1 Introduction - Main result
1.1 The joint ergodicity problem
Let
$(X,\mathcal {B},\mu)$
be a standard probability space equipped with an invertible measure-preserving transformation
$T:X\to X$
(that is,
$\mu (TA)=\mu (A)$
for every
$A\in \mathcal {B}$
). We say that the quadruple
$(X,\mathcal {B},\mu ,T)$
is a measure-preserving system. In particular, the latter is called ergodic or weakly mixing if the transformation T is ergodic (i.e., every T-invariant set
$A\in \mathcal {B}$
satisfies
$\mu (A)\in \{0,1\}$
) or weakly mixing (i.e., the transformation
$T\times T,$
acting on the Cartesian square
$X^2:=X\times X$
, is ergodic), respectively.
Given a weakly mixing measure-preserving system
$(X,\mathcal {B},\mu ,T)$
and distinct nonzero integers
$a_1,\ldots ,a_d,$
we have the following independence property of the sequences
$(T^{a_i n})_n, 1\leq i\leq d.$
Footnote
1
Theorem 1.1 [Reference Furstenberg25].
Let
$(X,\mathcal {B},\mu ,T)$
be a weakly mixing measure-preserving system. Then, for any
$d\in \mathbb {N}, $
any distinct nonzero integers
$a_1,\ldots ,a_d,$
and any
$f_1,\ldots ,f_d\in L^\infty (\mu)$
, we have
$$ \begin{align}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T^{a_1 n} f_1\cdot\ldots\cdot T^{a_d n}f_d=\int_X f_1\;d\mu\cdot\ldots\cdot\int_X f_d\;d\mu,\end{align} $$
where the convergence takes place in
$L^2(\mu).$
This result, in particular its recurrence reformulation on
$\mathcal {B}$
-measurable sets of positive measure, is a crucial ingredient in Furstenberg’s approach in proving Szemerédi’s theorem (that is, every subset of natural numbers of positive upper density contains arbitrarily long arithmetic progressions) by recasting it as a recurrence problem.
Later, in [Reference Bergelson5], Bergelson extended Theorem 1.1 to essentially distinct integer polynomial iterates. The convergence of general multiple ergodic averages for various classes of iterates to the right-hand side of (1.1), also known as the ‘expected limit’, developed to be a topic on its own (e.g., see [Reference Bergelson5, Reference Bergelson, Moreira and Richter7, Reference Chu, Frantzikinakis and Host9, Reference Donoso, Ferré Moragues, Koutsogiannis and Sun12, Reference Donoso, Koutsogiannis, Kuca, Tsinas and Sun13, Reference Donoso, Koutsogiannis and Sun14, Reference Donoso, Koutsogiannis and Sun15, Reference Frantzikinakis19, Reference Frantzikinakis20, Reference Frantzikinakis21, Reference Frantzikinakis and Kra22, Reference Karageorgos and Koutsogiannis30, Reference Koutsogiannis31, Reference Koutsogiannis32, Reference Koutsogiannis and Sun33, Reference Tsinas39] for various results on polynomial and Hardy field functions of polynomial growth); the one of joint ergodicity.
Definition 1.2. Let
$(X,\mathcal {B},\mu ,T_{1},\dots ,T_{d})$
be a measure-preserving system with commuting and invertible transformationsFootnote
2
and
$(a_1(n))_n,\ldots ,(a_d(n))_n$
be integer-valued sequences. We say that
$(T_1^{a_1(n)})_n,\ldots , (T_d^{a_d(n)})_n$
are jointly ergodic (for
$\mu $
) if for any
$f_1,\ldots , f_d\in L^\infty (\mu)$
, we have
$$ \begin{align} \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T_1^{a_1(n)}f_1\cdot\ldots\cdot T_d^{a_d(n)} f_d=\int_X f_1\;d\mu\cdot\ldots\cdot\int_X f_d\;d\mu,\end{align} $$
where the convergence takes place in
$L^2(\mu)$
(i.e., the
$L^2(\mu)$
-limit of the left-hand side of (1.2) exists, and it takes the value of the right-hand side).
