Sharpness for C1 linearization of planar hyperbolic diffeomorphisms

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:257

Article

Zhang, Wenmeng¹  Zhang, Weinian^2,3 

[1] Chongqing Normal Univ, Coll Math Sci, Chongqing 400047, Peoples R China

[2] Sichuan Univ, Yangtze Ctr Math, Chengdu 610064, Sichuan, Peoples R China

[3] Sichuan Univ, Dept Math, Chengdu 610064, Sichuan, Peoples R China

关键词: C-1 linearization;    Hyperbolic diffeomorphism;    Invariant manifold;    Functional equation;    Whitney extension theorem;   

DOI  :  10.1016/j.jde.2014.08.014

来源: Elsevier

【 摘 要 】

C-1 linearization preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions. Planar hyperbolic diffeomorphisms are the most elementary ones of representatively technical difficulties in the study of C-1 linearization. In the Poincare domain (both eigenvalues inside the unit circle S-1) a lower bound alpha(0) was given such that C-1,C-alpha smoothness with alpha(0) < alpha <= 1 admits C-1 linearization. Our first purpose of this paper is to prove the sharpness of cep and give a weaker linearization for a < cep. In the Siegel domain (one eigenvalue inside S-1 but the other outside S-1) it is known that C-1,C-alpha smoothness admits C-1 linearization for all alpha is an element of (0,1]. The second purpose is to prove that the C-1 linearization is actually a C-1,C-beta linearization and give sharp estimates for beta. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free