Smooth dependence of nonautonomous linearization on parameters

Abstract

Barreira and Valls (2010) proved in [4] the C1 dependence of C0 linearization on parameters for nonuniformly hyperbolic diffeomorphisms, whose nonlinear parts depend on the parameters but linear parts do not. In this paper, we further allow the linear parts to depend on the parameters, where only Hölder dependence is possible and a difficulty arises: the projections in the nonuniform exponential dichotomy do not have explicit expressions. We will overcome the difficulty by giving a new result on the Hölder dependence of nonuniform exponential dichotomy and constructing the conjugacy in a different way. Moreover, under Barreira and Valls's framework we will improve the C1 dependence to Cr, where another difficulty arises: without the non-resonant condition, the systems do not have small derivatives of orders ≥2 in general. We will introduce functional spaces with small bounds to overcome the difficulty. Besides, our different constructing method ensures that the conjugacy is invertible, which was not considered by Barreira and Valls.