Invariant curves of low smooth quasi-periodic reversible mappings

Abstract

In this paper, we obtain the invariant curves of quasi-periodic reversible mappings with finite smoothness. Since the reversible property is difficult to maintain in the process of approximating smooth functions by analytical ones, R\"{u}ssmann's method in \cite{HR} is invalid. Inspired by the recent work of Li, Qi and Yuan in \cite{LJ}, we turn to regard the reversible mapping as the Poincar\'{e} map of a reversible differential equation. By constructing a KAM theorem for a reversible differential equation which is quasi-periodic in time, we obtain the invariant curves of the reversible mapping. Beyond that, we establish some variants of invariant curve theorems for quasi-periodic reversible mappings.