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

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 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 $$\sum\limits_n {{{\left\| {{f_m} - {A_\infty }} \right\|}_{L\infty \left( B \right)}} < \infty } $$ ∑ n ‖ f m − A ∞ ‖ L ∞ ( B ) < ∞ $${A_\infty } = diag\left( {{e^{2\pi i{\omega _1}}},...,{e^{2\pi i{\omega _d}}}} \right)\omega = \left( {{\omega _1},...,{\omega _q}} \right) \in {\mathbb{R}^d},$$ A ∞ = d i a g ( e 2 π i ω 1 , ... , e 2 π i ω d ) ω = ( ω 1 , ... , ω q ) ∈ 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 $${h_{n + 1}} \circ {f_n} = {A_\infty }{h_n}.$$ h n + 1 ∘ f n = A ∞ 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 $${\sum\nolimits_n {\left\| {{h_n} - Id} \right\|} _{L\infty (B)}} < \infty .$$ ∑ n ‖ h n − I d ‖ L ∞ ( B ) < ∞ . 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Regular and Chaotic Dynamics Springer Journals

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

de la Llave, Rafael
Regular and Chaotic Dynamics , Volume 22 (6) – Dec 9, 2017
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References (56)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Pleiades Publishing, Ltd.
Subject
Mathematics; Dynamical Systems and Ergodic Theory
ISSN
1560-3547
eISSN
1468-4845
DOI
10.1134/S1560354717060053
Publisher site
See Article on Publisher Site

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 $$\sum\limits_n {{{\left\| {{f_m} - {A_\infty }} \right\|}_{L\infty \left( B \right)}} < \infty } $$ ∑ n ‖ f m − A ∞ ‖ L ∞ ( B ) < ∞ $${A_\infty } = diag\left( {{e^{2\pi i{\omega _1}}},...,{e^{2\pi i{\omega _d}}}} \right)\omega = \left( {{\omega _1},...,{\omega _q}} \right) \in {\mathbb{R}^d},$$ A ∞ = d i a g ( e 2 π i ω 1 , ... , e 2 π i ω d ) ω = ( ω 1 , ... , ω q ) ∈ 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 $${h_{n + 1}} \circ {f_n} = {A_\infty }{h_n}.$$ h n + 1 ∘ f n = A ∞ 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 $${\sum\nolimits_n {\left\| {{h_n} - Id} \right\|} _{L\infty (B)}} < \infty .$$ ∑ n ‖ h n − I d ‖ L ∞ ( B ) < ∞ . 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.

Journal

Regular and Chaotic Dynamics – Springer Journals

Published: Dec 9, 2017

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