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

Abstract

Planar hyperbolic diffeomorphisms can be referred to two cases: Poincaré 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 Poincaré domain it was proved that $C^{1,α}$ smoothness with $α_0:=1-\log|λ_2|/\log|λ_1|<α\le 1$, where $λ_1$ and $λ_2$ are both eigenvalues such that $0<|λ_1|<|λ_2|<1$, admits $C^1$ linearization and the linearization is actually $C^{1,β}$. While a sharp Hölder exponent $β>0$ is given, an interesting problem is: Is the exponent $α_0$ also sharp? On the other hand, in Siegel domain we only know that $C^{1,α}$ smoothness with $α\in (0,1]$ admits $C^1$ linearization. In this paper we further study the sharpness for $C^1$ linearization in both cases.