Sharpness for $C^1$ linearization of planar hyperbolic diffeomorphisms

Abstract

Planar hyperbolic diffeomorphisms can be referred to two cases: Poincar\'{e} domain (both eigenvalues lie inside the unit circle $S^1$) and Siegel domain (one eigenvalue inside $S^1$ but the other outside $S^1$). In Poincar\'{e} domain it was proved that $C^{1,\alpha}$ smoothness with $\alpha_0:=1-\log|\lambda_2|/\log|\lambda_1|<\alpha\le 1$, where $\lambda_1$ and $\lambda_2$ are both eigenvalues such that $0<|\lambda_1|<|\lambda_2|<1$, admits $C^1$ linearization and the linearization is actually $C^{1,\beta}$. While a sharp H\"older exponent $\beta>0$ is given, an interesting problem is: Is the exponent $\alpha_0$ also sharp? On the other hand, in Siegel domain we only know that $C^{1,\alpha}$ smoothness with $\alpha\in (0,1]$ admits $C^1$ linearization. In this paper we further study the sharpness for $C^1$ linearization in both cases.