Topological characteristic factors and nilsystems

We prove that the maximal infinite step pro-nilfactor X_∞ of a minimal dynamical system (X‚ T) is the topological characteristic factor in a certain sense. Namely, we show that by an almost one-to-one modification of π: X → X_∞, the induced open extension π ^∗ : X ^∗ → X_∞^∗ has the following property: for x in a dense G_δ subset of X ^∗, the orbit closure L_x = O((x‚... ‚ x)‚ T × T ² × ‧ ‧ ‧ × T ^d ) is (π ^∗)^(d)-saturated, i.e., L_x = ((π ^∗)(d))^-1(π ^∗)^(d)(L_x). Using results derived from the above fact, we are able to answer several open questions: (1) if (X‚ T ^k ) is minimal for some k ≥ 2, then for any d 2 ℕ and any 0 ≤ j < k there is a sequence {n_i } of ℤ with n_i ≡ j (mod k) such that T ⁿⁱ x → x‚ T ²ⁿⁱ x → x‚... ‚ T ^{d ni} x → x for x in a dense G_δ subset of X; (2) if (X‚ T) is totally minimal, then {T ⁿ² x: n 2 ℤ} is dense in X for x in a dense G_δ subset of X; (3) for any d 2 ℕ and any minimal t.d.s. which is an open extension of its maximal distal factor, RP^[d] = AP^[d], where the former is the regionally proximal relation of order d and the latter is the regionally proximal relation of order d along arithmetic progressions.