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Sharp regularity of linearization for C^{1,1} hyperbolic diffeomorphisms
Authors:Wenmeng Zhang  Weinian Zhang  Witold Jarczyk
Institution:1. Department of Mathematics, Yangtze Center of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, People’s Republic of China
2. College of Mathematics Science, Chongqing Normal University, Chongqing, 400047, People’s Republic of China
3. Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516, Zielona Góra, Poland
Abstract: $C^1$ linearization is of special significance because it preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions. However, $C^1$ smoothness is not enough to guarantee $C^1$ linearization. For $C^{1,1}$ hyperbolic diffeomorphisms on Banach spaces $C^1$ linearization was proved under a gap condition together with a band condition of the spectrum. In this paper, the result of $C^1$ linearization in Banach spaces is strengthened to $C^{1,\beta }$ linearization with a constant $\beta >0$ under a weaker band condition by a decomposition with invariant foliations. The weaker band condition allows the spectrum to be a union of more than two but finitely many bands but restricts those bands to be bounded by a number depending on the supremum of contractive spectrum and the infimum of expansive spectrum. Furthermore, we give an estimate for the exponent $\beta $ and prove that the estimate is sharp in the planar case.
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