Sharpness for C1C1 linearization of planar hyperbolic diffeomorphisms

C1C1 linearization preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions. Planar hyperbolic diffeomorphisms are the most elementary ones of representatively technical difficulties in the study of C1C1 linearization. In the Poincaré domain (both eigenvalues inside the unit circle S1S1) a lower bound α0α0 was given such that C1,αC1,α smoothness with α0<α≤1α0<α≤1 admits C1C1 linearization. Our first purpose of this paper is to prove the sharpness of α0α0 and give a weaker linearization for α≤α0α≤α0. In the Siegel domain (one eigenvalue inside S1S1 but the other outside S1S1) it is known that C1,αC1,α smoothness admits C1C1 linearization for all α∈(0,1]α∈(0,1]. The second purpose is to prove that the C1C1 linearization is actually a C1,βC1,β linearization and give sharp estimates for β.