We consider the general second order difference equation $ x_{n+1} = F(x_n, x_{n-1}) $ in which $ F $ is continuous and of mixed monotonicity in its arguments. In equations with negative terms, a persistent set can be a proper subset of the positive orthant, which motivates studying global stability with respect to compact invariant domains. In this paper, we assume that $ F $ has a semi-convex compact invariant domain, then make an extension of $ F $ on a rectangular domain that contains the invariant domain. The extension preserves the continuity and monotonicity of $ F. $ Then we use the embedding technique to embed the dynamical system generated by the extended map into a higher dimensional dynamical system, which we use to characterize the asymptotic dynamics of the original system. Some illustrative examples are given at the end.

中文翻译：

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二维单调映射的扩展，嵌入和全局稳定性

我们考虑一般的二阶差分方程$ x_ {n + 1} = F（x_n，x_ {n-1}）$，其中$ F $是连续的，并且在其参数中具有混合单调性。在带有负项的方程中，持久集可以是正正态的适当子集，这会激发人们对紧不变域的全局稳定性的研究。在本文中，我们假定$ F $具有半凸紧致不变域，然后在包含不变域的矩形域上扩展$ F $。扩展保留了$ F的连续性和单调性。$然后，我们使用嵌入技术将扩展图生成的动力学系统嵌入到高维动力学系统中，以用来表征原始系统的渐近动力学。最后给出一些说明性的例子。