# On Hom-Leibniz and Hom-Lie-Yamaguti Superalgebras

Donatien Gaparayi ${}^{1},$ Sylvain Attan 2 and A. Nourou Issa 3

1 Ecole Normale Supérieure (E.N.S),
BP 6983 Bujumbura, Burundi
2,3 Département de Mathématiques
Université d’Abomey-calavi
01 BP 4521, Cotonou 01, Bénin
1<EMAIL_ADDRESS>2<EMAIL_ADDRESS>3<EMAIL_ADDRESS>
###### Abstract.

In this paper some characterizations of Hom-Leibniz superalgebras are given
and some of their basic properties are found. These properties can be seen as
a generalization of corresponding well-known properties of Hom-Leibniz
algebras. Considering the Hom-Akivis superalgebra asso-
ciated to a given Hom-Leibniz superalgebra, it is observed that the Hom-super
Akivis identity leads to an additional property of Hom-Leibniz superalgebras,
which in turn gives a necessary and sufficient condition for Hom-super Lie
admissibility of Hom-Leibniz superalgebras. We show also that every (left)
Hom-Leibniz superalgebra has a natural Hom-Lie-Yamaguti superalgebra
structure.

Key words: Hom-Leibniz superalgebras, Hom-Akivis superalgebras, Hom-Lie-
Yamaguti superalgebras
AMS Subject Class. (2010): 17A30, 17A32, 17D99

##  1\. Introduction

A (left) Leibniz superalgebra is an superalgebra $(L=L_{0}\oplus L_{1},\ast)$
satisfying the identity

(1) $x\ast(y\ast z)=(x\ast y)\ast z+(-1)^{|x||y|}y\ast(x\ast z).$

As Leibniz algebras which are been introduced by J. -L. Loday [16] (and so
they are sometimes called Loday algebras) as a noncommutative analogue of Lie
algebras, Leibniz superalgebras are also seen as supernoncommutative analogues
of Lie superalgebras. Indeed, if the operation $"\ast"$ of a given Leibniz
superalgebra $(L,\ast)$ is skew-supersymmetric, then $(L,\ast)$ is a Lie
superalgebra.

One of the problems in the general theory of a given class of (binary or
binary-ternary) nonassociative algebras is the study of relationships between
that class of algebras and the one of Lie algebras. In the same rule, the
search of relationships between a class of nonassociative algebras and the one
of Leibniz algebras is of interest (at least for constructing concrete
examples of the given class of nonassociative algebras). In this setting, the
existence of a Lie-Yamaguti structure on any (left) Leibniz algebra pointed
out in [15] is a good illustration. The counterpart of this construction in
the Hom-algebra setting has investigated in [7]. Indeed, the authors show that
every multiplicative left Hom-Leibniz algebra has a natural Hom-Lie-Yamaguti
structure. Hom-Lie-Yamaguti algebras (Hom-LY algebras) is introduced in [6].
Recall that a (left) Hom-Leibniz algebra [18] is a Hom-algebra $(L,.,\alpha)$
satisfying the (left) Hom-Leibniz identity

(2) $\alpha(x)\cdot(y\cdot z)=(x\cdot y)\cdot\alpha(z)+\alpha(y)\cdot(x\cdot
z)$

for all $x,y,z$ in $L$ and Hom-Lie Yamaguti algebra [6] (Hom-LY algebra for
short) is a quadruple $(L,\ast,\\{,,\\},\alpha)$ in which $L$ is
$\mathbb{K}$-vector space, $"\ast"$ a binary operation and $"\\{,,\\}"$ a
ternary operation on $L$, and $\alpha:L\rightarrow L$ a linear map such that

$\displaystyle(HLY1)\,\,\alpha(x\ast y)=\alpha(x)\ast\alpha(y),$
$\displaystyle(HLY2)\,\,\alpha(\\{x,y,z\\})=\\{\alpha(x),\alpha(y),\alpha(z)\\},$
$\displaystyle(HLY3)\,\,x\ast y=-y\ast x,$
$\displaystyle(HLY4)\,\,\\{x,y,z\\}=-\\{y,x,z\\},$
$\displaystyle(HLY5)\,\,\circlearrowleft_{(x,y,z)}[(x\ast
y)\ast\alpha(z)+\\{x,y,z\\}]=0,$
$\displaystyle(HLY6)\,\,\circlearrowleft_{(x,y,z)}[\\{x\ast
y,\alpha(z),\alpha(u)\\}]=0,$
$\displaystyle(HLY7)\,\,\\{\alpha(x),\alpha(y),u\ast
v\\}=\\{x,y,u\\}\ast\alpha^{2}(v)+\alpha^{2}(u)\ast\\{x,y,v\\},$
$\displaystyle(HLY8)\,\,\\{\alpha^{2}(x),\alpha^{2}(y),\\{u,v,w\\}\\}=\\{\\{x,y,u\\},\alpha^{2}(v),\alpha^{2}(w)\\}+\\{\alpha^{2}(u),\\{x,y,v\\},\alpha^{2}(w)\\}$
$\displaystyle\hskip
184.9429pt+\,\,\\{\alpha^{2}(u),\alpha^{2}(v),\\{x,y,w\\}\\},\,\,\forall\,\,u,v,w,x,y,z\in
L.$

Observe that if $x\ast y=0$, for all $x,y$ in $L$, then Hom-Lie-Yamaguti
algebras $(L,\ast,\\{,,\\},\alpha)$ reduce to a Hom-Lie triple system
$(L,\\{,,\\},\alpha)$ as defined in [25] and if $\\{x,y,z\\}=0$ for all
$x,y,z$ in $L$, then $(L,\ast,\\{,,\\},\alpha)$ is a Hom-Lie algebra
$(L,\ast,\alpha)$ [11].

The Hom-algebra structures arise first in quasi-deformation of Lie algebras of
vector fields. Discrete modifications of vector fields via twisted derivations
lead to Hom-Lie and quasi-Hom-Lie structures in which the Jacobi condition is
twisted [11]. A general study and construction of Hom-Lie algebras have been
considered (see [14],[20],[26]). Many researchers investigated Hom-structures
on different objects. Yau introduced Hom-Malcev algebras and gave the
connections between Hom-alternative algebras and Hom-Malcev algebras [22]. For
further more information on other Hom-type algebras, one may refer to, e.g.,
[4], [10], [17], [23], [24], [25].

In this paper, after considering some characterizations and basic properties
of Hom-Leibniz superalgebras, we show that every (left) Hom-Leibniz
superalgebra has a natural Hom- Lie-Yamaguti superalgebra structure. We
observe that the even part of this investigation recovers to the works done in
[7] and in [12].

The rest of this paper is organized as follows. In Section two, we recall
basic definitions in the Hom-superalgebras theory and useful results about
Hom-associative superalgebras and Hom-Lie superalgebras. In [1], the authors
show that the supercommutator bracket defined using the multiplication in a
Hom-associative superalgebra leads naturally to Hom-Lie superalgebra. In
particular, we recall the notion of Hom-Leibniz, Hom-Akivis and Hom-Lie-
Yamaguti superalgebras. In fact, in Proposition 2.6 we show that The
supercommutator-Hom-associator superalgebra of a multiplicative non-Hom-
associative superalgebra is a Hom-Akivis superalgebra. In the third Section,
super versions of some well-known properties of (left) Hom-Leibniz algebras
[12] are displayed. Consi-
dering the specific properties of the binary and ternary operations of the
Hom-Akivis superalgebra associated to a given Hom-Leibniz superalgebra, we
infer a characteristic property of Hom-Leibniz superalgebras. This property in
turn allows to give a necessary and sufficient condition for the super Hom-Lie
admissibility of these Hom-superalgebras. In the last Section, we prove the
existence of a Hom-Lie-Yamaguti superalgebra structure on any (multiplicative)
left Hom-Leibniz superalgebra. Indeed, every multiplicative (left) Hom-Leibniz
superalgebra has a natural Hom-Lie-Yamaguti superalgebra structure.

##  2\. Preliminaries

In this section, we recall useful definitions and results that are for further
use. In particular, we recall the notion of Hom-Leibniz, Hom-Akivis and Hom-
Lie-Yamaguti superalgebras.

######  Definition 2.1.

  * $(i)$

Let $f:(A,\ast,\alpha)\rightarrow(A^{\prime},\ast^{\prime},\alpha^{\prime})$
be a linear map, where $A=A_{0}\oplus A_{1}$ and
$A^{\prime}=A^{\prime}_{0}\oplus A^{\prime}_{1}$ are $\mathbf{Z}_{2}-$graded
vector spaces ( $\mathbb{K}$-vectors superspaces). The map $f$ is called an
even (resp. odd ) map if $f(A_{i})\subset A^{\prime}_{i}$ (resp.
$f(A_{i})\subset A^{\prime}_{i+1})$, for $i=0,1.$

  * $(ii)$

A (multiplicative)  $n$-ary Hom-superalgebra is a triple
$(A,\\{,\cdots,\\},\alpha)$ in which $A=A_{0}\oplus A_{1}$ is a
$\mathbb{K}$-vectors superspace, $\\{,\cdots,\\}:A^{n}\rightarrow A$ is an
$n$-linear map such that $\\{A_{i_{1}},A_{i_{2}},\cdots,A_{i_{n}}\\}\subseteq
A_{i_{1}+i_{2}+\cdots+i_{n}}$ and $\alpha:A\rightarrow A$ is an even linear
map such that
$\alpha(\\{x_{1},x_{2},\cdots,x_{n}\\})=\\{\alpha(x_{1}),\alpha(x_{2}),\cdots,\alpha(x_{n})\\}$
(multiplicativity).

We will be interested in binary ($n=2$), ternary ($n=3$) and binary-ternary
Hom-algebras (i.e. Hom-algebras with binary and ternary operations). For
convenience, throughout this paper we assume that all (binary, ternary and
binary-ternary) Hom-superalgebras are multiplicative and for a given vector
superspace $A$, the set of all homogeneous elements will be denoted by
$\mathcal{H}(A).$

######  Definition 2.2.

Let $(A,\ast,\alpha)$ be a Hom-superalgebra.

  * $(i)$

The Hom-associator of $A$ [2] is the trilinear map $as_{\alpha}:A\times
A\times A\rightarrow A$ defined as

(3) $as_{\alpha}=\ast\circ(\ast\otimes\alpha-\alpha\otimes\ast).$

In terms of homogeneous elements, the map $as_{\alpha}$ is given by

$as_{\alpha}(x,y,z)=(x\ast y)\ast\alpha(z)-\alpha(x)\ast(y\ast z).$

  * $(ii)$

A Hom-associative superalgebra [1] is a multiplicative Hom-superalgebra
$(A,\ast,\alpha)$ such that

$as_{\alpha}(x,y,z)=0,\,\,\,\forall\,\,x,y,z\in\mathcal{H}(A).$

However, the Hom-associativity is not always assumed, i.e. $as(x,y,z)\neq 0$
in general. In this case $(A,\cdot,\alpha)$ is said non-Hom-associative [13]
(or Hom-nonassociative [23]; in [19], $(A,\cdot,\alpha)$ is also called a
nonassociative Hom-superalgebra). This matches the generalization of
associative superalgebras by the nonassociative ones.

  * $(iii)$

The Hom-super-Jacobian of $A$ [2] is the trilinear map $J_{\alpha}:A\times
A\times A\rightarrow A$ defined as $J_{\alpha}(x,y,z):=(x\ast
y)\ast\alpha(z)+(-1)^{|x|(|y|+|z|)}(y\ast
z)\ast\alpha(x)+(-1)^{|z|(|x|+|y|)}(z\ast x)\ast\alpha(y)$ for all homogeneous
elements $x,y,z\in A.$ If $\alpha=Id$, then the Hom-super-Jacobian reduces to
the usual super-Jacobian.

  * $(iv)$

A Hom-Lie superalgebra [1] is a multiplicative Hom-superalgebra
$(A,\ast,\alpha)$ such that

$\displaystyle x\ast y=-(-1)^{|x||y|}y\ast x$ (4) $\displaystyle
J_{\alpha}(x,y,z)=0\,\,\,\mbox{(the Hom-super-Jacobi identity)}$

for all homogeneous elements $x,y,z\in A.$ If $\alpha=Id,$ a Hom-Lie
superalgebra reduces to an usual Lie superalgebra.

######  Definition 2.3.

[21] A (left) Hom-Leibniz superalgebra is a multiplicative Hom-superalgebra
$(L,\ast,\alpha)$ satisfying the (left) Hom-Leibniz superalgebra identity

(5) $\alpha(x)\ast(y\ast z)=(x\ast
y)\alpha(z)+(-1)^{|x||y|}\alpha(y)\ast(x\ast z)$

for all homogeneous elements $x,y,z\in L.$

######  Remark 2.4.

The dual superidentity of (5) is given by

(6) $(x\cdot y)\cdot\alpha(z)=\alpha(x)\cdot(y\cdot z)+(-1)^{|y||z|}(x\cdot
z)\cdot\alpha(y)$

which defines a (right) Hom-Leibniz superalgebra. Indeed, if the operation
$"\ast"$ verifies (5), then the operation $"\cdot"$ defined by $x\cdot y=y\ast
x$ verifies also (6) and then from a (left) Hom-Leibniz superalgebra, one can
get the (right) Hom-Leibniz superalgebra.

