We study fibrations in the category of cubespaces/nilspaces. We show
that a fibration of finite degree
$f\rightarrow Y$ 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 Furstenberg's and Bronshtein's
structure theorems.
Joint work with Bingbing Liang.