C1 linearization for planar contractions

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