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Embedding diffeomorphisms in flows in Banach spaces

Published online by Cambridge University Press:  01 August 2009

XIANG ZHANG
XIANG ZHANG*
Affiliation:
Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, People’s Republic of China (email: xzhang@sjtu.edu.cn)
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Abstract

This paper concerns the problem of embedding, in the flow of an autonomous vector field, a local diffeomorphism near a hyperbolic fixed point in a Banach space. To solve the problem, we first extend Floquet theory to Banach spaces, and then prove that two C∞ hyperbolic diffeomorphisms are formally equivalent if and only if they are C∞-equivalent. The latter result is a version, in the Banach space context, of a classical theorem by Chen.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 4 , August 2009 , pp. 1349 - 1367
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Beyer, W. A. and Channell, P. J.. A Functional Equation for the Embedding of a Homeomorphism of the Interval into a Flow (Lecture Notes in Mathematics, 1163). Springer, New York, 1985, pp. 1–13.Google Scholar
[2]Bibikov, Yu. N.. Local Theory of Nonlinear Analytic Ordinary Differential Equations (Lecture Notes in Mathematics, 702). Springer, Berlin, 1979.CrossRefGoogle Scholar
[3]Chen, K. T.. Equivalence and decomposition of vector fields about an elementary critical point. Amer. J. Math. 85 (1963), 693–722.CrossRefGoogle Scholar
[4]Deimling, K.. Ordinary Differential Equations in Banach Spaces (Lecture Notes in Mathematics, 596). Springer, Berlin, 1977.CrossRefGoogle Scholar
[5]de la Llave, R.. Invariant manifolds associated to nonresonant spectral subspaces. J. Stat. Phys. 87 (1997), 211–249.CrossRefGoogle Scholar
[6]Dowson, H. R.. Spectral Theory of Linear Operators. Academic Press, London, 1978.Google Scholar
[7]ElBialy, M. S.. Local contractions of Banach spaces and spectral gap conditions. J. Funct. Anal. 182 (2001), 108–150.CrossRefGoogle Scholar
[8]Henry, D.. Geometric Theory of Parablic Equations (Lecture Notes in Mathematics, 840). Springer, New York, 1981.CrossRefGoogle Scholar
[9]Ilyashenko, Yu. S. and Yakovenko, S. Yu.. Finitely smooth normal forms of local families of diffeomorphisms and vector fields. Russian Math. Surveys 46(1) (1991), 1–43.CrossRefGoogle Scholar
[10]Kuksin, S. and Pöschel, J.. On the inclusion of analytic symplectic maps in analytic Hamiltonian flows and its applications. Seminar on Dynamical Systems. Eds. S. Kuksin, V. Lazutkin and J. Pöschel. Birkhäuser, Basel, 1978, pp. 96–116.Google Scholar
[11]Lam, P. F.. Embedding a differential homeomorphism in a flow. J. Differential Equations 30 (1978), 31–40.CrossRefGoogle Scholar
[12]Lam, P. F.. Embedding homeomorphisms in C 1-flows. Ann. Math. Pura Appl. 123 (1980), 11–25.CrossRefGoogle Scholar
[13]Li, W.. Normal Form Theory and its Applications. Science Press, Beijing, 2000.Google Scholar
[14]Li, W. and Lu, K.. Sternberg theorems for random dynamical systems. Comm. Pure Appl. Math. 58 (2005), 941–988.CrossRefGoogle Scholar
[15]Li, W., Llibre, J. and Zhang, X.. Extension of Floquet’s theory to nonlinear periodic differential systems and embedding diffeomorphisms in differential flows. Amer. J. Math. 124 (2002), 107–127.CrossRefGoogle Scholar
[16]Li, W., Llibre, J. and Zhang, X.. Local first integrals of differential systems and diffeomorphisms. Z. Angew. Math. Phys. 54 (2003), 235–255.CrossRefGoogle Scholar
[17]Rodrigues, H. M. and Solà-Morales, J.. Smooth linearization for a saddle on Banach spaces. J. Dynam. Differential Equations 16 (2004), 767–793.CrossRefGoogle Scholar
[18]Palis, J.. Vector fields generate few diffeomorphisms. Bull. Amer. Math. Soc. 80 (1974), 503–505.CrossRefGoogle Scholar
[19]Rudin, W.. Functional Analysis, 2nd edn. McGraw-Hill, New York, 1991.Google Scholar
[20]Zhang, X.. Analytic normalization of analytic integrable systems and the embedding flows. J. Differential Equations 244 (2008), 1080–1092.CrossRefGoogle Scholar