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arXiv · 1305.4121

Sharp regularity of linearization for $C^{1,1}$ hyperbolic diffeomorphisms

Wenmeng Zhang·Weinian Zhang·Witold Jarczyk
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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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Wenmeng Zhang, Weinian Zhang, Witold Jarczyk. 2013-05-17. Sharp regularity of linearization for $C^{1,1}$ hyperbolic diffeomorphisms. https://arxiv.org/abs/1305.4121

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