前向或后向自相似系统的相互作用上同调
Interaction cohomology of forward or backward self-similar systems
摘要：
我们研究了前向或后向自相似系统（迭代函数系统）的动力学及其不变集的拓扑结构。我们定义了一个新的上同调理论（相互作用上同调）的前向或后向自相似系统。我们表明，在一定的条件下，不变集的连接组件的空间是同构的逆极限的连接组件的神经的实现的有限覆盖${\cal U} $不变集，其中每个${\cal U}$由（向后）图像的不变集有限字长的元素。特别地，我们给出了不变集连通的一个判据。此外，我们给出了第一上同调群具有无穷秩的一个充分条件。作为应用，我们得到了关于复平面上多项式生成半群动力学的许多结果。此外，我们定义了后无分支系统，并研究了这类系统的相互作用上同调群。列举了许多实例。
英文摘要：
We investigate the dynamics of forward or backward self-similar systems (iterated function systems) and the topological structure of their invariant sets. We define a new cohomology theory (interaction cohomology) for forward or backward self-similar systems. We show that under certain conditions, the space of connected components of the invariant set is isomorphic to the inverse limit of the spaces of connected components of the realizations of the nerves of finite coverings${\cal U} $ of the invariant set, where each ${\cal U}$ consists of (backward)images of the invariant set under elements of finite word length. Inparticular, we give a criterion for the invariant set to be connected. Moreover, we give a sufficient condition for the first cohomology group to have infinite rank. As an application, we obtain many results on the dynamics of finitely generated semigroups of polynomials on the complex plane. Moreover, we define postunbranched systems and we investigate the interaction cohomology groups of such systems. Many examples are given.
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