Invariant Curves of Analytic Reversible Mappings Under Brjuno–Rüssmann’s Non-resonant Condition

In this paper we prove the existence of invariant curves for analytic reversible mappings under Brjuno–Rüssmann’s non-resonant condition. In the proof we use the polynomial structure of function to truncate, introduce a parameter q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document} and make the steps of KAM iteration infinitely small in the speed of function qnϵ,0<q<1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{n}\epsilon ,0 <q<1, $$\end{document} rather than super exponential function.