In terms of the Hom-associator, the identity (5) is written as

(7) $as(x,y,z)=-(-1)^{|x||y|}\alpha(y)\ast(x\ast z)$

Therefore, from Definition 2.2 in assertion $(ii)$, we see that Hom-Leibniz
superalgebras are examples of non-Hom-associative superalgebras.

######  Definition 2.5.

[8] A Hom-Akivis superalgebra is a multiplicative binary-ternary Hom-
superalgebra $(A,[,],\\{,,\\},\alpha)$ such that

$\displaystyle[x,y]$ $\displaystyle=$ $\displaystyle(-1)^{|x||y|}[y,x]$ (8)
$\displaystyle J_{\alpha}(x,y,z)$ $\displaystyle=$
$\displaystyle\circlearrowleft_{(x,y,z)}(-1)^{|x||z|}[x,y,z]-\circlearrowleft_{(x,y,z)}(-1)^{(|y|+|z|)|x|}\\{y,x,z\\},$

for all homogeneous elements $x,y,z$ in $A$. The identity (2.5) is called the
Hom-super Akivis identity.

Observe that when $\alpha=Id,$ the Hom-super Akivis identity (2.5) reduces to
the usual super Akivis identity (see in [3]).

######  Proposition 2.6.

The supercommutator-Hom-associator superalgebra of a multiplicative non-Hom-
associative superalgebra is a Hom-Akivis superalgebra.

Proof. Let $(A,\ast,\alpha)$ be a multiplicative non-Hom-associative
superalgebra. For any homogeneous elements $x,y,z\in A$, define the operations

$[x,y]:=x\ast y-(-1)^{|x||y|}y\ast x$ (supercommutator)

$\\{x,y,z\\}_{\alpha}:=as_{\alpha}(x,y,z)$ (Hom-associator; see (3)).
Then

$[[x,y],\alpha(z)]\\\ =(x\ast y)\ast\alpha(z)-(-1)^{|x||y|}(y\ast
x)\ast\alpha(z)-(-1)^{|z|(|x|+|y|)}\alpha(z)\ast(x\ast
y)+(-1)^{|x|(|z|+|y|)+|y||z|}\alpha(z)\ast(y\ast x)$ and

$\\{x,y,z\\}_{\alpha}-(-1)^{|x||y|}\\{y,x,z\\}_{\alpha}\\\ =(x\ast
y)\ast\alpha(z)-\alpha(x)\ast(y\ast z)-(-1)^{|x||y|}(y\ast
x)\ast\alpha(z)+(-1)^{|x||y|}\alpha(y)\ast(x\ast z).$ Expanding
$\circlearrowleft_{(x,y,z)}[[x,y],\alpha(z)]$ and
$\circlearrowleft_{(x,y,z)}(\\{x,y,z\\}_{\alpha}-(-1)^{|x||y|}\\{y,x,z\\}_{\alpha})$
respectively, where $\circlearrowleft_{(x,y,z)}$ denotes the sum over cyclic
permutation of $x,y,z,$ one gets (2.5) and so
$(A,[,],\\{,,\\}_{\alpha},\alpha)$ is a Hom-Akivis superalgebra. The
multiplicativity of $(A,[,],\\{,,\\}_{\alpha},\alpha)$ follows from the one of
$(A,\ast,\alpha).$ $\square$

Observed that to each non-Hom-associative superalgebra is associated a Hom-
Akivis superalgebra (this is the Hom-analogue of a similar relationship
between nonassociative superalgebras and Akivis superalgebras [3]). In this
note, in the third Section we use the specific properties of the Hom-Akivis
superalgebra associated to a given Hom-Leibniz superalgebra to derive a
property characterizing Hom-Leibniz superalgebras. Before this, we recall the
following.

######  Definition 2.7.

[9] A Hom-Lie Yamaguti superalgebra (Hom-LY superalgebra for short) is a
binary-ternary Hom-superalgebra $(L,\ast,\\{,,\\},\alpha)$ such that

$\displaystyle(SHLY1)\,\,\alpha(x\ast y)=\alpha(x)\ast\alpha(y),$
$\displaystyle(SHLY2)\,\,\alpha(\\{x,y,z\\})=\\{\alpha(x,)\alpha(y),\alpha(z)\\},$
$\displaystyle(SHLY3)\,\,x\ast y=-(-1)^{|x||y|}y\ast x,$
$\displaystyle(SHLY4)\,\,\\{x,y,z\\}=-(-1)^{|x||y|}\\{y,x,z\\},$
$\displaystyle(SHLY5)\,\,\circlearrowleft_{(x,y,z)}(-1)^{|x||z|}[(x\ast
y)\ast\alpha(z)+\\{x,y,z\\}]=0,$
$\displaystyle(SHLY6)\,\,\circlearrowleft_{(x,y,z)}(-1)^{|x||z|}[\\{x\ast
y,\alpha(z),\alpha(u)\\}]=0,$
$\displaystyle(SHLY7)\,\,\\{\alpha(x),\alpha(y),u\ast
v\\}=\\{x,y,u\\}\ast\alpha^{2}(v)+(-1)^{|u|(|x|+|y|)}\alpha^{2}(u)\ast\\{x,y,v\\},$
$\displaystyle(SHLY8)\,\,\\{\alpha^{2}(x),\alpha^{2}(y),\\{u,v,w\\}\\}=\\{\\{x,y,u\\},\alpha^{2}(v),\alpha^{2}(w)\\}$
$\displaystyle\hskip
184.9429pt+\,\,(-1)^{|u|(|x|+|y|)}\\{\alpha^{2}(u),\\{x,y,v\\},\alpha^{2}(w)\\}$
$\displaystyle\hskip
184.9429pt+\,\,(-1)^{(|x|+|y|)(|u|+|v|)}\\{\alpha^{2}(u),\alpha^{2}(v),\\{x,y,w\\}\\},$

for all homogeneous elements $u,v,w,x,y,z\in L$ and where
$\circlearrowleft_{(x,y,z)}$ denotes the sum over cyclic permutation of
$x,y,z.$

Note that the conditions $(SHLY1)$ and $(SHLY2)$ mean the multiplicativity of
$(L,\ast,\\{,,\\},\alpha).$

##  3\. Some characterizations of Hom-Leibniz superalgebras

In this section, super-versions of some well-known properties of (left) Hom-
Leibniz algebras [12] are displayed. Considering the specific properties of
the binary and ternary operations of the Hom-Akivis superalgebra associated to
a given Hom-Leibniz superalgebra, we infer a characteristic property of Hom-
Leibniz superalgebras. This property in turn allows to give a necessary and
sufficient condition for the Hom-super Lie admissibility of these Hom-
superalgebras.

######  Proposition 3.1.

Let $(L,\ast,\alpha)$ be a Hom-Leibniz superalgebra. Consider on
$(L,\ast,\alpha)$ the operations

(9) $\displaystyle[x,y]:=x\ast y-(-1)^{|x||y|}y\ast x$ (10)
$\displaystyle\\{x,y,z\\}:=as(x,y,z).$

Then $(L,[,],\\{,,\\})$ is a Hom-Akivis superalgebra.

Proof. The proof follows if observe that Proposition 3.1 is Proposition 2.6
using (7). Then, we have

(11) $J_{\alpha}(x,y,z)=\,\,\circlearrowleft_{(x,y,z)}(-1)^{|x||z|}(x\ast
y)\ast\alpha(z),$

for all homogeneous elements $x,y,z\in A.$ $\square$

One observes that (11) is the specific form of the Hom-super Akivis identity
(8) in case of Hom-Leibniz superalgebras. The relation above will be used in
the Section 4. The skew-supersymmetry of the operation $"\ast"$ of a Hom-
Leibniz superalgebra $(L,\ast,\alpha)$ is a condition for $(L,\ast,\alpha)$ to
be a Hom-Lie superalgebra [1]. From Proposition 3.1 one gets the following
necessary and sufficient condition for the Hom-super Lie admissibility [1] of
a given Hom-Lie superalgebra.

######  Corollary 3.2.

A Hom-Leibniz superalgebra $(L,\ast,\alpha)$ is Hom-super Lie admissible if
and only if $\circlearrowleft_{(x,y,z)}(-1)^{|x||z|}(x\ast y)\ast\alpha(z)=0$,
for all homogeneous elements $x,y,z$ in $L$.

In the following, some properties of Hom-Leibniz superalgebras are found.

######  Proposition 3.3.

Let $(L,\ast,\alpha)$ be a Hom-Leibniz superalgebra. Then

(12) $\displaystyle(i)\,\,\,(x\ast y+(-1)^{|x||y|}y\ast x)\ast\alpha(z)=0,$
(13) $\displaystyle(ii)\,\,\,\alpha(x)\ast[y,z]=[x\ast
y,\alpha(z)]+(-1)^{|x||y|}[\alpha(y),x\ast z],$

for all homogeneous elements $x,y,z\in L.$

Proof. The identity (5) implies that $(x\ast
y)\ast\alpha(z)=\alpha(x)\ast(y\ast z)-(-1)^{|x||y|}\alpha(y)\ast(x\ast z).$
Likewise, interchanging $x$ and $y$, we have $(y\ast
x)\ast\alpha(z)=\alpha(y)\ast(x\ast z)-\alpha(x)\ast(y\ast z).$ Then, adding
memberwise the equalities above, we come to the property $(i)$. Next we have

$\displaystyle[x\ast y,\alpha(z)]+(-1)^{|x||y|}[\alpha(y),x\ast z]$
$\displaystyle=$ $\displaystyle(x\ast
y)\ast\alpha(z)-(-1)^{|z|(|x|+|y|)}\alpha(z)\ast(x\ast y)$ $\displaystyle+$
$\displaystyle(-1)^{|x||y|}\alpha(y)\ast(x\ast z)-(-1)^{|y||z|}(x\ast
z)\ast\alpha(y)$ $\displaystyle=$ $\displaystyle\alpha(x)\ast(y\ast
z)-(-1)^{|z|(|x|+|y|)}\alpha(z)\ast(x\ast y)$ $\displaystyle-$
$\displaystyle(-1)^{|y||z|}(x\ast z\ast\alpha(y)\,\,\,\mbox{(by
(\ref{LLSI}))}$ $\displaystyle=$ $\displaystyle\alpha(x)\ast(y\ast
z)-(-1)^{|z|(|x|+|y|)}(z\ast x)\ast\alpha(y)$ $\displaystyle+$
$\displaystyle(-1)^{|x||z|}\alpha(x)\ast(z\ast y)-(-1)^{|y||z|}(x\ast
z)\ast\alpha(y)\,\,\,\mbox{(by (\ref{LLSI}))}$ $\displaystyle=$
$\displaystyle\alpha(x)\ast(y\ast z)-(-1)^{|y||z|}\alpha(x)\ast(z\ast
y)\,\,\,\mbox{(by (i))}$ $\displaystyle=$ $\displaystyle\alpha(x)\ast[y,z].$

and so we get $(ii)$. $\hfill\square$

######  Remark 3.4.

In Proposition 3.3, if $x,y,z$ are in $L_{0}$, then one recovers the well-
known properties of Hom-Leibniz algebras (see the Proposition 3.1. in [12]).

##  4\. Hom-Lie-Yamaguti superalgebra on Hom-Leibniz superalgebra

In this section, we prove the existence of a Hom-Lie-Yamaguti superalgebra
structure on any (multiplicative) left Hom-Leibniz superalgebra. The proof is
based on a specific ternary operation that can be considered on a given Hom-
Leibniz superalgebra (this product generalizes the ternary operation
considered in [7] on a left Hom-Leibniz algebra which generalizes the one in
[15] on a left Leibniz algebra $L$ that produces, along with the skew-
symmetrization, a Lie-Yamaguti algebra structure on $L$). Also note that our
proof below essentially relies on some properties characterizing Hom-Leibniz
superalgebras, obtained in Proposition 3.3. In the following we have

######  Theorem 4.1.

Every left Hom-Leibniz superalgebra has a natural Hom-Lie-Yamaguti
superalgebra structure.

