This article is cited in
2 papers
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 30332-0160, 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 = \mathop{\rm diag}( 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 $\sum_n \|h_n - \mathop{\rm Id} \|_{\mathbf{L}^\infty(B)} < \infty$.
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.
MSC: 37C60,
34C35,
37F50,
30D05,
47J07 Received: 17.08.2017
Accepted: 02.10.2017
Language: English
DOI:
10.1134/S1560354717060053
References