In this paper, we investigate the dynamical properties associated with planar maps which can be represented as a composition of twist maps together with expansive–contractive homeomorphisms. The class of maps we consider present some common features both with those arising in the context of the Poincaré–Birkhoff theorem and those studied in the theory of topological horseshoes. In our main theorems, we show that the multiplicity results of fixed points and periodic points typical of the Poincaré–Birkhoff theorem can be recovered and improved in our setting. In particular, we can avoid assuming area-preserving conditions and we also obtain higher multiplicity results in the case of multiple twists. Applications are given to periodic solutions for planar systems of non-autonomous ODEs with sign-indefinite weights, including the non-Hamiltonian case. The presence of complex dynamics is also discussed. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

中文翻译：

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具有多个扭曲的平面地图的固定点，适用于具有不确定权重的非线性方程

在本文中，我们研究了与平面映射相关的动力学特性，它可以表示为扭曲映射与膨胀-收缩同胚的组合。我们考虑的地图类别呈现出一些共同特征，这些特征既有在庞加莱-伯克霍夫定理的背景下产生的特征，也有在拓扑马蹄铁理论中研究的特征。在我们的主要定理中，我们展示了 Poincaré-Birkhoff 定理典型的不动点和周期点的多重性结果可以在我们的设置中恢复和改进。特别是，我们可以避免假设区域保持条件，并且在多次扭曲的情况下我们也可以获得更高的多重性结果。应用了具有符号不定权重的非自治 ODE 平面系统的周期解，包括非哈密顿情况。还讨论了复杂动力学的存在。本文是主题问题“微分和差分方程中的拓扑度和不动点理论”的一部分。