Proof. In a left Hom-Leibniz algebra $(L,\ast,\alpha)$ consider the skew-
supersymmetrization

$[x,y]:=x\ast y-(-1)^{|x||y|}y\ast x$

for all homogeneous elements $x,y$ in L. In the following, consider the left
translations $\Lambda_{a}b:=a\ast b$ in $(L,\ast,\alpha)$. Then the identities
(5) and (13) can be written respectively as

(14) $\displaystyle\Lambda_{\alpha(x)}(y\ast
z)=(\Lambda_{x}y)\ast\alpha(z)+(-1)^{|x||y|}\alpha(y)\ast(\Lambda_{x}z),$ (15)
$\displaystyle\Lambda_{\alpha(x)}[y,z]=[\Lambda_{x}y,\alpha(z)]+(-1)^{|x||y|}[\alpha(y),\Lambda_{x}z].$

In $(L,\ast,\alpha)$ consider the following ternary operation:

(16) $\\{x,y,z\\}:=as_{\alpha}(y,x,z)-as_{\alpha}(x,y,z)$

for all homogeneous elements $x,y,z$ in $L.$ Then (16), (5) and (7) imply

(17) $\\{x,y,z\\}=-(x\ast y)\ast\alpha(z).$

Moreover, we have

(18) $\displaystyle[x,y]\ast\alpha(z)$ $\displaystyle=$ $\displaystyle(x\ast
y-(-1)^{|x||y|}y\ast x)\ast\alpha(z)$ $\displaystyle=$ $\displaystyle 2(x\ast
y)\ast\alpha(z)\,\,\mbox{(by (\ref{Prop2}))}$ $\displaystyle=$
$\displaystyle-2\\{x,y,z\\}\,\,\mbox{(see (\ref{TLSR3}))}$

so that

(19) $\\{x,y,z\\}=-\frac{1}{2}[x,y]\ast\alpha(z)$

Thus (16), (17) and (19) are different expressions of the operation
”$\\{,,\\}$” that are for use in what follows. Now we proceed to verify the
validity on $(L,\ast,\alpha)$ of the set of identities $(SHLY1)-(SHLY8).$ The
multiplicativity of $(L,\ast,\alpha)$ implies $(SHLY1)$ and $(SHLY2)$ while
$(SHLY3)$ is the skew-supersymmetrization and $(SHLY4)$ clearly follows from
(16) (or (17)). Next, observe that $(HLY5)$ is just the Hom-super Akivis
identity (2.5) for $(L,\ast,\alpha).$

Consider now $\circlearrowleft_{(x,y,z)}\\{[x,y],\alpha(z),\alpha(u)\\}$. Then

$\displaystyle\circlearrowleft_{(x,y,z)}(-1)^{|x||z|}\\{[x,y],\alpha(z),\alpha(u)\\}$
$\displaystyle=$
$\displaystyle\circlearrowleft_{(x,y,z)}(-1)^{|x||z|}(-[x,y]\ast\alpha(z))\ast\alpha^{2}(u)\quad(\mbox{by
(\ref{TLSR3})})$ $\displaystyle=$ $\displaystyle
2(\circlearrowleft_{(x,y,z)}(-1)^{|x||z|}\\{x,y,z\\})\ast\alpha^{2}(u)\quad(\mbox{by
(\ref{TLSR5})})$ $\displaystyle=$ $\displaystyle-2((-1)^{|x||z|}(x\ast
y)\ast\alpha(z)+(-1)^{|x||y|}(y\ast z)\ast\alpha(x)+(-1)^{|y||z|}(z\ast
x)\ast\alpha(y))\ast\alpha^{2}(u)$ $\displaystyle=$
$\displaystyle-2((-1)^{|x||z|}\alpha(x)\ast(y\ast
z)-(-1)^{|x|(|y|+|z|)}\alpha(y)\ast(x\ast z)+(-1)^{|x||y|}(y\ast
z)\ast\alpha(x)$ $\displaystyle+(-1)^{|y||z|}(z\ast
x)\ast\alpha(y))\ast\alpha^{2}(u)\quad(\mbox{by (\ref{LLSI})})$
$\displaystyle=$ $\displaystyle-2(-1)^{|x||z|}(\alpha(x)\ast(y\ast
z)+(-1)^{|x|(|y|+|z|)}(y\ast z)\ast\alpha(x))\ast\alpha^{2}(u)$
$\displaystyle-2(-(-1)^{|x|(|y|+|z|)}\alpha(y)\ast(x\ast
z)+(-1)^{|y||z|}(z\ast x)\ast\alpha(y))\ast\alpha^{2}(u)$ $\displaystyle=$
$\displaystyle-2(-(-1)^{|x|(|y|+|z|)}\alpha(y)\ast(x\ast
z)+(-1)^{|y||z|}(z\ast x)\ast\alpha(y))\ast\alpha^{2}(u)\quad(\mbox{by
(\ref{Prop2})})$ $\displaystyle=$
$\displaystyle-2(-(-1)^{|x|(|y|+|z|)}\alpha(y)\ast(x\ast
z)-(-1)^{|z|(|x|+|y|)}(x\ast z)\ast\alpha(y))\ast\alpha^{2}(u)\quad(\mbox{by
(\ref{Prop2})})$ $\displaystyle=$ $\displaystyle
2(-1)^{|x|(|y|+|z|)}(\alpha(y)\ast(x\ast z)+(-1)^{|y|(|x|+|z|)}(x\ast
z)\ast\alpha(y))\ast\alpha^{2}(u)$ $\displaystyle=$ $\displaystyle
0\quad(\mbox{by (\ref{Prop2})})$

so that we get $(SHLY6).$ Next

$\displaystyle\\{\alpha(x),\alpha(y),[u,v]\\}$ $\displaystyle=$
$\displaystyle-\alpha(x\ast y)\ast\alpha([u,v])\quad(\mbox{by (\ref{TLSR3})
and multiplicativity})$ $\displaystyle=$ $\displaystyle\Lambda_{(-\alpha(x\ast
y))}[\alpha(u),\alpha(v)]$ $\displaystyle=$ $\displaystyle[\Lambda_{(-x\ast
y)}\alpha(u),\alpha^{2}(v)]+(-1)^{|u|(|x|+|y|)}[\alpha^{2}(u),\Lambda_{(-x\ast
y)}\alpha(v)]\quad(\mbox{by (\ref{TLSR0})})$ $\displaystyle=$
$\displaystyle[\\{x,y,u\\},\alpha^{2}(v)]+(-1)^{|u|(|x|+|y|)}[\alpha^{2}(u),\\{x,y,v\\}]\quad(\mbox{by
(\ref{TLSR3})})$

which is $(SHLY7).$ Finally, we compute

$\displaystyle\\{\\{x,y,u\\},\alpha^{2}(v),\alpha^{2}(w)\\}+(-1)^{|u|(|x|+|y|)}\\{\alpha^{2}(u),\\{x,y,v\\},\alpha^{2}(w)\\}$
$\displaystyle+(-1)^{(|u|+|v|)(|x|+|y|)}\\{\alpha^{2}(u),\alpha^{2}(v),\\{x,y,w\\}\\}$
$\displaystyle=$ $\displaystyle\\{-\Lambda_{x\ast
y}\alpha(u),\alpha^{2}(v),\alpha^{2}(w)\\}+(-1)^{|u|(|x|+|y|)}\\{\alpha^{2}(u),-\Lambda_{x\ast
y}\alpha(v),\alpha^{2}(w)\\}$
$\displaystyle+(-1)^{(|u|+|v|)(|x|+|y|)}\\{\alpha^{2}(u),\alpha^{2}(v),-\Lambda_{x\ast
y}\alpha(w)\\}$ $\displaystyle=$ $\displaystyle((-\Lambda_{x\ast
y}\alpha(u))\ast\alpha^{2}(v)).\alpha^{3}(w)-(-1)^{|u|(|x|+|y|)}(\alpha^{2}(u)\ast(-\Lambda_{x\ast
y}\alpha(v)))\ast\alpha^{3}(w)$
$\displaystyle-(-1)^{(|u|+|v|)(|x|+|y|)}(\alpha^{2}(u)\ast\alpha^{2}(v))\alpha(-\Lambda_{x\ast
y}\alpha(w))$ $\displaystyle=$ $\displaystyle(\Lambda_{\alpha(x\ast
y)}\alpha(u\ast v))\ast\alpha^{3}(w)+(-1)^{(|u|+|v|)(|x|+|y|)}\alpha^{2}(u\ast
v)\ast\Lambda_{\alpha(x\ast y)}\alpha^{2}(w)\quad(\mbox{by (\ref{TLSR0}) and
multiplicativity})$ $\displaystyle=$ $\displaystyle\Lambda_{\alpha^{2}(x\ast
y)}(\alpha(u\ast v)\ast\alpha^{2}(w))\quad(\mbox{by (\ref{TLSR1})})$
$\displaystyle=$ $\displaystyle-\alpha^{2}(x\ast y)\ast(-\alpha(u\ast
v)\ast\alpha^{2}(w))$ $\displaystyle=$
$\displaystyle-(\alpha^{2}(x)\ast\alpha^{2}(y))\ast\alpha(-(u\ast
v)\ast\alpha(w))\quad(\mbox{by multiplicativity})$ $\displaystyle=$
$\displaystyle\\{\alpha^{2}(x),\alpha^{2}(y),\\{u,v,w\\}\\}\quad(\mbox{by
(\ref{TLSR3})})$

Therefore $(L,[,],\\{,,\\},\alpha)$ is a Hom-LY superalgebra. This completes
the proof. $\square$

## References

  * [1] , K. Abdaoui, F. Ammar and A. Makhlouf, Hom-Lie Superalgebras and Hom-Lie admissible Superalgebras, Journal of algebra, Vol. 324, Issue 7, 1513-1528 (2010).
  * [2] , K. Abdaoui, F. Ammar and A. Makhlouf, Hom-alternative, Hom-Malcev and Hom-Jordan Superalgebras, arXiv: 1304.1579v1 [math.RA] Apr 2013.
  * [3] H. Albuquerque A. P. Santana Akivis superalgebras and speciality, arXiv: 0710.4535v1 [math.RA] 24 oct 2007.
  * [4] H. Ataguema, A. Makhlouf and S.D. Silvestrov, Generalization of n-ary Nambu algebras and beyond, J. Math. Phys., 50 (2009), 083501.
  * [5] Elhamdadi M, Makhlouf A. Deformations of Hom-Alternative and Hom-Malcev algebras, Algebras Groups Geom, 2011, 28(2): 117–145
  * [6] D. Gaparayi and A. N. Issa, A Twisted Generalization of Lie-Yamaguti Algebras, Int. J. Algebra, Vol. 6 (7) (2012), 339-352.
  * [7] D. Gaparayi and A. N. Issa,Hom-Lie-Yamaguti structures on Hom-Leibniz algebras, Extracta Math., vol. 28 (2013), Num. 1, 1-12.
  * [8] D. Gaparayi and A. N. Issa, Hom-Akivis superalgebras, Journal of Algebra and computational applications 6(2017), 36-51.
  * [9] D. Gaparayi and S. Attan, Hom-Lie-Yamaguti superalgebras, arXiv[math.RA]
  * [10] Y. Fregier, A. Gohr, Unital algebras of Hom-associative type and surjective or injective twistings, J. of Gen. Lie Theo. and Appl. Vol. 3, (2009), No. 4, 285-295.
  * [11] J.T Hartwig, D. Larsson and S.D. Silvestrov, Deformations of Lie algebras using $\sigma$-derivations, J. Algebra,
  * [12] A. N. Issa, Some characterizations of Hom-Leibniz algebras, International Electronic Journal of Algebra, 14 (2013), 1-9.
  * [13] A. N. Issa, Hom-Akivis algebras, Comment. Math. Univ. Carolin. 52, (2011).
  * [14] Jin Q Q, Li X C. Hom-Lie algebra structures on semi-simple Lie algebras, J Algebra, 2008, 319(4): 1398-1408.
  * [15] M. K. Kinyon and A. Weinstein, Leibniz algebras, Courant algebroids and multiplications on reductive homogeneous spaces,
  * [16] J-L. Loday. Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math., 39 (1993), 269-293.
  * [17] A. Makhlouf, Hom-alternative algebras and Hom-Jordan algebras, Int. Elect. J. Alg., 8 (2010), 177-190.
  * [18] A. Makhlouf and S.D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl., 2 (2008),51-64.
  * [19] A. Makhlouf, Paradigm of nonassociative Hom-algebras and Hom-superalgebras, Proceedings of Jordan structures in algebra and Analysis Meeting, 143-177, Editional Circulo Rojo, Almeria, 2010.
  * [20] I.P. Shestakov, Prime Mal’tsev superalgebras, Mat Sb, 1991, 182(9): 1357-1366
  * [21] C. Wang, Q. Zhang and Z. Wei  Hom-Leibniz superalgebras and Hom-Leibniz poisson superalgebras, Hacettepe Journal of Mathematics and Statistics., 44 (5), 1163-1179, 2015.
  * [22] D. Yau, Hom-Malcev, Hom-alternative, and Hom-Jardan algebras, International Elctronic Journal of algebras, 11 (2012), 177-217.
  * [23] D. Yau, Hom-algebras and Homology, J. Lie Theory, 19 (2009), 409-421.
  * [24] Yau D., Hom-Novikov algebras, J. Phys. A 44 (2011), 085202.
  * [25] Yau D., On n-ary Hom-Nambu and Hom-Nambu-Lie algebras, J. Geom. Phys. 62 (2012), 506-522.
  * [26] Yuan J X, Sun L P, Liu W D. Multiplicative Hom-Lie superalgebra structures on infinite dimensional simple Lie superalgebras of vector fields, arXiv: 1304.5616


††footnotetext: 2020 Mathematics Subject Classification. Primary: 37B05

# A refined saturation theorem for polynomials and applications

Xiangdong Ye and Jiaqi Yu  School of Mathematical Sciences, University of
Science and Technology of China, Hefei, Anhui, 230026, P.R. China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

For a dynamical system $(X,T)$, $d\in\mathbb{N}$ and distinct non-constant
integral polynomials $p_{1},\ldots,p_{d}$ vanishing at $0$, the notion of
regionally proximal relation along $C=\\{p_{1},\ldots,p_{d}\\}$ (denoted by
${\bf RP}_{C}^{[d]}(X,T)$) is introduced.

