Topological characteristic factors along cubes of minimal systems

In this paper we study the topological characteristic factors along cubes of minimal systems. It is shown that up to proximal extensions the pro-nilfactors are the topological characteristic factors along cubes of minimal systems. In particular, for a distal minimal system, the maximal $(d-1)$-step pro-nilfactor is the topological cubic characteristic factor of order $d$.


Introduction
This paper is motivated by the work of Glasner on topological characteristic factors in topological dynamics [7] and the work of Host and Kra on the multiple ergodic averages [9]. In [7], Glasner studied the topological characteristic factors along arithmetic progressions, and his work is the counterpart of Furstenberg's work [4] in topological dynamics. The present work is dedicated to the topological characteristic factors along cubes, which may be considered as the counterpart of [9] in topological dynamics.

Characteristic factors in ergodic theory.
The connection between ergodic theory and additive combinatorics was built in the 1970's with Furstenberg's beautiful proof of Szemerédi's theorem via ergodic theory [4]. Furstenberg [4] proved Szemerédi's theorem via the following multiple recurrence theorem: let T be a measure-preserving transformation on the probability space (X, X , µ), then for every integer d ≥ 1 and A ∈ X with positive measure, So it is natural to ask about the convergence of these averages, or more generally about the convergence in L 2 (X, µ) of the multiple ergodic averages (or called nonconventional averages) where f 1 , . . . , f d ∈ L ∞ (X, µ). After nearly 30 years' efforts of many researchers, this problem was finally solved in [9] (see [15] for an another proof).
In the study of multiple ergodic averages, the idea of characteristic factors play an very important role. This idea was suggested by Furstenberg in [4], and the notion of "characteristic factors" was first introduced in a paper by Furstenberg and Weiss [6]. Definition 1.1. [6] Let (X, X , µ, T ) be a measurable system and (Y, Y, µ, T ) be a factor of X. Let {p 1 , . . . , p d } be a family of integer valued polynomials, d ∈ N. We say that Y is a L 2 (resp. a.e.)-characteristic factor of X for the scheme {p 1 , . . . , p d } if for all f 1 , . . . , f d ∈ L ∞ (X, X , µ), in L 2 (X, X , µ) (resp. almost everywhere).
Finding a characteristic factor for a scheme often gives a reduction of the problem of evaluating limit behavior of multiple ergodic averages to special systems. The structure theorem of [9,15] states that for an ergodic system (X, X , µ, T ) if one wants to understand the multiple ergodic averages one can replace each function f i by its conditional expectation on some d − 1step pro-nilsystem (0-step system is a trivial system and 1-step pro-nilsystem is the Kroneker's one). Thus one can reduce the problem to the study of the same average in a nilsystem, i.e. reducing the average in an arbitrary system to a more tractable question.
In [9], lots of useful tools, such as dynamical parallelepipeds, ergodic uniformity seminorms etc., were introduced in the study of dynamical systems. One of main results of [9] is the following theorem of multiple ergodic averages along cubes. Theorem 1.2. [9, Theorem 1.2] Let (X, X , µ, T ) be an measure preserving probability system, and d ∈ N. Then for functions f ǫ ∈ L ∞ (µ), ǫ ∈ {0, 1} d , ǫ = (0, . . . , 0), the averages One may define the characteristic factor of (1.1) as defined in Definition 1.1. To prove theorem above the authors in [9] showed that the d-dimensional average along cubes has the same characteristic factor as the average along arithmetic progressions of length d − 1, which is a d − 1-step pro-nilsystem. The main result of the paper is to give the topological counterpart of this fact, that is, to show that pro-nilfactors are the topological characteristic factors along cubes of minimal systems.
Here is the definition of topological characteristic factors along arithmetic progressions: Let (X, T ) be a system and d ∈ N. Let π : (X, T ) → (Y, T ) be a factor map and σ d = T ×T 2 ×. . .×T d . (Y, T ) is said to be a topological characteristic factor (along arithmetic progressions) of order d if there exists a dense G δ set X 0 of X such that for each x ∈ X 0 the orbit closure L . In [7] it is said that this notation was suggested by Furstenberg and it is systematically studied. In [7], it is shown that up to a canonically defined proximal extension, a characteristic family for T × T 2 × . . . × T d is the family of canonical PI flows of class d − 1. In particular, if (X, T ) is a distal minimal system, then its largest class d − 1 distal factor is its topological characteristic factor of order d; if (X, T ) is a weakly mixing system (X, T ), then the trivial system is its topological characteristic factor. For more related results and details please refer to [7].
A unsolved problem is: Cojecture 1.5. If (X, T ) is a distal minimal system, then its maximal (d − 1)-step pro-nilfactor is its topological characteristic factor along arithmetic progressions of order d.