The first characterization of joint ergodicity is due to Berend and Bergelson.
Theorem 1.3 [Reference Berend and Bergelson4].
Let
$(X,\mathcal {B},\mu ,T_{1},\dots ,T_{d})$
be a measure-preserving system with commuting and invertible transformations. Then
$(T_{1}^n)_n,\dots ,(T_{d}^n)_n$
are jointly ergodic (for
$\mu $
) if and only if both of the following conditions are satisfied:
-
(i)
$T_{i}T^{-1}_{j}$
is ergodic (for
$\mu $
) for all
$1\leq i,j\leq d, i\neq j$
; and -
(ii)
$T_{1}\times \dots \times T_{d}$
is ergodic (for
$\mu ^{\otimes d},$
where
$\mu ^{\otimes d}:=\mu \times \cdots \times \mu $
is the product measure in
$X^d$
).
There is a plethora of analogous joint ergodicity characterizations for generalized linear functions [Reference Bergelson, Leibman and Son6], polynomial functions [Reference Donoso, Ferré Moragues, Koutsogiannis and Sun12, Reference Donoso, Koutsogiannis and Sun15, Reference Frantzikinakis and Kuca23, Reference Frantzikinakis and Kuca24] and Hardy field functions [Reference Donoso, Koutsogiannis, Kuca, Tsinas and Sun13, Reference Donoso, Koutsogiannis and Sun16].
The objective of this article is to prove a topological counterpart of Theorem 1.3.
1.2 The joint transitivity problem
A
$\mathbb {Z}^{k}$
-system is a tuple
$(X,S_1,\ldots ,S_k)$
, where X is a compact metric space and
$S_{1},\dots ,S_{k}\colon X\to X$
are homeomorphisms with
$S_i S_j= S_j S_i$
for all
$1\leq i,j\leq k$
. We let
$\langle S_1,\ldots ,S_k\rangle $
denote the group generated by
$S_1,\ldots ,S_k$
.
In order to state the result corresponding to Theorem 1.3 in the topological setting, we start with the notion of joint transitivity; recall that a
$G_\delta $
set is a subset of a topological space that is a countable intersection of open sets.
Definition 1.4. Let
$(X,S_1,\ldots ,S_k)$
be a
$\mathbb {Z}^k$
-system,
$T_1,\ldots ,T_d\in \langle S_1,\ldots ,S_k\rangle $
and
$(a_1(n))_n,\ldots , (a_d(n))_n$
be integer-valued sequences. We say that
$(T_1^{a_1(n)})_n,\ldots ,(T_d^{a_d(n)})_n$
are jointly transitive if there is a
$G_{\delta }$
-dense subset
$X_0\subseteq X$
such that for all
$x\in X_0$
, the set
$$\begin{align*}\{ (T_1^{a_1(n)}x,\ldots,T_d^{a_d(n)}x) : n\in \mathbb{Z}\} \end{align*}$$
is dense in
$X^d=X\times \cdots \times X$
(d times).Footnote
3
In the
$d=1$
case, we say that
$(T_1^{a_1(n)})_n$
is a transitive sequence.
We call a
$\mathbb {Z}^k$
-system
$(X,S_1,\ldots ,S_k)$
minimal if, for any point
$x\in X$
, its orbit
$\{ S_1^{m_1}\cdot \ldots \cdot S_k^{m_k} x: (m_1,\ldots ,m_k)\in \mathbb {Z}^k\}$
is dense in X. A system is transitive if there exists a point
$x\in X$
whose orbit is dense; we say that any such point x is a transitive point (of
$(X,S_1,\ldots ,S_k)$
). The pioneer work of Glasner [Reference Glasner26] established joint transitivity for sequences given by iterates of powers of a transformation in a minimal and (topologically) weakly mixing
$\mathbb {Z}$
-system
$(X,T)$
(meaning that the product system
$(X\times X, T\times T)$
is transitive).