It turns out that for a minimal system, ${\bf RP}_{C}^{[d]}(X,T)=\Delta$
implies that $X$ is an almost one-to-one extension of $X_{k}$ for some
$k\in\mathbb{N}$ only depending on a set of finite polynomials associated with
$C$ and has zero entropy, where $X_{k}$ is the maximal $k$-step pro-nilfactor
of $X$.

Particularly, when $C$ is a collection of linear polynomials, it is proved
that ${\bf RP}_{C}^{[d]}(X,T)=\Delta$ implies $(X,T)$ is a $d$-step pro-
nilsystem, which answers negatively a conjecture in [10]. The results are
obtained by proving a refined saturation theorem for polynomials.

###### Key words and phrases:

saturation theorem, integral polynomials, regionally proximal relation

This research is supported by NNSF of China 12031019.

##  1\. Introduction

In this section we give the motivation of our study and state the main results
of the paper.

###  1.1. The motivation

To study the convergence of the multiple ergodic averages, the notion of
characteristic factors was introduced by Furstenberg and Weiss in [6]. By the
results of Host-Kra [11] and Ziegler [21], the characteristic factor of the
multiple ergodic averages $\frac{1}{N}\sum_{n=0}^{N-1}f_{1}(T^{n}x)\cdots
f_{d}(T^{dn}x)$ is some $(d-1)$-step pro-nilsystem.

The motivation of this paper comes from the consideration of characteristic
factors in topological dynamics. In the topological setting, the corresponding
notion was first introduced by Glasner in [8]. To be precise, let $(X,T)$ be a
dynamical system and $d\in\mathbb{N}$, set $\tau_{d}=T\times
T^{2}\times\cdots\times T^{d}$. $(Y,T)$ is a topological characteristic factor
(TCF for short) of order $d$ if there exists a dense $G_{\delta}$ set $\Omega$
of $X$ such that for each $x\in\Omega$ the orbit closure
$L_{x}^{d}=\overline{O}(x^{(d)},\tau_{d})$ is
$\pi^{(d)}=:\pi\times\cdots\times\pi$ ($d$ times) saturated, i.e.,
$(\pi^{(d)})^{-1}\pi^{(d)}(L_{x}^{d})=L_{x}^{d}$, where $x^{(d)}=(x,\ldots,x)$
($d$ times) and $\pi:X\to Y$ is the factor map.

Glasner, Hunag, Shao, Weiss and Ye in [10] obtained the following theorem.
Here we state it in the equivalence form.

Theorem GHSWY: Let $(X,T)$ be a minimal system and $\pi:X\rightarrow
X_{\infty}$ be the factor map, where $X_{\infty}$ is the maximal $\infty$-step
pro-nilfactor of $X$. Then there is a dense $G_{\delta}$ set $\Omega$ of $X$
such that for any $d\in\mathbb{N}$ and $x\in\Omega$,

$(\pi^{(d)})^{-1}\pi^{(d)}(x^{(d)})\subset
L_{x}^{d}=\overline{O}(x^{(d)},\tau_{d}).$

Particularly, if $\pi$ is open then $X_{\infty}$ is the TCF of order $d$ for
any $d\in\mathbb{N}$.

Now let $p_{1},\ldots,p_{d}$ be integral polynomials vanishing at $0$. It is
easy to extend the notion of TCF of order $d$ to general polynomials
$p_{1},\ldots,p_{d}$. A natural question is how to generalize the Theorem
GHSWY to general polynomials. In the recent nice work by Qiu [18], the author
proved a version of the saturation theorem for polynomials, which allows him
to give a complete answer to a well known question concerning the density of
polynomial orbits in a totally minimal systems. The result of Qiu was
strengthened in [15] by Huang, Shao and Ye, and now we state the equivalence
form.

Theorem HSY: Let $(X,T)$ be a minimal system, and $\pi:X\rightarrow
X_{\infty}$ be the factor map. Let $d\in\mathbb{N}$ and
$p_{1},p_{2},\ldots,p_{d}$ be distinct non-constant integral polynomials
vanishing at $0$. Then there is a dense $G_{\delta}$ set $\Omega$ of $X$ such
that for any $x\in\Omega$,

$(\pi^{(d)})^{-1}\pi^{(d)}(x^{(d)})\subset
L_{x}^{C}=:\overline{\\{(T^{p_{1}(n)}x,\ldots,T^{p_{d}(n)}x):n\in{\mathbb{Z}}\\}},$

where $C=\\{p_{1},\ldots,p_{d}\\}$. Particularly, if $\pi$ is open then
$X_{\infty}$ is the TCF of order $d$ for C, i.e.,
$(\pi^{(d)})^{-1}\pi^{(d)}(L_{x}^{C})=L_{x}^{C}.$

###  1.2. Main results

One of the main results of the paper is a refinement of Theorem HSY.

Theorem A: Let $(X,T)$ be a minimal system, $d\in\mathbb{N}$ and
$p_{1},p_{2},\ldots,p_{d}$ be distinct non-constant integral polynomials
vanishing at $0$. Then there are $k\in\mathbb{N}$ (depending only on the
polynomials) and a dense $G_{\delta}$ set $\Omega$ of $X$ such that for any
$x\in\Omega$,

(1) $(\pi_{k}^{(d)})^{-1}\pi_{k}^{(d)}(x^{(d)})\subset
L_{x}^{C}=:\overline{\\{(T^{p_{1}(n)}x,\ldots,T^{p_{d}(n)}x):n\in{\mathbb{Z}}\\}},$

where $C=\\{p_{1},\ldots,p_{d}\\}$, $X_{k}$ is the maximal $k$-step pro-
nilfactor and $\pi_{k}:X\rightarrow X_{k}$ is the factor map. Particularly, if
$\pi_{k}$ is open then $X_{k}$ is the TCF of order $d$ for $C$.

We remark that when $p_{1},\ldots,p_{d}$ are linear polynomials, we can choose
$k=d-1$.

Moreover, when $d=1$, we can choose $k=1$ and $X_{1}$ can be replaced by
$X_{rat}$, where $X_{rat}$ is a factor of $X_{1}$ associated with all
(topological) rational eigenvalues. This implies that if $(X,T)$ is totally
minimal, then there is a dense $G_{\delta}$ set $\Omega$ of $X$ such that for
each $x\in\Omega$, $\\{T^{p_{1}(n)}x:n\in{\mathbb{Z}}\\}$ is dense in $X$ (as
in this case $X_{rat}$ is trivial), which gives another approach to the result
in [10, 18] for a single polynomial.

Theorem A is proved by using Theorem HSY, an ergodic argument to deal with the
distal case, and a method to put them together.

Theorem A has applications to the regionally proximal relation along
polynomials we now introduce, which is a natural generalization of the
regionally proximal relation along linear polynomials (denoted by ${\bf
AP}^{[d]}(X,T)$) introduced in [9]. Precisely, let $(X,T)$ be a topological
system, and $C=\\{p_{1},\ldots,p_{d}\\}$ be a set of distinct non-constant
integral polynomials vanishing at $0$. We say $(x,y)\in X\times X$ is
regionally proximal along $C$ (denoted by $(x,y)\in{\bf RP}_{C}^{[d]}(X,T)$)
if for a given $\varepsilon>0$ and a neighborhood $U\times V$ of $(x,y)$,
there are a pair $(x^{\prime},y^{\prime})\in U\times V$ and $n\in{\mathbb{Z}}$
such that $\rho(T^{p_{i}(n)}x^{\prime},T^{p_{i}(n)}y^{\prime})<\varepsilon$
for any $1\leq i\leq d$. In a certain sense, this definition was inspired by
the study of the multiple ergodic averages

$\frac{1}{N}\sum_{n=0}^{N-1}f_{1}(T^{p_{1}(n)}x)\cdots f_{d}(T^{p_{d}(n)}x).$

Using Theorem A and the properties of ${\bf RP}^{[d]}_{C}$ we obtain the other
main result of the paper.

Theorem B: Let $(X,T)$ be a minimal system, $d\in\mathbb{N}$ and
$C=\\{p_{1},p_{2},\ldots,p_{d}\\}$ be distinct non-constant integral
polynomials vanishing at $0$. Then ${\bf RP}_{C}^{[d]}(X,T)=\Delta$ implies
that $X$ is an almost one to one extension of $X_{k}$ for some
$k\in\mathbb{N}$ only depending on a set of finite polynomials derived from
$C$, and has zero entropy. Particularly, if $(X,T)$ is distal then it is a
$k$-step pro-nilsystem.

As a corollary we get that if ${\bf AP}^{[d]}(X,T)=\Delta$ then $(X,T)$ is a
$d$-step pro-nilsystem, which gives a negative answer to a conjecture in [10,
Conjecture 4].

###  1.3. Organization of the paper

We organize the paper as follows. In Section 1, we give the motivation of this
paper and state the main results. In Section 2 we introduce some necessary
notions and some known facts to be used in the paper. In Section 3, we prove
the refined saturation theorem. In Section 4, we give some applications to
regionally proximal relation along polynomials and give a negative answer to a
conjecture in [10]. Moreover, we ask several open questions at the end of the
paper.

Acknowledgement: We thank Jiahao Qiu, Song Shao and Hui Xu for useful
discussions.

##  2\. Preliminary

In this section we give some necessary notions and some known facts used in
the paper.

###  2.1. Topological dynamical systems

#### 2.1.1.

By a topological dynamical system (for short t.d.s.) we mean a pair $(X,T)$,
where $X$ is a compact metric space $X$ with a metric $\rho$ and $T:X\to X$ is
a homeomorphism. Let $(X,T)$ be a t.d.s. and $x\in X$. Then
$O(x,T)=\\{T^{n}x:n\in{\mathbb{Z}}\\}$ denotes the orbit of $x$. A subset
$A\subseteq X$ is called invariant (or $T$-invariant) if $TA=A$. When
$Y\subseteq X$ is a closed and invariant subset of the system $(X,T)$, we say
that the system $(Y,T|_{Y})$ is a subsystem of $(X,T)$. Usually we will omit
the subscript, and denote $(Y,T|_{Y})$ by $(Y,T)$. If $(X,T)$ and $(Y,S)$ are
two t.d.s., their product system is the system $(X\times Y,T\times S)$.

When there are more than one t.d.s. involved, usually we should use different
symbols to denote different transformations on different spaces, for example,
$(X,T),(Y,S),(Z,H)$ etc. But when no confusing, it is convenient to use only
one symbol $T$ for all transformations in all t.d.s. involved, for example,
$(X,T),(Y,T),(Z,T)$ etc. In this paper, we use the same symbol $T$ for the
transformations in all t.d.s.

#### 2.1.2.

Let $X,Y$ be compact metric spaces and $\phi:X\to Y$ be a map. For $n\geq 2$,
let $\phi^{(n)}=\phi\times\cdots\times\phi:X^{n}\rightarrow Y^{n}.$ We write
$(X^{n},T^{(n)})$ for the $n$-fold product system $(X\times\cdots\times
X,T\times\cdots\times T)$. The diagonal of $X^{n}$ is
$\Delta_{n}(X)=\\{(x,\ldots,x)\in X^{n}:x\in X\\}.$ When $n=2$ we write
$\Delta(X)=\Delta_{2}(X)$.

#### 2.1.3.

A t.d.s. $(X,T)$ is called minimal if $X$ contains no proper non-empty closed
invariant subsets. It is easy to verify that a t.d.s. is minimal if and only
if every orbit is dense. In a t.d.s. $(X,T)$ we say that a point $x\in X$ is
minimal if $(\overline{O(x,T)},T)$ is minimal.

#### 2.1.4.

Let $(X,T)$ be a t.d.s. A pair $(x,y)\in X^{2}$ is proximal if
$\inf_{n\in{\mathbb{Z}}}\rho(T^{n}x,T^{n}y)=0$; and it is distal if it is not
proximal. Denote by ${\bf P}(X,T)$ the set of all proximal pairs of $(X,T)$. A
t.d.s. $(X,T)$ is called distal if $(x,x^{\prime})$ is distal whenever
$x,x^{\prime}\in X$ are distinct.

