
Simple Proofs and Extensions of a Result of L. D. Pustylnikov on the Nonautonomous Siegel Theorem
Rafael de la Llave^{} ^{} Georgia Institute of Technology, School of Mathematics, 686 Cherry St., Atlanta GA 303320160, USA
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
${\mathbb C}^d$ has a common fixed point $f_n(0) = 0$,
and the maps $f_n$ converge to a linear mapping
$A_\infty$ so fast that
\begin{equation*}
\sum_n \f_m  A_\infty\_{\mathbf{L}^\infty(B)} < \infty
\end{equation*}
\begin{equation*}
A_\infty = \mathopdiag( e^{2 \pi i \omega_1}, \ldots, e^{2 \pi i \omega_d})
\qquad \omega = (\omega_1, \ldots, \omega_q) \in {\mathbb R}^d,
\end{equation*}
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 $h_n$ are
defined in a large domain and they are bounded.
We show that $\sum_n \h_n  \mathopId \_{\mathbf{L}^\infty(B)} < \infty$.
We also provide results when $f_n$ converges to a nonlinearizable mapping
$f_\infty$ 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 $h_n$ 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.
Keywords:
nonautonomous linearization, scattering theory, implicit function theorem, deformations
Funding Agency 
Grant Number 
National Science Foundation 
DMS1500943 
The work of the author was supported in part by NSF grant DMS1500943. 
DOI:
https://doi.org/10.1134/S1560354717060053
References:
PDF file
HTML file
Bibliographic databases:
MSC: 37C60, 34C35, 37F50, 30D05, 47J07 Received: 17.08.2017 Accepted:02.10.2017
Language:
Citation:
Rafael de la Llave, “Simple Proofs and Extensions of a Result of L. D. Pustylnikov on the Nonautonomous Siegel Theorem”, Regul. Chaotic Dyn., 22:6 (2017), 650–676
Citation in format AMSBIB
\Bibitem{De 17}
\by Rafael de la Llave
\paper Simple Proofs and Extensions of a Result of L. D. Pustylnikov on the Nonautonomous Siegel Theorem
\jour Regul. Chaotic Dyn.
\yr 2017
\vol 22
\issue 6
\pages 650676
\mathnet{http://mi.mathnet.ru/rcd281}
\crossref{https://doi.org/10.1134/S1560354717060053}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3736466}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000417697500004}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2s2.085037615801}
Linking options:
http://mi.mathnet.ru/eng/rcd281 http://mi.mathnet.ru/eng/rcd/v22/i6/p650
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles

Number of views: 
This page:  53  References:  14 
