Simple Proofs and Extensions of a Result of L. D. Pustylnikov on the Nonautonomous Siegel Theorem

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 $$ \sum_n \|f_m - A_\infty\|_{\mathbf{L}^\infty(B)} < \infty $$ $$ A_\infty = \mathop{\rm diag}( e^{2 \pi i \omega_1}, \ldots, e^{2 \pi i \omega_d}) \qquad \omega = (\omega_1, \ldots, \omega_q) \in {\mathbb R}^d, $$ 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.