#### 2.1.5.

Let $(X,T)$ be a t.d.s. and $\mathcal{M}(X)$ be the set of all Borel
probability measures on $X$. A measure $\mu\in\mathcal{M}(X)$ is $T$-invariant
if for any Borel set $B\in\mathcal{X}$, $\mu(T^{-1}B)=\mu(B)$ and we call
$(X,\mathcal{X},\mu,T)$ a measure-preserving system (m.p.s. for short). Denote
by $\mathcal{M}(X,T)$ the set of invariant elements of $\mathcal{M}(X)$. A
measure $\mu\in\mathcal{M}(X,T)$ is ergodic if for any Borel set $B$ of $X$
satisfying $\mu(T^{-1}B\triangle B)=0$ we have $\mu(B)=0$ or $\mu(B)=1$. The
system $(X,T)$ is uniquely ergodic if $\mathcal{M}(X,T)$ consists of only one
element, and it is strictly ergodic if in addition it is minimal.

Let $(X,T)$ be a t.d.s, $\mu\in\mathcal{M}(X)$. The support of $\mu$ is
defined to be

$Supp(\mu)=\\{x\in X:\text{for each neighborhood}\ U\ \text{of}\ x,\
\mu(U)>0\\}.$

###  2.2. Factor maps

#### 2.2.1.

A factor map $\pi:X\rightarrow Y$ between two t.d.s. $(X,T)$ and $(Y,T)$ is a
continuous onto map which intertwines the actions (i.e. $\pi\circ
T=T\circ\pi$); one says that $(Y,T)$ is a factor of $(X,T)$ and that $(X,T)$
is an extension of $(Y,T)$.

#### 2.2.2.

Let $\pi:(X,T)\rightarrow(Y,T)$ be a factor map. Then
$R_{\pi}=\\{(x_{1},x_{2}):\pi(x_{1})=\pi(x_{2})\\}$ is a closed invariant
equivalence relation, and $Y=X/R_{\pi}$. Let $(X,T)$ and $(Y,T)$ be t.d.s. and
let $\pi:(X,T)\to(Y,T)$ be a factor map. One says that:

  * •

$\pi$ is a proximal extension if $\pi(x_{1})=\pi(x_{2})$ implies
$(x_{1},x_{2})\in{\bf P}(X,T)$;

  * •

$\pi$ is a distal extension if $\pi(x_{1})=\pi(x_{2})$ and $x_{1}\neq x_{2}$
implies $(x_{1},x_{2})\not\in{\bf P}(X,T)$;

  * •

$\pi$ is an almost one to one extension if there exists a dense $G_{\delta}$
set $X_{0}\subseteq X$ such that $\pi^{-1}(\\{\pi(x)\\})=\\{x\\}$ for any
$x\in X_{0}$;

Note that an almost one to one extension is a proximal one.

###  2.3. Nilsystems

Let $G$ be a group. For $g,h\in G$ and $A,B\subseteq G$, we write
$[g,h]=ghg^{-1}h^{-1}$ for the commutator of $g$ and $h$ and $[A,B]$ for the
subgroup spanned by $\\{[a,b]:a\in A,b\in B\\}$. The commutator subgroups
$G_{j}$, $j\geq 1$, are defined inductively by setting $G_{1}=G$ and
$G_{j+1}=[G_{j},G]$. Let $d\geq 1$ be an integer. We say that $G$ is $d$-step
nilpotent if $G_{d+1}$ is the trivial subgroup.

Let $d\in\mathbb{N}$, $G$ be a $d$-step nilpotent Lie group and $\Gamma$ be a
discrete cocompact subgroup of $G$. The compact manifold $X=G/\Gamma$ is
called a $d$-step nilmanifold. The group $G$ acts on $X$ by left translations
and we write this action as $(g,x)\mapsto gx$. The Haar measure $\mu$ of $X$
is the unique Borel probability measure on $X$ invariant under this action.
Fix $t\in G$ and let $T$ be the transformation $x\mapsto tx$ of $X$, i.e.
$t(g\Gamma)=(tg)\Gamma$ for each $g\in G$. Then $(X,\mu,T)$ 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. A $d$-step pro-nilsystems is an
inverse limit of $d$-step nilsystems.

###  2.4. Regionally proximal relation of order $d$

######  Definition 2.1.

[13, Definition 3.2] Let $(X,T)$ be a t.d.s. and let $d\in\mathbb{N}$. The
points $x,y\in X$ are said to be regionally proximal of order $d$ if for any
$\delta>0$, there exist $x^{\prime},y^{\prime}\in X$ and a vector ${\bf
n}=(n_{1},\ldots,n_{d})\in{\mathbb{Z}}^{d}$ such that
$\rho(x,x^{\prime})<\delta,\rho(y,y^{\prime})<\delta$, and

$\rho(T^{{\bf n}\cdot\varepsilon}x^{\prime},T^{{\bf
n}\cdot\varepsilon}y^{\prime})<\delta\ \text{for every
$\varepsilon=(\varepsilon_{1},\ldots,\varepsilon_{d})\in\\{0,1\\}^{d}\setminus\\{(0,0,\cdots,0)\\}$},$

where ${\bf n}\cdot\varepsilon=\sum_{i=1}^{d}n_{i}\varepsilon_{i}$. The set of
regionally proximal pairs of order $d$ is denoted by ${\bf RP}^{[d]}$ (or by
${\bf RP}^{[d]}(X,T)$ in case of ambiguity), and is called the regionally
proximal relation of order $d$.

It is easy to see that ${\bf RP}^{[d]}$ is a closed and invariant relation,
and

${\bf P}(X,T)\subseteq\ldots\subseteq{\bf RP}^{[d+1]}\subseteq{\bf
RP}^{[d]}\subseteq\ldots{\bf RP}^{[2]}\subseteq{\bf RP}^{[1]}.$

The following theorems were proved in [13] and [17].

######  Theorem 2.2.

Let $(X,T)$ be a minimal t.d.s. and let $d\in\mathbb{N}$. Then

  1. (1)

${\bf RP}^{[d]}$ is an equivalence relation.

  2. (2)

$(X,T)$ is a $d$-step pro-nilsystem if and only if ${\bf
RP}^{[d]}=\Delta_{X}$.

The regionally proximal relation of order $d$ allows us to construct the
maximal $d$-step pro-nilfactor of a system.

######  Theorem 2.3.

Let $\pi:(X,T)\rightarrow(Y,T)$ be a factor map between minimal t.d.s. and let
$d\in\mathbb{N}$. Then,

  1. (1)

$\pi\times\pi({\bf RP}^{[d]}(X,T))={\bf RP}^{[d]}(Y,T)$.

  2. (2)

$(Y,T)$ is a $d$-step pro-nilsystem if and only if ${\bf
RP}^{[d]}(X,T)\subseteq R_{\pi}$.

In particular, $X_{d}=X/{\bf RP}^{[d]}(X,T)$, the quotient of $(X,T)$ under
${\bf RP}^{[d]}(X,T)$, is the maximal $d$-step pro-nilfactor of $X$.

###  2.5. $\infty$-step pro-nilsystems

By Theorem 2.2 for any minimal t.d.s. $(X,T)$, ${\bf
RP}^{[\infty]}=\bigcap_{d\geq 1}{\bf RP}^{[d]}$ is a closed invariant
equivalence relation (we write ${\bf RP}^{[\infty]}(X,T)$ in case of
ambiguity). Now we formulate the definition of $\infty$-step pro-nilsystems. A
minimal t.d.s. $(X,T)$ is an $\infty$-step pro-nilsystem (see [4, Definition
3.4]), if the equivalence relation ${\bf RP}^{[\infty]}$ is trivial, i.e.,
coincides with the diagonal.

######  Remark 2.4.

  1. (1)

Similar to Theorem 2.3, one can show that the quotient of a minimal system
$(X,T)$ under ${\bf RP}^{[\infty]}$ is the maximal $\infty$-step pro-nilfactor
of $(X,T)$. We denote the maximal $\infty$-step pro-nilfactor of $(X,T)$ by
$X_{\infty}$.

  2. (2)

A minimal system is an $\infty$-step pro-nilsystem if and only if it is an
inverse limit of minimal nilsystems [4, Theorem 3.6].

  3. (3)

An $\infty$-step pro-nilsystem is distal.

###  2.6. Dense sets

Let $\pi:X\rightarrow Y$ be a map between topological spaces. We say $\pi$ is
semi-open if $U$ is open and non-empty in $X$, then $\pi(U)$ has non-empty
interiors; and $\pi$ is open if it sends open sets to open sets. It is known
that if $\pi$ is a factor map between minimal systems, then $\pi$ is semi-
open. Moreover, the factor map between distal systems is open.

######  Lemma 2.5.

[20, Proposition3.1] Let $\pi:X\longrightarrow Y$ be a factor between minimal
t.d.s. If $\Omega$ is a dense $G_{\delta}$ set of $X$, then there is a dense
$G_{\delta}$ set $Y_{0}$ of $Y$ such that for each $y\in Y_{0}$,
$\Omega\cap\pi^{-1}(y)$ is a dense $G_{\delta}$ set of $\pi^{-1}(y).$

##  3\. The proof of Theorem A

In this section we will show Theorem A. As we said before, Theorem A is proved
by using Theorem HSY, an ergodic argument to deal with the distal case, and a
method to put them together. We start with some preparation.

###  3.1. Preparation

Note that we say polynomials $p$ and $q$ are distinct if $p-q$ is not constant
(it is called essentially distinct in [16]). Let $C=\\{p_{1},\ldots,p_{d}\\}$
be a set of distinct non-constant integral polynomials vanishing at $0$. Given
a factor map $\pi:(X,T)\rightarrow(Y,T)$ and $d\geq 2$, the t.d.s. $(Y,T)$ is
said to be a TCF of order d for $C$ of $(X,T)$, if there exists a dense
$G_{\delta}$ subset $\Omega$ of $X$ such that for each $x\in\Omega$ the orbit
closure
$L_{x}^{C}=\overline{\\{(T^{p_{1}(n)}x,\ldots,T^{p_{d}(n)}x):n\in{\mathbb{Z}}\\}}$
is $\pi^{(d)}=\pi\times\cdots\times\pi$ ($d$-times) saturated. The following
was proved by in [15].

######  Theorem 3.1.

Let $(X,T)$ be a minimal t.d.s., and $\pi:X\rightarrow X_{\infty}$ be the
factor map from $X$ to its maximal $\infty$-step pro-nilfactor $X_{\infty}$ of
$X$. Then there are minimal t.d.s. $X^{*}$ and $X_{\infty}^{*}$ which are
almost one to one extensions of $X$ and $X_{\infty}$ respectively, an open
factor map $\pi^{*}$ and a commuting diagram below

$\begin{CD}X@<{\varsigma^{*}}<{}<X^{*}\\\ @V{}V{\pi}V@V{}V{\pi^{*}}V\\\
X_{\infty}@<{\varsigma}<{}<X_{\infty}^{*}\end{CD}$

such that there is a $T$-invariant dense $G_{\delta}$ subset $X^{*}_{0}$ of
$X^{*}$ having the following property: for all $x\in X^{*}_{0}$, for any non-
empty open subsets $V_{1},\ldots,V_{d}$ of $X^{*}$ with
$\pi(x)\in\bigcap_{i=1}^{d}\pi^{*}(V_{i})$ and distinct non-constant integral
polynomials $p_{1},p_{2},\ldots,p_{d}$ with $p_{i}(0)=0$, $i=1,2,\ldots,d$,
there is some $n\in\mathbb{N}$ such that

$x\in T^{-p_{1}(n)}V_{1}\cap T^{-p_{2}(n)}V_{2}\cap\cdots\cap
T^{-p_{d}(n)}V_{d}.$

Now we give the following lemma to explain the equivalence between Theorem 3.1
and Theorem HSY.

######  Lemma 3.2.

Theorem 3.1 and Theorem HSY are equivalent.

###### Proof.

It is clear that Theorem HSY implies Theorem 3.1, as the maximal $\infty$-step
pro-nilfactor of $X^{*}$ is also $X_{\infty}$, see for example [10].

To show the other implication, let us assume the commuting diagram in Theorem
3.1. As $X^{*}$ and $X_{\infty}^{*}$ are almost one to one extensions of $X$
and $X_{\infty}$ respectively,

$\Omega_{1}=\\{y\in X_{\infty}:|\varsigma^{-1}(y)=1|\\}\ \text{and}\
\Omega_{2}=\\{y\in X:|\varsigma^{*-1}(y)=1|\\}$

are dense $G_{\delta}$ sets. Let $X^{*}_{0}$ be the dense $G_{\delta}$ subset
of $X^{*}$ in Theorem 3.1, and set
$\Omega=\pi^{-1}(\Omega_{1})\cap\Omega_{2}\cap\varsigma^{*}(X^{*}_{0})$, which
is a dense $G_{\delta}$ subset of $X$ (as $\varsigma^{*}$ is semi-open,
$\varsigma^{*}(X_{0}^{*})$ is also a dense $G_{\delta}$ subset of $X$).

