We study fibrations in the category of cubespaces/nilspaces. We show that a fibration of finite degree \(f :X\rightarrow Y\) between compact ergodic gluing cubespaces (in particular nilspaces) factors as a (possibly countable) tower of compact abelian Lie group principal fiber bundles over Y. If the structure groups of f are connected then the fibers are (uniformly) isomorphic (in a strong sense) to an inverse limit of nilmanifolds. In addition we give conditions under which the fibers of f are isomorphic as subcubespaces. We introduce regionally proximal equivalence relations relative to factor maps between minimal topological dynamical systems for an arbitrary acting group. We prove that any factor map between minimal distal systems is a fibration and conclude that if such a map is of finite degree then it factors as a (possibly countable) tower of principal abelian Lie compact group extensions, thus achieving a refinement of both the Furstenberg’s and the Bronstein–Ellis structure theorems in this setting.

我们研究立方空间/零空间类别中的纤维化。我们表明，有限度的纤维化\（F：X \ RIGHTARROW Y \）紧凑遍历胶合cubespaces之间（特别是nilspaces）因素，如紧凑阿贝尔李群主纤维束的上方（可能可数）塔ÿ。如果连接了f的结构组，则纤维（均匀地）同构（在强烈意义上）为反硝苯胺极限。另外，我们给出f的纤维的条件同构为子多维空间。我们介绍了相对于任意动作组的最小拓扑动力学系统之间的因子映射的区域近端等效关系。我们证明最小远端系统之间的任何因素图都是纤维化，并得出结论，如果这样的图是有限度的，则它作为主要的阿贝尔李氏密群扩展的（可能是可数的）塔，从而实现了两种Furstenberg的改进在这种情况下，Bronstein-Ellis构造定理。