1.3.
Topological characteristic factors along cubes and main results of the paper.
First we define topological characteristic factors along cubes. The transformation group related to (1.1) is the face group F [d] . Please refer to next section for precise definition. Note that the group F [d] acts on X 2 d and it acts on the first coordinate as an identity map. Definition 1.6. Let (X, T ) be a system and d ∈ N. Let π : (X, T ) → (Y, T ) be a factor map. The system (Y, T ) is said to be a topological cubic characteristic factor of order d or topological characteristic factor along cubes of order d if there exists a dense G δ set X 0 of X such that for each x ∈ X 0 the set is the projection on the later 2 d − 1 coordinates. That is, for each x ∈ X 0 , One of the main results of the paper is that up to proximal extensions the maximal (d − 1)-step pro-nilfactor is the topological cubic characteristic factor of order d. To be precise, we will show the following theorem: Theorem 1.7. Let (X, T ) be a minimal system and d ∈ N. Let π : (X, T ) → (Z d−1 , T ) be the factor map to the maximal (d − 1)-step pro-nilfactor. Then there is a commutative diagram of homomorphisms of minimal flows is the topological cubic characteristic factor of order d of (X ′ , T ), where θ, θ ′ are proximal extensions.
When X is distal or weakly mixing, the proximal extensions θ, θ ′ in theorem are trivial (i.e. isomorphisms). That is: (1) Let (X, T ) be a minimal distal system and d ∈ N. Then the maximal (d − 1)step pro-nilfactor is the topological cubic characteristic factor of order d.
(2) Let (X, T ) be a minimal weakly mixing system and d ∈ N. Then the trivial system is the topological cubic characteristic factor of order d.
We do not know whether one may remove proximal extensions in Theorem 1.7, that is, we have the following question: Question 1.9. Let (X, T ) be a minimal system and d ∈ N. Is the maximal (d − 1)step pro-nilfactor the topological cubic characteristic factor of order d?

Preliminaries
In the article, integers, nonnegative integers and natural numbers are denoted by Z, Z + and N respectively. In the following subsections we give the basic background in topological dynamics necessary for the article.
2.1. Topological dynamical systems. By a topological dynamical system (TDS for short) we mean a pair (X, T ) where X is a compact metric space (with metric d) and T : X → X is a homeomorphism. For n ≥ 2 one writes (X n , T (n) ) for the n-fold product system (X × · · · × X, T × · · · × T ). The diagonal of X n is ∆ n (X) = {(x, . . . , x) ∈ X n : x ∈ X}.
When n = 2 one writes ∆ 2 (X) = ∆(X). The orbit of x ∈ X is given by O(x, T ) = {T n x : n ∈ Z}. For convenience, sometimes one denotes the orbit closure of A factor map π : X → Y between the TDS (X, T ) and (Y, S) is a continuous onto map which intertwines the actions; one says that (Y, S) is a factor of (X, T ) and that (X, T ) is an extension of (Y, S).
Generally, a topological dynamical systems is a triple X = (X, G, Π), where X is a compact T 2 space, G is a T 2 topological group and Π : G × X → X is a continuous map such that Π(e, x) = x and Π(s, Π(t, x)) = Π(st, x). We should fix G and suppress the action symbol. In lots of literatures, X is also called a topological transformation group or a flow. An analogous definition can be given if G is a semigroup. Also, the notions of transitivity, minimality and weak mixing are naturally generalized to group actions.