Now let $x\in\Omega$, and $x_{1},\cdots,x_{d}\in\pi^{-1}\pi(x).$ Since
$\varsigma\pi^{*}=\pi\varsigma^{*}$, we get that

$(\pi^{*})^{-1}\varsigma^{-1}\pi(x)=(\varsigma^{*})^{-1}\pi^{-1}\pi(x).$

Let $\\{y\\}=(\varsigma^{*})^{-1}(x)$. Note that since $\pi(x)\in\Omega_{1}$,
we have $|\varsigma^{-1}\pi(x)|=1$ and in fact,
$\varsigma^{-1}\pi(x)=\\{\pi^{*}y\\}$. So there are
$y_{1},\ldots,y_{d}\in(\pi^{*})^{-1}\pi^{*}(y)$ with
$\varsigma^{*}(y_{i})=x_{i},1\leq i\leq d$.

By Theorem 3.1,
$(y_{1},\ldots,y_{d})\in\overline{\\{(T^{p_{1}(n)}y,\ldots,T^{p_{d}(n)}y):n\in{\mathbb{Z}}\\}}$
which implies that

$(x_{1},\ldots,x_{d})\in\overline{\\{(T^{p_{1}(n)}x,\ldots,T^{p_{d}(n)}x):n\in{\mathbb{Z}}\\}}.$

∎

Let $G$ be a locally compact group, written with multiplicative notation, and
let $m_{G}$ be its Haar measure. A Følner sequence in $G$ is a sequence
$\bm{\Phi}=(\Phi_{N})_{N\in\mathbb{N}}$ of compact subsets of $G$ each of
which has positive Haar measure and such that for any $g\in G$,

$\frac{m_{G}(g\Phi_{N}\Delta\Phi_{N})}{m_{G}(\Phi_{N})}\to 0,\ as\
N\to\infty,$

where $\Delta$ denotes the symmetric difference and
$g\Phi_{N}=\\{gh:h\in\Phi_{N}\\}.$

For the definition of the semi-norm of order $k$ see [11]. The convergence of
the multiple ergodic averages in $L^{2}$ was studied in [12, 16]. We need a
result from [16, Theorem 3].

######  Theorem 3.3.

Let $(X,\mathcal{X},\mu,T)$ be an ergodic system. For any $r,b\in\mathbb{N}$
there exists $k\in\mathbb{N}$ such that for any system of distinct non-
constant polynomials
$p_{1},\ldots,p_{r}:{\mathbb{Z}}^{d}\longrightarrow{\mathbb{Z}}$ of degree
$\leq b$ and any $f_{1},\ldots,f_{r}\in L^{\infty}$ with $|||f_{1}|||_{k}=0$,
one has

$\lim_{N\rightarrow\infty}\frac{1}{|\Phi_{N}|}\sum_{u\in\Phi_{N}}T^{p_{1}(u)}f_{1}\cdots
T^{p_{r}(u)}f_{r}=0$

in $L^{2}$ for any Følner sequence $\Phi_{N}$ in ${\mathbb{Z}}^{d}$.

For an ergodic system $(X,\mathcal{X},\mu,T)$, the authors in [11] showed that
it admits a maximal $k$-step pro-nilfcator, denoted by $Z_{k}$. We remark that
for an $\infty$-step minimal pro-nilsystem, $X_{k}=Z_{k}$, where the measure
is the unique measure on $Z_{k}$.

Let $(Z_{k},\mathcal{Z}_{k},\mu_{k},T)$ be the maximal $k$-step pro-nilfcator
of an ergodic system $(X,\mathcal{X},\mu,T)$. It is known in [11] that for
$f\in L^{\infty}$,
$\mathbb{E}(f|\mathcal{Z}_{k-1})=0\Leftrightarrow|||f|||_{k}=0,$ where
$\mathbb{E}(f|\mathcal{Z}_{k-1})$ is the conditional expectation. For $f\in
L^{\infty}$, write
$f=(f-\mathbb{E}(f|\mathcal{Z}_{k-1})+\mathbb{E}(f|\mathcal{Z}_{k-1})$. It is
easy to see that $\mathbb{E}((f-\mathbb{E}(f|\mathcal{Z}_{k-1}))|Z_{k-1})=0$.
So in Theorem 3.3, we can replace $f_{i}$ by
$\mathbb{E}(f_{i}|\mathcal{Z}_{k-1})$. Thus, we can rewrite the conclusion of
Theorem 3.3 as:

(2)
$||\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}(\prod_{i=0}^{d}T^{p_{i}(n)}f_{i}-\prod_{i=0}^{d}T^{p_{i}(n)}\mathbb{E}(f_{i}|\mathcal{Z}_{k}))||_{L^{2}(\mu)}=0,$

for some $k$ only depending on the the polynomials $p_{1},\cdots,p_{d}$.

The other result we need is the following theorem from [1, Theorem A].

######  Theorem 3.4.

Let $(X,\mathcal{X},\mu,T)$ be a measure preserving transformation and let
$A\in\mathcal{X}$ be a set with positive measure, then for each
$C=\\{p_{1},\ldots,p_{k}\\}$ of distinct non-constant integral polynomials
vanishing at 0,

$\liminf_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\mu(A\cap T^{-p_{1}(n)}A\cap
T^{-p_{2}(n)}A\cap\cdots\cap T^{-p_{k}(n)}A)>0.$

###  3.2. The distal case

To show the distal case we need a lemma from [9].

######  Lemma 3.5.

[9, Proposition 5.5] Let $(X,T)$ be a strictly ergodic system with a unique
invariant measure $\mu$ and let $\pi:(X,T)\to(Y,T)$ be a distal extension. Let
$\mu=\int_{Y}\mu_{y}d\nu$ be the disintegration of $\mu$ over $\nu$, where
$\nu=\pi_{*}(\mu)$. Then there is $Y_{0}\subset Y$ with full measure such that
for each $y\in Y_{0}$, $supp(\mu_{y})=\pi^{-1}(y)$.

Now we are able to show

######  Theorem 3.6.

Let $(X,T)$ be a minimal distal system and $C=\\{p_{1},\ldots,p_{d}\\}$ be a
collection of distinct non-constant integral polynomials vanishing at $0$.
Then there is $k\in\mathbb{N}$ (only depending on $C$) such that $(X_{k},T)$
is the TCF of order $d$ for $C$ of $(X,T)$.

###### Proof.

First we know that $(X_{\infty},T)$ is the TCF of order $d$ for $C$ of $(X,T)$
by Theorem HSY, since the factor map $X\rightarrow X_{\infty}$ is open. It
remains to show $(X_{k},T)$ is TCF of order $d$ for $C$ of $(X_{\infty},T)$,
as the composition of two factor maps with TCF keeps the TCF property (one can
check this directly, or see it from the proof of Theorem A).

Let $\pi_{k}:X_{\infty}\to X_{k}$ be the factor map, and $k$ be the number
defined in (2). Set $\mu$ be the uniquely ergodic measure of $(X_{\infty},T)$,
and $(Z_{k},\mathcal{Z}_{k},\mu_{k},T)$ be the maximal $k$-step pro-nilfactor
of $(X_{\infty},\mathcal{X}_{\infty},\mu,T)$. It is known that $Z_{k}=X_{k}$,
and so $\pi_{k}$ can be viewed as the continuous factor map from
$(X_{\infty},\mu,T)$ to $(Z_{k},\mu_{k},T)$.

Let $\mu=\int_{Z_{k}}\mu_{z}d\mu_{k}(z)$ be the disintegration of $\mu$ over
$\mu_{k}$. For $d\in\mathbb{N}$, let $L_{d,k}^{\mu}={\rm
Supp}(\lambda_{d}^{k})$, where

$\lambda_{d}^{k}=\int_{Z_{k}}\prod_{j=1}^{d}\mu_{y}\ d\mu_{k}(y).$

Similar to the argument in [9, Theorem 5.9], we now show that ${\rm
Supp}(\lambda_{d}^{k})=R_{\pi_{k}}^{d}$. First we note that
$\lambda_{d}^{k}(R_{\pi_{k}}^{d})=1$, so ${\rm Supp}(\lambda_{d}^{k})\subset
R_{\pi_{k}}^{d}$. By Lemma 3.5 there is a measurable set $Y_{0}\subset X_{d}$
with full measure such that for any $y\in Y_{0}$, ${\rm
Supp}(\mu_{y})={\pi_{k}}^{-1}(y)$. Let $W={\rm Supp}(\lambda_{d}^{k})$. Since

$\lambda_{d}^{k}(W)=\int_{Y_{0}}\prod_{j=1}^{d}\mu_{y}(W)\ d\mu_{k}(y)=1,$

we have that for a.e. $y\in Y$, $\prod_{j=1}^{d}\mu_{y}(W)=1$. This implies
that $\prod_{j=1}^{d}{\rm Supp}(\mu_{y})\subset W$, a.e. $y\in Y$. Thus by the
distality of $\pi_{k}$, the map $y\mapsto\pi_{k}^{-1}(y)$ (from $X_{k}$ to
$2^{X_{\infty}}$) is continuous and we conclude that
$R_{\pi_{k}}^{d}\subset{\rm Supp}(\lambda_{d}^{k})$. Thus, we get that ${\rm
Supp}(\lambda_{d}^{k})=R_{\pi_{k}}^{d}$.

Now let $C=\\{p_{1},\ldots,p_{d}\\}$ be the polynomials mentioned in the
theorem. We now show a claim:

$\mathbf{Claim}$: For any non-empty open subsets $U_{0},\cdots,U_{d}$ of
$X_{\infty}$ with $\bigcap_{i=0}^{d}\pi_{k}(U_{i})\neq\emptyset$,

$U_{0}\cap T^{-p_{1}(n)}U_{1}\cap T^{-p_{2}(n)}U_{2}\cap\cdots\cap
T^{-p_{d}(n)}U_{d}\neq\emptyset.$

###### Proof.

By (2) we have

$||\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}(\prod_{i=1}^{d}T^{p_{i}(n)}f_{i}-\prod_{i=1}^{d}T^{p_{i}(n)}\mathbb{E}_{\mu}(f_{i}|\mathcal{Z}_{k}))||_{L^{2}(\mu)}=0.$

Let $(x_{0},x_{1},\cdots,x_{d})\in R_{\pi_{k}}^{d+1}={\rm
Supp}(\lambda_{d+1}^{k})$, and let $(U_{0},U_{1},\cdots,U_{d})$ be a
neighborhood of $(x_{0},x_{1},\cdots,x_{d})$, then it follows that

(3) $\lambda_{k}(U_{0}\times U_{1}\times\cdots\times
U_{d})=\int_{Z_{k}}{\mathbb{E}}(1_{U_{0}}|\mathcal{Z}_{k}){\mathbb{E}}(1_{U_{1}}|\mathcal{Z}_{k})\cdots{\mathbb{E}}(1_{U_{d}}|\mathcal{Z}_{k})d\mu_{{k}}>0.$

By (2), we have

$\begin{split}&\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\mu(U_{0}\cap
T^{-p_{1}(n)}U_{1}\cap T^{-p_{2}(n)}U_{2}\cap\cdots\cap T^{-p_{d}(n)}U_{d})\\\
=&\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}1_{U_{0}}(x)1_{U_{1}}(T^{p_{1}(n)}x)1_{U_{2}}(T^{p_{2}(n)}x)\cdots
1_{U_{d}}(T^{p_{d}(n)}x)d\mu(x)\\\
=&\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{Z_{k}}\mathbb{E}(1_{U_{0}}|\mathcal{Z}_{k})(z)\mathbb{E}(1_{U_{1}}|\mathcal{Z}_{k})(T^{p_{1}(n)}z)\cdots\mathbb{E}(1_{U_{d}}|\mathcal{Z}_{k})(T^{p_{d}(n)})d\mu_{k}(z)\\\
\geq&\liminf_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}a^{d+1}\int_{Z_{k}}1_{A_{a}}(z)1_{A_{a}}(T^{p_{1}(n)}z)\cdots
1_{A_{a}}(T^{p_{d}}(n)z)d\mu_{k}(z)\\\
=&\liminf_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}a^{d+1}\mu_{k}(A_{a}\cap
T^{-p_{1}(n)}A_{a}\cap T^{-p_{2}(n)}A_{a}\cap\cdots\cap
T^{-p_{d}(n)}A_{a}),\end{split}$

where $a>0$ and

$A_{a}=\\{z\in
Z_{k}:\mathbb{E}(1_{U_{0}}|\mathcal{Z}_{k})(z)>a,\mathbb{E}(1_{U_{1}}|\mathcal{Z}_{k})(z)>a,\cdots,\mathbb{E}(1_{U_{d}}|\mathcal{Z}_{k})(z)>a\\}.$

