In this paper we study the cohomological Conley index of arbitrary isolated invariant continua for continuous maps $$f :U \subseteq {\mathbb {R}}^d \rightarrow {\mathbb {R}}^d$$ f : U ⊆ R d → R d by analyzing the topological structure of their unstable manifold. We provide a simple dynamical interpretation for the first cohomological Conley index, describing it completely, and relate it to the cohomological Conley index in higher degrees. A number of consequences are derived, including new computations of the fixed point indices of isolated invariant continua in dimensions 2 and 3. Our approach exploits a certain attractor–repeller decomposition of the unstable manifold, reducing the study of the cohomological Conley index to the relation between the cohomology of an attractor and its basin of attraction. This is a classical problem that, in the present case, is particularly difficult because the dynamics is discrete and the topology of the unstable manifold can be very complicated. To address it we develop a new method that may be of independent interest and involves the summation of power series in cohomology: if Z is a metric space and $$K \subseteq Z$$ K ⊆ Z is a compact, global attractor for a continuous map $$g :Z \rightarrow Z$$ g : Z → Z , we show how to interpret series of the form $$\sum _{j \ge 0} a_j (g^*)^j$$ ∑ j ≥ 0 a j ( g ∗ ) j as endomorphisms of the cohomology group of the pair ( Z , K ).

中文翻译：

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上同调中的无限级数：吸引子和康利指数

在本文中，我们研究了连续映射的任意孤立不变连续体的上同调 Conley 指数 $$f :U \subseteq {\mathbb {R}}^d \rightarrow {\mathbb {R}}^d$$ f : U ⊆ R d → R d 通过分析其不稳定流形的拓扑结构。我们为第一个上同调 Conley 指数提供了一个简单的动力学解释，完整地描述了它，并在更高程度上将它与上同调 Conley 指数联系起来。推导出了许多结果，包括对维度 2 和 3 中孤立不变连续体的不动点指数的新计算。我们的方法利用了不稳定流形的某种吸引子 - 排斥子分解，将上同调康利指数的研究减少到关系吸引子的上同调与其吸引盆之间的关系。这是一个经典的问题，在目前的情况下，特别困难，因为动力学是离散的，不稳定流形的拓扑结构可能非常复杂。为了解决这个问题，我们开发了一种可能具有独立意义的新方法，并涉及上同调中幂级数的求和：如果 Z 是一个度量空间并且 $$K \subseteq Z$$ K ⊆ Z 是一个紧凑的全局吸引子连续映射 $$g :Z \rightarrow Z$$ g : Z → Z ，我们展示了如何解释 $$\sum _{j \ge 0} a_j (g^*)^j$$ ∑ j 形式的级数≥ 0 aj ( g ∗ ) j 作为对 ( Z , K ) 的上同调群的自同态。