MINIMAL DYNAMICAL SYSTEM FOR Rn

The paper proves that the phase space of the universal minimal flow for R^n is homeomorphic to (M(Z^n) × I^n)/h by first identifying the universal ambit S(R^n) with (β(Z^n) × I^n)/h and then taking a minimal subflow of the form M × I^n/h with M ⊆ β(Z^n) minimal for the Z^n-action; since any minimal subflow of the universal ambit is a universal minimal flow, this yields M(R^n) ≅ (M(Z^n) × I^n)/h (Theorem 3; the boundary-toggling equivalence h and the ambit isomorphism are established in detail) . The candidate model derives the same identification via an induced (suspension) flow R^n ×_{Z^n} M(Z^n) and then gives an explicit homeomorphism to (M(Z^n) × I^n)/h using floor/frac and the h-identifications. Both arguments agree on the result and the space/action. The model, however, overstates a general identity M(G) ≅ G×_H M(H) for arbitrary closed H; this requires additional hypotheses (e.g., G/H compact) to ensure compactness of the induced space and surjectivity to arbitrary minimal G-flows. In the present cocompact case R^n/Z^n ≅ T^n, the model’s approach is valid, so both are correct, though obtained by different methods.