As $\mathbb{E}(1_{U_{i}}|\mathcal{Z}_{k})\leq 1$ , $i=0,1,\cdots,d$, by (3) we
can get that

$\begin{split}0<b&:=\int_{Z_{k}}\mathbb{E}(1_{U_{0}}|\mathcal{Z}_{k})\mathbb{E}(1_{U_{1}}|\mathcal{Z}_{k})\cdots\mathbb{E}(1_{U_{d}}|\mathcal{Z}_{k})d\mu_{k}\\\
&=\int_{A_{a}}\mathbb{E}(1_{U_{0}}|\mathcal{Z}_{k})\cdots\mathbb{E}(1_{U_{d}}|\mathcal{Z}_{k})d\mu_{k}+\int_{Z_{k}\setminus
A_{a}}\mathbb{E}(1_{U_{0}}|\mathcal{Z}_{k})\cdots\mathbb{E}(1_{U_{d}}|\mathcal{Z}_{k})d\mu_{k}\\\
&\leq\mu_{k}(A_{a})+a\mu_{k}(Z_{k}\setminus A_{a})=a+(1-a)\mu_{k}(A_{a}),\\\
\end{split}$

here we use that fact that for $z\in Z_{k}\setminus A_{a}$,
$\mathbb{E}(1_{U_{d}}|\mathcal{Z}_{k})(z)\leq a$. Hence there exists $a>0$
such that $\mu_{k}(A_{a})>0$. So by Theorem 3.4

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\mu(U_{0}\cap
T^{-p_{1}(n)}U_{1}\cap\cdots\cap T^{-p_{d}(n)}U_{d})>0.$

Then there exists $n\in\mathbb{N}$ such that

$U_{0}\cap T^{-p_{1}(n)}U_{1}\cap T^{-p_{2}(n)}U_{2}\cap\cdots\cap
T^{-p_{d}(n)}U_{d}\neq\emptyset.$

This ends the proof of the claim. ∎

Now we show that $(X_{k},T)$ is the TCF of order $d$ for $C$ of
$(X_{\infty},T)$ using the claim we just proved. This is exact the proof of
[15, Theorem 3.6]. This ends the proof. ∎

###  3.3. Proof of Theorem A

In this subsection we show Theorem A. For a compact metric space $X$, we use
$2^{X}$ to denote the set of all non-empty closed subsets of $X$ (with
Hausdorff topology). We need a lemma which can be verified directly by results
in [3].

######  Lemma 3.7.

Let $\pi:(X,T)\rightarrow(Y,T)$ be a factor map between minimal t.d.s. and let
$\Pi:Y\rightarrow 2^{X}$ be defined by $y\mapsto\pi^{-1}(y)$. Then we have
$\\{y\in Y:\Pi\ \text{is continuous at}\ y\\}$ is a dense $G_{\delta}$ set of
$Y$.

Another lemma we need is a direct consequence of Theorem HSY

######  Lemma 3.8.

Let $d\in\mathbb{N}$, $(X,T)$ be a minimal system, and $\pi_{1}:X\rightarrow
X_{\infty}$ be the factor map. Then there is a dense $G_{\delta}$ set
$\Omega_{1}$ of $X$ such that for any $x\in\Omega_{1}$, any distinct non-
constant integral polynomials $p_{1},p_{2},\ldots,p_{d}$ vanishing at $0$ and
any $m\in Z$

$\prod_{i=1}^{d}\pi_{1}^{-1}\pi_{1}(T^{p_{i}(m)}x)\subset\overline{\\{(T^{p_{1}(n)}x,\ldots,T^{p_{d}(n)}x):n\in{\mathbb{Z}}\\}}.$

###### Proof.

Since the set of integral polynomials is countable, we may assume that the set
$\Omega$ in Theorem HSY is satisfied for any distinct non-constant integral
polynomials vanishing at $0$. Set $\Omega_{1}$ to be the set. For
$x\in\Omega_{1}$ and $m\in{\mathbb{Z}}$, let
$x_{i}\in\pi_{1}^{-1}\pi_{1}(T^{p_{i}(m)}x)$, and $V_{i}$ be a neighborhood of
$x_{i}$, $1\leq i\leq d$. It is clear that
$T^{-p_{i}(m)}V_{i}\cap\pi_{1}^{-1}\pi_{1}(x)\not=\emptyset$, $1\leq i\leq d$.
Applying Theorem HSY to polynomials
$\\{p_{1}(n+m)-p_{1}(m),\ldots,p_{d}(n+m)-p_{d}(m)\\}$ we get the result. ∎

Now we are ready to show Theorem A.

###### Proof of Theorem A.

Let $C=\\{p_{1},\ldots,p_{d}\\}$ be distinct non-constant integral polynomials
with $p_{i}(0)=0,1\leq i\leq d$. Let $\pi_{1}:X\to X_{\infty}$ and
$\pi_{2}:X_{\infty}\rightarrow X_{k}$ be the factor maps, where $k$ is the
number defined in Theorem 3.6.

Let $\pi_{k}=\pi_{2}\pi_{1}:X\rightarrow X_{k}$ and $\Omega_{1}$ be the set
defined in Lemma 3.8.

By Theorem 3.6, there is a dense $G_{\delta}$ set $\Omega_{2}$ of $X_{\infty}$
such that for any $y\in\Omega_{2}$,

(4)
$(\pi_{2}^{-1}\pi_{2}(y))^{d}\subset\overline{\\{(T^{p_{1}(n)}y,\ldots,T^{p_{d}(n)}y):n\in{\mathbb{Z}}\\}}.$

Let $\Omega_{3}$ be the dense $G_{\delta}$ subset of $X_{\infty}$ such that if
$y\in\Omega_{3}$, then $y$ is a continuous point of $y\mapsto\pi_{1}^{-1}(y)$
(see Lemma 3.7).

Let $\Omega^{\prime}=\Omega_{1}\cap\pi_{1}^{-1}(\Omega_{2}\cap\Omega_{3})$.
Then $\Omega^{\prime}$ is a dense $G_{\delta}$ subset of $X$. Then there is a
dense $G_{\delta}$ set $\Omega_{4}$ of $X_{k}$ such that for each
$y\in\Omega_{4}$, $\pi_{k}^{-1}(y)\cap\Omega^{\prime}$ is a dense $G_{\delta}$
set of $\pi_{k}^{-1}(y)$ (see Lemma 2.5). Recall that
$\pi_{k}=\pi_{2}\pi_{1}:X\rightarrow X_{k}$.

Set $\Omega=\pi_{k}^{-1}\Omega_{4}\cap\Omega^{\prime}$. We claim $\Omega$ is
the set we need. To show the claim let $x\in\Omega$ and

$x_{1},\ldots,x_{d}\in(\pi_{k}^{-1}\pi_{k}(x))\cap\Omega^{\prime}=(\pi_{1}^{-1}\pi_{2}^{-1}\pi_{2}\pi_{1}(x))\cap\Omega^{\prime}.$

Then $\pi_{1}(x_{1}),\ldots,\pi_{1}(x_{d})\in\pi_{2}^{-1}\pi_{2}\pi_{1}(x).$
It is clear that $\pi_{2}(\pi_{1}(x_{i}))=\pi_{2}\pi_{1}(x)$, $1\leq i\leq d$
and $\pi_{1}(x)\in\Omega_{2}\subset X_{\infty}$. Thus by (4) there is a
sequence $\\{n_{i}\\}$ of ${\mathbb{Z}}$ such that

$(T^{p_{1}(n_{i})}\pi_{1}(x),\ldots,T^{p_{d}(n_{i})}\pi_{1}(x))\rightarrow(\pi_{1}(x_{1}),\ldots,\pi_{1}(x_{d})),\
as\ i\to\infty.$

That is,

$(\pi_{1}(T^{p_{1}(n_{i})}x),\ldots,\pi_{1}(T^{p_{d}(n_{i})}x))\rightarrow(\pi_{1}(x_{1}),\ldots,\pi_{1}(x_{d})),\
as\ i\to\infty.$

Since $x_{i}\in\Omega^{\prime}$ we have that $\pi_{1}(x_{i})\in\Omega_{3}$,
which implies that $\pi_{1}(x_{i})$ is a continuous point for
$y\mapsto\pi_{1}^{-1}(y)$, $1\leq i\leq d$. By Lemma 3.8 we know that for any
$i\in\mathbb{N}$

$\prod_{j=1}^{d}\pi_{1}^{-1}\pi_{1}(T^{p_{j}(n_{i})}x)\subset\overline{\\{(T^{p_{1}(n)}x,\ldots,T^{p_{d}(n)}x):n\in{\mathbb{Z}}\\}}.$

So, we get that
$(x_{1},\ldots,x_{d})\in\overline{\\{(T^{p_{1}(n)}x,\ldots,T^{p_{d}(n)}x):n\in{\mathbb{Z}}\\}}.$
Since such tuples $(x_{1},\ldots,x_{d})$ are dense in
$\pi_{k}^{-1}\pi_{k}(x)$, we conclude that

$(\pi_{k}^{-1}\pi_{k}(x))^{d}\subset\overline{\\{(T^{p_{1}(n)}x,\ldots,T^{p_{d}(n)}x):n\in{\mathbb{Z}}\\}}.$

This ends the proof. ∎

###  3.4. Remarks

It is clear that if $C$ is a set of linear polynomials, the number $k$ can be
chosen to be $d-1$ by [11] or [21].

Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. Let

${\mathcal{H}}_{\rm rat}=\overline{\\{f\in L^{2}(X,\mu):\exists
a\in\mathbb{N}\ s.t.\ T^{a}f=f\\}},$

and let $\mathcal{X}_{\rm rat}\subseteq\mathcal{X}$ such that
${\mathcal{H}}_{\rm rat}=L^{2}(X,\mathcal{X}_{\rm rat},\mu).$

######  Theorem 3.9.

[5, Lemma 3.14] Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. and let $q(n)$ be a
non-constant integral polynomial. Then for all $f\in L^{2}(X,\mu)$,

$\frac{1}{N}\sum_{n=0}^{N-1}T^{q(n)}f-\frac{1}{N}\sum_{n=0}^{N-1}T^{q(n)}{\mathbb{E}}(f|\mathcal{X}_{\rm
rat})\longrightarrow 0,\ N\to\infty,$

in $L^{2}(X,\mu)$. In particular, if $(X,\mathcal{X},\mu,T)$ is totally
ergodic ††A m.p.s. $(X,\mathcal{X},\mu,T)$ is totally ergodic if
$(X,\mathcal{X},\mu,T^{k})$ is ergodic for all
$k\in{\mathbb{Z}}\setminus\\{0\\}$., then

$\frac{1}{N}\sum_{n=0}^{N-1}T^{q(n)}f\longrightarrow\int_{X}fd\mu,\
N\to\infty,$

in $L^{2}(X,\mu)$.

For a minimal system $(X,T)$ we may define $X_{rat}$, a topological version of
$Z_{rat}$ (where $(Z_{rat},\mathcal{X}_{\rm rat},\mu_{rat},T)$ is the factor),
which is called rational topological Kronecker factor in [7]. It is an adding
machine, and maximal in the sense that if $Y$ is an adding machine and is a
factor of $X$ then $Y$ is a factor of $X_{rat}$. $X_{rat}$ reflects the number
of minimal subsystems of $(X,T^{n})$, $n\in{\mathbb{Z}}\setminus\\{0\\}$. Note
that for an $\infty$-step pronilsystem, $X_{rat}=Z_{rat}$.

By using Lemma 3.8, the arguments in Subsection 3.2 (replacing Theorem 3.3 by
Theorem 3.9) we get that if $(X,T)$ is minimal and $p$ is a non-constant
integral polynomial vanishing at $0$, then there is a dense $G_{\delta}$ set
$\Omega$ of $X$ such that for each $x\in\Omega$,
$\pi_{rat}^{-1}\pi_{rat}(x)\subset\overline{\\{T^{p(n)}x:n\in{\mathbb{Z}}\\}}$,
where $\pi_{rat}:X\rightarrow X_{rat}$ is the factor map.

Particularly, when $(X,T)$ is totally minimal ††$(X,T)$ is totally minimal if
$(X,T^{n})$ is minimal for any $n\in{\mathbb{Z}}\setminus\\{0\\}$. $X_{rat}$
is trivial. Thus, the above conclusion gives another approach to the result in
[10, 18] for a single polynomial.

##  4\. Applications

In this section we will provide applications of Theorem A, i.e. prove Theorem
B. Namely, we will use it to study the regionally proximal relation along
polynomials. We begin with the general case and then obtain a corollary when
the polynomials are linear.

