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Abelian group Assume Borel closed subgroup coboundary cocycle cocycle extensions compact Abelian group continuous map Corollary decomposition defined Definition denote disjoint dynamical system equivalent ergodic cocycle ergodic theory essential values exists factor relations family of factors G a locally g G G graph group extension group of essential Haar measure Hausdorff space hence homomorphism idempotent integer irrational rotation isomorphism Lebesgue space Lemma Let X,T linear subspace locally compact Abelian measure-theoretic metric monothetic group minimal ideal minimal rotation minimal set minimal subset Moreover nonempty open numbers open sets p x p o periodic point Proposition proximal quotient regular relatively weakly mixing rigidity time nt)t^i S G C(T satisfying second countable self-joinings semisimple semisimple automorphisms sequence subspace Suppose T-flow T-invariant Theorem topological dynamics topological group topologically ergodic Tv,g U C X uniformly