On ergodicity and symmetry for affine systems

Let f be a volume-preserving affine diffeomorphism of a compact homogeneous space with smooth centralizer Z∞(f). We establish the following two equivalences: (1) f is ergodic if and only if Z∞(f) is a finite-dimensional Lie group; (2) f is non-ergodic if and only if Z∞(f) contains either Cc∞((0,1),R) or Diffc∞((0,1)). This strengthens the corresponding perturbative version established in arXiv:2504.09084. We also prove analogues of the above equivalences for affine-induced diffeomorphisms on bi-homogeneous spaces, frame flows over negatively curved manifolds, and automorphisms of Kummer-type complex surfaces.