## Simple proofs and extensions of a result of L. D. Pustylnikov on the nonautonomous Siegel theorem.(English)Zbl 1385.37022

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.

### MSC:

 37B55 Topological dynamics of nonautonomous systems 37C60 Nonautonomous smooth dynamical systems 37F50 Small divisors, rotation domains and linearization in holomorphic dynamics 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 47J07 Abstract inverse mapping and implicit function theorems involving nonlinear operators

Zbl 0324.34051
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### References:

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