On the structure theory of cubespace fibrations

TL;DR

This paper investigates the structure of fibrations in cubespaces and nilspaces, revealing they can be decomposed into towers of abelian Lie group bundles, with implications for dynamical systems and their factor maps.

Contribution

It provides a detailed structural decomposition of fibrations in cubespaces and nilspaces, extending classical theorems to a broader dynamical systems context.

Findings

Fibrations of finite degree factor into towers of abelian Lie group principal bundles.

Fibers are isomorphic to inverse limits of nilmanifolds under certain conditions.

Factor maps between minimal distal systems are fibrations and decompose into abelian Lie group extensions.

Abstract

We study fibrations in the category of cubespaces/nilspaces. We show that a fibration of finite degree $f \colon X\rightarrow Y$ 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.