Cubes and faces.
Cube groups and face groups are introduced by Host and Kra in dynamical systems. Please refer to [9,10] for more details.
can be written as Hence x 0 is the first coordinate of x. As examples, points in X [2] are like (x 00 , x 10 , x 01 , x 11 ).
. We can also isolate the first coordinate, writing X

Dynamical parallelepipeds.
Definition 2.1. Let (X, T ) be a topological dynamical system and let d ≥ 1 be an integer. We define Q [d] (X) to be the closure in X [d] of elements of the form where n = (n 1 , . . . , n d ) ∈ Z d and x ∈ X. When there is no ambiguity, we write As examples, Q [2] is the closure in X [2] = X 4 of the set x ∈ X, m, n ∈ Z} and Q [3] is the closure in X [3] = X 8 of the set Definition 2.3. Face transformations are defined inductively as follows: Let For convenience, we denote the orbit closure of If (X, T ) is minimal, then for all

Nilmanifolds and nilsystems.
Let G be a group. For g, h ∈ G and A, B ⊂ G, we write [g, h] = ghg −1 h −1 for the commutator of g and h and [A, B] for the subgroup spanned by {[a, b] : a ∈ A, b ∈ B}. The commutator subgroups G j , j ≥ 1, are defined inductively by setting Let G be a d-step nilpotent Lie group and Γ be a discrete cocompact subgroup of G. The compact manifold X = G/Γ is called a d-step nilmanifold. The group G acts on X by left translations and we write this action as (g, x) → gx. The Haar measure µ of X is the unique probability measure on X invariant under this action. Let τ ∈ G and T be the transformation x → τ x of X. Then (X, µ, T ) is called a d-step nilsystem. In the topological setting we omit the measure and just say that (X, T ) is a d-step nilsystem.
We will need to use inverse limits of nilsystems, so we recall the definition of a sequential inverse limit of systems. If (X i , T i ) i∈N are systems with diam(X i ) ≤ 1 and π i : X i+1 → X i are factor maps, the inverse limit of the systems is defined to be the compact subset of i∈N X i given by where ρ i is the metric in X i . We note that the maps T i induce naturally a transformation T on the inverse limit.
A transitive system satisfying one of the equivalent properties above is called a system of order (d − 1) or a (d − 1)-step pro-nilsystem.
it is a distal pair if it is not proximal. Denote by P(X, T ) the set of proximal pairs of (X, T ). It is also called the proximal relation. A TDS (X, T ) is equicontinuous if for every ǫ > 0 there exists δ > 0 such that d(x, y) < δ implies d(T n x, T n y) < ǫ for every n ∈ Z. It is distal if P(X, T ) = ∆(X). Any equicontinuous system is distal.
Let (X, T ) be a minimal system. The regionally proximal relation RP(X, T ) is defined as: (x, y) ∈ RP if there are sequences x i , y i ∈ X, n i ∈ Z such that x i → x, y i → y and (T × T ) n i (x i , y i ) → (z, z), i → ∞, for some z ∈ X. It is well known that RP(X, T ) is an invariant closed equivalence relation and this relation defines the maximal equicontinuous factor X eq = X/RP(X, T ) of (X, T ) (for example see [14]).

2.6.
Regionally proximal relation of order d.
Definition 2.5. Let (X, T ) be a system and let d ∈ N. The points x, y ∈ X are said to be regionally proximal of order d if for any δ > 0, there exist x ′ , y ′ ∈ X and a vector n = (n 1 , . . . , n d ) ∈ Z d such that ρ(x, x ′ ) < δ, ρ(y, y ′ ) < δ, and In other words, there exists S ∈ F [d] such that ρ(S ǫ x ′ , S ǫ y ′ ) < δ for every ǫ ∈ {0, 1} d \ {0}. The set of regionally proximal pairs of order d is denoted by RP [d] (or by RP [d] (X, T ) in case of ambiguity), and is called the regionally proximal relation of order d. [2] ⊆ RP [1] = RP(X, T ).

