Simple proofs and extensions of a result of L. D. Pustylnikov on the nonautonomous Siegel theorem

Abstract

We present simple proofs of a result of L.D. Pustylnikov extending to nonautonomous dynamics the Siegel theorem of linearization of analytic mappings. We show that if a sequence f_n of analytic mappings of C^d has a common fixed point f_n(0) = 0, and the maps f_n converge to a linear mapping A∞ so fast that

\(\sum\limits_n {{{\left\| {{f_m} - {A_\infty }} \right\|}_{L\infty \left( B \right)}} < \infty } \)

\({A_\infty } = diag\left( {{e^{2\pi i{\omega _1}}},...,{e^{2\pi i{\omega _d}}}} \right)\omega = \left( {{\omega _1},...,{\omega _q}} \right) \in {\mathbb{R}^d},\)

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 of the result is that the functions hn are defined in a large domain and they are bounded. We show that

\({\sum\nolimits_n {\left\| {{h_n} - Id} \right\|} _{L\infty (B)}} < \infty .\)

We also provide results when f_n converges to a nonlinearizable mapping f∞ or to a nonelliptic linear mapping. In the case that the mappings f_n preserve a geometric structure (e. g., symplectic, volume, contact, Poisson, etc.), we show that the hn can be chosen so that they preserve the same geometric structure as the f_n. We present five elementary proofs based on different methods and compare them. Notably, we consider the results in the light of scattering theory. We hope that including different methods can serve as an introduction to methods to study conjugacy equations.