C1 linearization for planar contractions

摘要

C~1 linearization is of special interests because it can distinguish characteristic directions of dynamical systems. It is known that planar C~(1,α) contractions with a fixed point at the origin O admit C~(1,β) linearization with sufficiently small β>0 if α=1 and admit C1,α linearization if (log|λ_1|/log|λ_2|)-1<α≤1, where λ1 and λ2 are eigenvalues of the linear parts of the contractions at O with 0<|λ_1|≤|λ_2|<1. In this paper we improve the lower bound of α to lower the condition of C_1 linearization for planar contractions. Furthermore, we prove that the derivatives of transformations in our C_1 linearization are H?lder continuous and give estimates for the H?lder exponent. Finally, we give a counter example to show that those estimates cannot be improved anymore.