It is easy to see that RP [d] is a closed and invariant relation. Observe that
The following theorems proved in [11] (for minimal distal systems) and in [13] (for general minimal systems) tell us conditions under which (x, y) belongs to RP [d] and the relation between RP [d] and d-step pro-nilsystems. In particular, the quotient of (X, T ) under RP [d] (X, T ) is the maximal d-step pronilfactor of X (i.e. the maximal factor of order d).
Let Z d = X/RP [d] (X, T ) and π d : (X, T ) → (Z d , T d ) be the factor map. Z 0 is the trivial system and Z 1 is the maximal equicontinuous factor X eq .

Some fundamental extensions.
Let (X, T ) and (Y, S) be TDS and let π : X → Y be a factor map. One says that An extension π is proximal if π(x 1 ) = π(x 2 ) implies (x 1 , x 2 ) ∈ P(X, T ), and π is distal if π(x 1 ) = π(x 2 ) and x 1 = x 2 implies (x 1 , x 2 ) ∈ P(X, T ). An extension π is almost one to one if there exists a dense G δ set X 0 ⊆ X such that π −1 ({π(x)}) = {x} for any x ∈ X 0 . It is easy to see that any almost one to one extension between minimal systems is proximal.

An extension π between minimal systems is called a relatively incontractible (RIC) extension if it is open and for every n ≥ 1 the minimal points are dense in the relation
R n π = {(x 1 , . . . , x n ) ∈ X n : π(x i ) = π(x j ), ∀ 1 ≤ i ≤ j ≤ n}. A distal extension between minimal systems is RIC and that a RIC extension is open. Every factor map between minimal systems can be lifted to a RIC extension by proximal extensions (see [3] or [14, Chapter VI]).
Theorem 2.8. Given a factor map π : X → Y between minimal systems (X, T ) and (Y, S) there exists a commutative diagram of factor maps (called RIC-diagram or EGS-diagram 1 ) π(x) = θ(y)} and θ ′ and π ′ are the restrictions to X ′ of the projections of X × Y ′ onto X and Y ′ respectively.

Topological characteristic factors along cubes
Let (X, T ) be a minimal system and d ∈ N. By Theorem 2.7, (Z d , T d ) = (X/RP [d] (X), T d ) is the maximal d-step pro-nilfactor of (X, T ). For convenience, we still use the symbol T as action on Z d instead of T d , that is, (Z d , T ) is the maximal d-step pro-nilfactor of (X, T ). Let π d : (X, T ) → (Z d , T ) be the factor map.
In this section we will prove the main results of the paper. First we will show that modulo proximal extensions d−1 -saturated. Then using this result we will prove that modulo proximal extensions the maximal (d − 1)-step pro-nilfactor (Z d−1 , T ) is the topological cubic characteristic factor of order d of (X, T ).

Parallelepiped Q [d] .
First we have the following useful lemma, which gives a condition when a point Lemma 3.1. Let (X, T ) be a minimal system and d ∈ N. Let π : (X, T ) → (Z d−1 , T ) be the factor map to the maximal (d−1)-step pro-nilfactor. If points x 1 , x 2 , . . . , x 2 d ∈ X satisfy the following coditions: Proof. We first prove the following claim.
Proof of Claim. Fix an i 0 ∈ {1, 2, . . . , 2 d }, we will show that ( By the definition of V 1 , it is easy to see that So there exists n 0 ∈ Z such that From the hypothesis, ( It follows that Thus there exists n 1 ∈ Z such that

