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

TL;DR

This paper improves the understanding of linearization of hyperbolic diffeomorphisms in Banach spaces, showing that under weaker spectral conditions, $C^{1,eta}$ linearization is possible with sharp estimates for the exponent.

Contribution

It extends previous $C^1$ linearization results to $C^{1,eta}$ linearization under weaker spectral band conditions using invariant foliations.

Findings

Achieves $C^{1,eta}$ linearization under relaxed spectral conditions.

Provides sharp estimates for the Hölder exponent $eta$ in the planar case.

Shows the spectrum can be a union of multiple bounded bands.

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.