欧几里得空间中伸缩映射的不动点与混沌动力学及其应用
Fixed points and chaotic dynamics for expansive-contractive maps in Euclidean spaces, with some applications
摘要：
本文介绍了在有限维欧几里得空间中寻找定义在广义矩形上的连续映射的不动点和周期点的一种拓扑方法。我们将我们的技术命名为“沿着路径拉伸”方法，因为我们处理的是沿着一个方向扩展弧的地图。从动力学的角度来看，我们的技术也很重要，因为它可以检测复杂的动力学。特别是，我们能够证明半共轭的伯努利位移，从而积极的拓扑熵，存在的拓扑传递性和灵敏度相对于初始条件，密度的周期点。此外，我们的方法，虽然数学上严格，避免使用复杂的拓扑理论，它是相对容易适用于特定的模型中产生的应用。例如，我们在这里采用了沿路径拉伸方法来研究经济学和生物学中的离散和连续时间模型。
英文摘要：
In this work we introduce a topological method for the search of fixed points and periodic points for continuous maps defined on generalized rectangles in finite dimensional Euclidean spaces. We name our technique "Stretching Along the Paths" method, since we deal with maps that expand the arcs along one direction. Our technique is also significant from a dynamical point of view, as it allows to detect complex dynamics. In particular, we are able to prove semi-conjugacy to the Bernoulli shift and thus positivity of the topological entropy, the presence of topological transitivity and sensitivity with respect to initial conditions, density of periodic points. Moreover, our approach, although mathematically rigorous, avoids the use of sophisticated topological theories and it is relatively easy to apply to specific models arising in the applications. For example we have here employed the Stretching along the paths method to study discrete and continuous-time models arising from economics and biology.
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