It follows that
, one has that In particular, one has that Note that U i is arbitrary for each i ∈ {1, 2, . . . , 2 d }, by definition one has that (x i 0 , α * ) ∈ Q [d] (X). The proof of claim is completed.
By Lemma 3.1, one has the following corollary immediately.
Corollary 3.2. Let (X, T ) be a minimal system and d ∈ N. Let π : (X, T ) → (Z d−1 , T ) be the factor map to the maximal (d − 1)-step pro-nilfactor. If The result follows from Lemma 3.1. Now we have that if the factor map to the maximal (d − 1)-step pro-nilfactor is RIC, then Proposition 3.3. Let (X, T ) be a minimal system and d ∈ N. Let π : (X, T ) → (Z d−1 , T ) be the factor map to the maximal (d − 1)-step pro-nilfactor. If π is RIC, then -minimal point, then by Corollary 3.2, x ∈ Q [d] (X). Since π is RIC, the set of T [d] -minimal points is dense in R 2 d π and it follows that . The proof is completed.
We point out that we only use the fact that R π ⊂ RP [d −1] (X) in the proofs above. Since a distal extension is RIC, we have the following corollary.
Proposition 3.5. Z d−1 is the minimal factor such that Corollary 3.4 holds.
Generally, we have that modulo proximal extensions Proof. We only need to prove . The result follows from Proposition 3.3.

A counterexample.
Let π : (X, T ) → (Z d−1 , T ) be the factor map. We use the following classical system to show that without additional conditions, Q [d] (X) may not be π [d] -saturated.
Consider z ∈ T and define x ∈ {0, 1} Z by: for all n ∈ Z, x n = i if and only if R n α (z) ∈ A i . Let X ⊂ {0, 1} Z be the orbit closure of x under the shift map σ on {0, 1} Z , i.e. for any y ∈ {0, 1} Z , (σ(y)) n = y n+1 . This system is called Sturmian system. It is well known that (X, σ) is a minimal almost one-to-one extension of (T, R α ). Moreover, it is an asymptotic extension.
Let π : X → T be the former extension and consider (x 1 , x 2 ) ∈ R π \ ∆ X . Then (x 1 , x 2 ) is an asymptotic pair and thus (x 1 ,

Topological characteristic factors along cubes.
In this subsection we will use results developed above to show that up to proximal extensions the maximal (d − 1)-step pro-nilfactor is the topological cubic characteristic factor of order d. Before that, we use a different method to deal with distal systems.
First we need some lemmas. By the proof of [13, Theorem 3.1.] one can show the following lemma, which one can find another proof in [8].
It is easy to check that (x, x ′ ) ∈ R π . The proof is complete.
By Lemma 3.10 one can deal with distal systems. Corollary 3.11. Let (X, T ) be a minimal distal system and d ∈ N. Let π : (X, T ) → (Z d−1 , T ) be the factor map to the maximal (d − 1)-step pro-nilfactor. Then for each x ∈ X with y = π(x) one has that . In particular, Z d−1 is the topological cubic characteristic factor of order d.
Remark 3.12. In the definition of the topological cubic characteristic factor of order d, we require (3.1) holds for a dense G δ set. But for distal systems, Corollary 3.11 shows (3.1) holds for all x ∈ X.
By the method in [1] or [7,Section 4.], one can prove the following result. We omit the proof here, and please refer to [1] and [7] for more details about the methods. Lemma 3.13. Let (X, T ) be a minimal system and d ∈ N. There exists a dense G δ set X 0 ⊂ X such that for each x ∈ X 0 one has that Proposition 3.14. Let (X, T ) be a minimal system and d ∈ N. Let π : (X, T ) → (Z d−1 , T ) be the factor map to the maximal (d − 1)-step pro-nilfactor. If π is RIC, then Z d−1 is the topological cubic characteristic factor of order d. That is, there exists a dense G δ set X 0 ⊂ X such that for each x ∈ X 0 with y = π(x) one has that such that (Y ′ , T ) is the topological cubic characteristic factor of order d of (X ′ , T ), where θ, θ ′ are proximal extensions.
Proof. It follows from Theorem 3.6 and Proposition 3.14.
For weakly mixing systems, we have the following theorem, which was first proved in [13].
Corollary 3.16. [13] Let (X, T ) be a minimal weakly mixing system and d ∈ N.