###  4.1. General cases

Let us recall a definition. Let $(X,T)$ be a t.d.s., $d\in\mathbb{N}$ and
$C=\\{p_{1},\ldots,p_{d}\\}$ be distinct non-constant integral polynomials
with $p_{i}(0)=0$, $1\leq i\leq d$. We say $(x,y)$ is regionally proximal
along $C$ if for each neighborhood $U\times V$ of $(x,y)$ and $\varepsilon>0$
there are $(x^{\prime},y^{\prime})\in U\times V$ and $n\in{\mathbb{Z}}$ such
that

$\rho(T^{p_{i}(n)}x^{\prime},T^{p_{i}(n)}y^{\prime})<\varepsilon,\ 1\leq i\leq
d.$

The set of all such pairs is denoted by ${\bf RP}^{[d]}_{C}(X,T)$.

Then it’s clear that:

######  Proposition 4.1.

Let $(X,T)$ be a t.d.s., and $\pi:X\rightarrow Y$ be a factor map. Then we
have

  1. (1)

${\bf RP}^{[d]}_{C}$ is a closed $T\times T$-invariant relation.

  2. (2)

$(\pi\times\pi)({\bf RP}^{[d]}_{C}(X))\subset{\bf RP}^{[d]}_{C}(Y).$

For a t.d.s. $(X,T)$ we use $h(T)$ to denote the topological entropy of $T$.
To study the entropy property of the minimal systems with ${\bf
RP}^{[d]}_{C}(X)=\Delta$ we need the properties of an entropy pair which was
studied in [2]. It is known that for a t.d.s. $(X,T)$, $h(T)>0$ if and only if
there is an entropy pair $(x_{1},x_{2})$ with $x_{1}\not=x_{2}$. It was shown
in [14] that an entropy pair has an independence set of positive density
††$F\subset\mathbb{N}$ has positive density if
$\lim_{N\rightarrow\infty}\frac{1}{N}|F\cap\\{1,\ldots,N\\}|>0$.. Note that
for $F\subset\mathbb{N}$ we say $(x_{1},x_{2})$ has an independence set $F$ if
for any finite $S\subset F$ and any neighborhood $U_{1}\times U_{2}$ of
$(x_{1},x_{2})$ we have

$T^{-s_{1}}U_{i_{1}}\cap\cdots\cap T^{-s_{l}}U_{i_{l}}\not=\emptyset,$

for any $(i_{1},\ldots,i_{l})\in\\{1,2\\}^{l},$ if
$S=\\{s_{1},\ldots,s_{l}\\}$.

Now we are ready to show Theorem B.

###### Proof of Theorem B.

Let $l$ be an integral polynomial with $l(0)=0$ such that
$l,p_{1},\ldots,p_{d},l+p_{1},\ldots,l+p_{d}$ are distinct non-constant
polynomials. For example, any integral polynomial $l$ with $l(0)=0$ and
$deg(l)>\max\\{deg(p_{1}),\ldots,deg(p_{d})\\}$ is the one we need. Set

$q_{0}=l,q_{1}=p_{1},q_{2}=p_{2},\ldots,q_{d}=p_{d},q_{d+1}=l+p_{1},\ldots,q_{2d}=l+p_{d},$

and $C_{2d+1}=\\{q_{0},q_{1},\ldots,q_{2d}\\}$.

Then by the Theorem A, there exists a $k\in\mathbb{N}$ depending only on
$C_{2d+1}$ such that for a residual subset $\Omega$ of $X$, and $x\in\Omega$
we have $(\pi^{-1}_{k}(\pi_{k}(x)))^{2d+1}\subset L_{x}^{C_{2d+1}}$, where
$\pi_{k}:X\to X_{k}$ is the factor map.

Now let $x_{1}\in\Omega$, and $x_{2}\in\pi_{k}^{-1}(\pi_{k}(x_{1}))$. Let
$\varepsilon>0$ and $U_{1},U_{2}$ be neighborhoods of $x_{1},x_{2}$
respectively with $\operatorname{diam}(U_{2})<\varepsilon$. Since
$(\pi^{-1}_{k}(\pi_{k}(x_{1})))^{2d+1}\subset L_{x_{1}}^{C_{2d+1}}$ we get
that there is $n\in{\mathbb{Z}}$ such that

$T^{q_{0}(n)}x_{1},T^{q_{1}(n)}x_{1},\ldots,T^{q_{2d}(n)}x_{1}\in U_{2}.$

That is, we have

$T^{l(n)}x_{1},T^{p_{1}(n)}x_{1},\ldots,T^{p_{d}(n)}x_{1},T^{l(n)+p_{1}(n)}x_{1},T^{l(n)+p_{2}(n)}x_{1}\ldots,T^{l(n)+p_{d}(n)}x_{1}\in
U_{2}.$

Set $x^{\prime}=x_{1}\in U_{1},y^{\prime}=T^{l(n)}x_{1}\in U_{2}$. Then we
have

$\rho(T^{p_{i}(n)}x^{\prime},T^{p_{i}(n)}y^{\prime})<\varepsilon,\ 1\leq i\leq
d.$

This implies that $(x_{1},x_{2})\in{\bf RP}^{[d]}_{C}$, where
$C=\\{p_{1},\ldots,p_{d}\\}$. As ${\bf RP}^{[d]}_{C}=\Delta$ we conclude that
$x_{2}=x_{1}$, i.e. $\pi_{k}$ is almost one-to-one.

Particularly, if $(X,T)$ is distal then it is a $k$-step pro-nilsystem by what
we just proved.

Now we show $h(T)=0$. Assume the contrary that $h(T)>0$. So, there is an
entropy pair $(x_{1},x_{2})$ in $X\times X$ with $x_{1}\not=x_{2}$. Fix
$\varepsilon>0$. For any neighborhood $U_{1}\times U_{2}$ of $(x_{1},x_{2})$
with $\operatorname{diam}(U_{2})<\varepsilon$, there is an independent set $F$
of positive density associated to $U_{1}\times U_{2}$. For the polynomials
$l,l+p_{1},\ldots,l+p_{d}$, by Bergelson and Leibman’s theorem (see [1]) there
are $m,n\in{\mathbb{Z}}$ with

$m+l(n),m+l(n)+p_{1}(n),m+l(n)+p_{2}(n),\ldots,m+l(n)+p_{d}(n)\in F.$

Thus, we have

$T^{-m-l(n)}U_{1}\cap T^{-m-l(n)-p_{1}(n)}U_{2}\cap\cdots\cap
T^{-m-l(n)-p_{d}(n)}U_{2}\not=\emptyset$

and

$T^{-m-l(n)}U_{2}\cap T^{-m-l(n)-p_{1}(n)}U_{2}\cap\cdots\cap
T^{-m-l(n)-p_{d}(n)}U_{2}\not=\emptyset.$

This implies that

$U_{1}\cap T^{-p_{1}(n)}U_{2}\cap\cdots\cap T^{-p_{d}(n)}U_{2}\not=\emptyset$

and

$U_{2}\cap T^{-p_{1}(n)}U_{2}\cap\cdots\cap T^{-p_{d}(n)}U_{2}\not=\emptyset.$

Thus there are $y_{1}\in U_{1}$ with $T^{p_{i}(n)}y_{1}\in U_{2}$, and
$y_{2}\in U_{2}$ with $T^{p_{i}(n)}y_{2}\in U_{2}$, $1\leq i\leq d$. This
implies that $\rho(T^{p_{i}(n)}y_{1},T^{p_{i}(n)}y_{2})<\varepsilon$, $1\leq
i\leq d$. Since $\varepsilon$ is arbitrary, we get that $(x_{1},x_{2})\in{\bf
RP}^{[d]}_{C}$, a contradiction. This proves that $h(T)=0$. ∎

###  4.2. Linear case

To refine Theorem B for the linear case, we need a lemma from [9, Theorem
3.8].

######  Lemma 4.2.

Let $\pi:(X,T)\longrightarrow(Y,T)$ be a proximal extension between two t.d.s.
Then ${\bf AP}^{[d]}(X)\supset R_{\pi}=\\{(x,y)\in X^{2}:\pi(x)=\pi(y)\\}$ for
any $d\in\mathbb{N}$.

Now let $C$ be the polynomials $\\{n,2n,\cdots,dn\\}$. We have a corollary of
Theorem B.

######  Corollary 4.3.

Let $(X,T)$ be a minimal t.d.s. with ${\bf AP}^{[d]}=\Delta$ for some
$d\in\mathbb{N}$, then it is a $d$-step pro-nilsystem.

###### Proof.

As ${\bf AP}^{[d]}=\Delta$, Theorem B tells us that $(X,T)$ is an almost one-
to-one extension of some $X_{k}$ for some $k\in\mathbb{N}$.

By Lemma 4.2, we conclude that $R_{\pi}=\Delta$ (where $\pi:X\rightarrow
X_{k}$ is the factor map) and thus $(X,T)$ is a $k$-step pro-nilsystem. Then
we have ${\bf AP}^{[d]}={\bf RP}^{[d]}=\Delta$ (see [9, Theorem 5.9]) which
implies that $(X,T)$ is a $d$-step pro-nilsystem by Theorem 2.3. ∎

We note that Corollary 4.3 gives a negative answer to a conjecture in [10,
Conjecture 4], which states that there is a minimal t.d.s. $(X,T)$ such that
${\bf AP}^{[2]}=\Delta$, and $(X,T)$ is not distal.

###  4.3. Questions

We end the paper by stating some open questions.

Recently, Qiu and Yu [19] obtained a measure-theoretical version of the
saturation theorem along cubes. We believe that Theorem A should have a
measure-theoretical version similar to [19, Theorem A] for polynomials (then
we could conclude that ${\bf RP}^{[d]}_{C}(X,T)=\Delta$ implies that $(X,T)$
is uniquely ergodic). Unfortunately, we do not know how to prove it even the
polynomials are linear at this moment.

Another question is that: is there a minimal t.d.s. $(X,T)$ such that ${\bf
RP}^{[1]}_{\\{n^{2}\\}}=\Delta$ and $(X,T)$ is not an equicontinuous system?
We think that such a system exists.

## References

  * [1] V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden’s and Szemerédi’s theorems, J. Amer. Math. Soc., 9 (1996), 725–753.
  * [2] F. Blanchard, A disjointness theorem involving topological entropy, Bull. Soc. Math. France, 121 (1993), 465-478.
  * [3] G. Choquet, Convergences, Annales de l’université de Grenoble, 23 (1947–1948), 55–112.
  * [4] P. Dong, S. Donoso, A. Maass, S. Shao and X. Ye, Infinite-step nilsystems, independence and complexity, Ergod. Th. and Dynam. Sys., 33(2013), 118-143.
  * [5] H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, M. B. Porter Lectures. Princeton University Press, Princeton, N.J., 1981.
  * [6] H. Furstenberg and B. Weiss, A mean ergodic theorem for $\frac{1}{N}\sum_{n=1}^{N}f(T^{n}x)g(Tn^{2}x)$. In: Convergence in Ergodic Theory and Probability. Ohio State Univerrsity Mathematical Research Institute Publications, vol. 5. Berlin: de Gruyter, 1996, 193-227.
  * [7] D. Glasscock, A. Koutsogiannis and F. Richter, Multiplicative combinatorial properties of return time sets in minimal dynamical systems, Discrete Contin. Dyn. Syst. 39 (2019), 5891-5921.
  * [8] E. Glasner, Topological ergodic decompositions and applications to products of powers of a minimal transformation, J. Anal. Math., 64 (1994), 241–262.
  * [9] E. Glasner, W. Huang, S. Shao and X. Ye, Regionally proximal relation of order $d$ along arithmetic progressions and nilsystems. Sci. China Math, 63 (2020), no. 9, 1757-1776.
  * [10] E. Glasner, W. Huang, S. Shao, and B. Weiss, X. Ye, Topological characteristic factors and nilsystems, arXiv:2006.12385 [math.DS], JEMS, to appear.
  * [11] B. Host and B. Kra, Nonconventional averages and nilmanifolds, Ann. of Math., 161(2005), 398–488.
  * [12] B. Host and B. Kra, Convergence of polynomial ergodic averages, Israel J. Math. 149 (2005), 1-19.
  * [13] B. Host, B. Kra and A. Maass, Nilsequences and a structure theory for topological dynamical systems, Adv. Math., 224 (2010), 103–129.
  * [14] W. Huang and X. Ye, A local variational relation and applications, Israel J. Math. 151(2006). 237–280.
  * [15] W. Huang, S. Shao and X. Ye, Topological dynamical systems induced by polynomials and combinatorial consequences, arXiv: 2301.07873.
  * [16] A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math. 146 (2005), 303-315.
  * [17] S. Shao and X. Ye, Regionally proximal relation of order $d$ is an equivalence one for minimal systems and a combinatorial consequence, Adv. Math., 231(2012), 1786–1817.
  * [18] J. Qiu, Polynomial orbits for totally minimal systems, Adv. Math., 432 (2023), 109260.
  * [19] J. Qiu and J. Yu, Saturated theorem along cubes for a measure and applications, arXiv: 2311.14198.
  * [20] W. A. Veech, Point-distal flows, Amer. J. Math., 92(1970), 205-242.
  * [21] T. Ziegler, Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc. 20 (2007), no.1, 53–97.