Smooth linearization of nonautonomous dynamics on the line

Abstract

The aim of this paper is to study linearization for a sequence of $C^{r,\alpha }$ $(r\ge 1)$ maps on the line corresponding to a class of non-autonomous dynamics with discrete time. We obtain the following results: (i) if $r+\alpha >1$, then there exists a $C^{r,\alpha }$ linearization for $C^{r,\alpha }$ hyperbolic systems; (ii) for $C^1$ hyperbolic systems, then there is a $C^{0,\beta }$ linearization for any β with $0<\beta <1$. Moreover, by presenting a concrete example, we demonstrate that in case (ii), the result is the best. As a special case, we also present a detailed investigation on periodic difference equations in this paper.