Existence of invariant curves for degenerate almost periodic reversible mappings (Q2171889)

Consider an area-preserving twist mapping \((x_1, y_1) =(x+\alpha(y)+\phi_1(x,y), y+\phi_2(x,y))\) on the infinite annulus \(\mathbb{R}\times [a,b]\). In the case that \(\phi_1=\phi_2=0\), the system is completely integrable and the annulus is foliated by invariant curves. The problem about the existence of invariant curves under perturbations has been extensively studied since the 1960s. If the perturbations \(\phi_1\) and \(\phi_2\) are periodic in \(x\), the existence of invariant curves has been proved in various situations, see [\textit{J. Moser}, Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 1962, 1--20 (1962; Zbl 0107.29301); \textit{H. Rüssman}, Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 1970, 67--105 (1970; Zbl 0201.11202); \textit{M. R. Herman}, Sur les courbes invariantes par les difféomorphismes de l'anneau. Vol. 1. Complété par un appendice au chapitre 1 de Albert Fathi. Paris: Société Mathématique de France (SMF) (1983; Zbl 0532.58011); Sur les courbes invariantes par les difféomorphismes de l'anneau. Volume 2. Paris: Société Mathématique de France (SMF) (1986; Zbl 0613.58021)]. In the more general case that the perturbations \(\phi_1\) and \(\phi_2\) are quasi-periodic in \(x\), the existence of invariant curves has also been established in many cases, see [\textit{V. Zharnitsky}, Nonlinearity 13, No. 4, 1123--1136 (2000; Zbl 1004.37026); \textit{B. Liu}, Nonlinearity 18, No. 2, 685--701 (2005; Zbl 1067.37080); \textit{P. Huang} et al., Discrete Contin. Dyn. Syst. 38, No. 1, 131--154 (2018; Zbl 1372.37088)]. The even more general case that the perturbations \(\phi_1\) and \(\phi_2\) are almost periodic in \(x\) has been studied in [\textit{P. Huang} et al., J. Dyn. Differ. Equations 34, No. 3, 1997--2033 (2022; Zbl 1506.37059); \textit{D. Piao} and \textit{X. Zhang}, ``Invariant curves of almost periodic reversible mappings'', Preprint, \url{arXiv:1807.06304}]. In the present paper, the author studies almost periodic reversible mappings with higher order degeneracy of the twist condition and proves the existence of invariant curves for such mappings.