. When the mappings \(f_n\) preserve a geometric structure (for instance, symplectic, volume, contact) the same happens for the chosen \(h_n\). One proof is based on the Cook method of scattering theory; another proof uses just the elementary implicit function theorem; the third proof is only based on compactness arguments; the fourth proof is based on a very simple Nash-Moser theorem which takes advantage of a very subtle cancellation; and finally, the fifth proof is based on the deformation method.
scientific article; zbMATH DE number 6845455
scientific article