Topological characteristic factors and nilsystems

Abstract

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 set of X * , the orbit closureUsing 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 ∈ N and any 0 ≤ j < k there is a sequence(3) for any d ∈ N and any minimal system, which is an open extension of its maximal distal factor, RP [d] = AP [d] , where the latter is the regionally proximal relation of order d along arithmetic progressions.

On structure theorems and non-saturated examples

“…In [12], the authors further studied the topological characteristic factors and the dynamics of (N d (X ), G d (T )). Up to an almost 1-1 modification, they showed that the topological characteristic factors are the pro-nilfactors (see [12, Theorem A]), which are the analogies in the ergodic situation.…”

“…Another interesting and useful result they showed on the dynamics of (N d (X ), G d (T )) is the following theorem [12,Theorem C].…”

“…By this theorem, they showed that if (X , T 2 ) is minimal, then TMRT holds along odd numbers (actually they showed more, see [12,Theorem D]) and for a totally minimal system (X , T ) and an integral quadratic polynomial p(n) = an 2 + bn + c with a = 0, there is a dense G δ subset Ω of X such that for any x ∈ Ω, {T p (n) x : n ∈ Z} is dense in X (see [12,Theorem E]).…”

“…In this paper, we will give another approach to Theorem A. That is, starting from the fact that the maximal equicontinuous factor of N d (X ) is N d (X eq ), we use an equivalence condition for N d (T ) = N d (T n ) (Theorem 3.2) to the get the conclusion, instead of converting to the equicontinuous cases as did in [12].…”

“…Based on the results obtained in [12,17] one naturally expect that the maximal distal factor of N d (X ) should be N d (X dis ), where X dis is the maximal distal factor of X . We show that indeed it is the case.…”