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 $C^{1,\alpha }$ 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 Hölder 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.

33 pages

Cited by in corpus (1)

• Linearization and H\" older Continuity for Nonautonomous Systems