Uniform enveloping semigroupoids for groupoid actions

We establish new characterizations for (pseudo)isometric extensions of topological dynamical systems. For such extensions, we also extend results about relatively invariant measures and Fourier analysis that were previously only known in the minimal case to a significantly larger class, including all transitive To bypass reliance on minimality classical approaches isometric via Ellis semigroup, show systems can be described as groupoid actions then adapt concept enveloping semigroups construct uniform semigroupoid actions. This approach allows deal with more complex orbit structures nonminimal study semigroupoids general translate back special In particular, that, under appropriate assumptions, action is if actually compact groupoid. provide an operator theoretic characterization based abstract Peter—Weyl-type theorem representations compact, groupoids Banach bundles which independent interest.