Theorem 1.5 [Reference Glasner26].
Let
$(X,T)$
be a minimal weakly mixing
$\mathbb {Z}$
-system. Then, for any
$d\in \mathbb {N}$
, and distinct nonzero integers
$a_1,\ldots ,a_d$
, the sequences
$(T^{a_1 n})_n,\ldots ,(T^{a_d n})_n$
are jointly transitive.
Theorem 1.5 can be thought as the topological analogue of Theorem 1.1. This result was extended by Huang, Shao and Ye in [Reference Huang, Shao and Ye29], who obtained topogically joint ergodicity results under weakly mixing assumptions of several transformations but were able to deal with polynomial expressions and nilpotent group actions. Some follow-up works on this line are given in [Reference Cao and Shao8, Reference Zhang and Zhao41].
We are now ready to state our main result, which can be regarded as the topological version of Theorem 1.3.
Theorem 1.6. Let
$(X,S_1,\ldots ,S_k)$
be a minimal system and
$T_1,\ldots ,T_d \in \langle S_1,\ldots ,S_k\rangle $
. Then
$(T_1^n)_n,\ldots ,(T_d^n)_n$
are jointly transitive if and only if both of the following conditions are satisfied:
-
(i)
$(X,T_i^{-1}T_j)$
is transitive for all
$1\leq i, j\leq d,\; i\neq j$
; and -
(ii)
$(X^d,T_1\times \cdots \times T_d)$
is transitive.
Remark 1.7. As in the measurable case with Theorem 1.3, Theorem 1.6 provides a characterization for all linear iterates.
Indeed, assuming that the iterate of the
$T_i$
is
$a_i n+b_i, a_i\in \mathbb {Z}\setminus \{0\}, b_i\in \mathbb {Z}, 1\leq i\leq d,$
noting that the shifts by the
$b_i$
’s do not affect the denseness of the orbits, we can use the Theorem 1.6 for the functions
$T^{a_i}$
(which still belong to
$\langle S_1,\ldots ,S_k\rangle $
).
It is important to note that the problem in the topological setting differs significantly from the one in the measurable setting as the dense
$G_{\delta }$
subset of X might have zero measure. For example, a minimal topologically weakly mixing system
$(X,T)$
may exhibit a discrete spectrum with respect to some invariant measure. In such a system,
$(T^n)_n,\ldots (T^{dn})_n$
are jointly transitive but not jointly ergodic. In fact, any ergodic measure-preserving system is measurably isomorphic to a minimal and uniquely ergodic topologically (strongly) mixing system (see [Reference Lehrer35]).
We also want to emphasize that Theorem 1.6 fails without the minimality assumption. Indeed, in [Reference Moothathu36], Moothathu showed that there exists a nonminimal, strongly mixing shift
$(X,\sigma)$
such that, for every point
$x\in X$
, the set
$\{(\sigma ^n x, \sigma ^{2n}x):n\in \mathbb {Z}\}$
fails to be dense in
$X^2$
. Because
$(X,\sigma )$
is strongly mixing, the
$\mathbb {Z}^2$
-system
$(X,\sigma ,\sigma ^2)$
satisfies conditions
$(i)$
and
$(ii)$
of Theorem 1.6, but the sequences
$(\sigma ^n)_n,(\sigma ^{2n})_n$
are not jointly transitive. It should be noted that there are no commuting transformations
$S_1,\ldots ,S_k$
that generate a minimal action and such that
$\sigma \in \langle S_1,\ldots ,S_k\rangle $
. One reason for this is, of course, Theorem 1.6, but this can also be seen directly in Moothathus’s example, using the fact that the set of
$\sigma $
-periodic points of a given period have to be preserved by
$\langle S_1,\ldots ,S_k\rangle $
, which prevents minimality.
Due to Theorem 1.6 and recent developments in the theory of topological factors, we believe that there will be numerous results in the joint transitivity problem in the near future.
Problem 1.8. Analogously to Theorem 1.6, obtain joint transitivity characterizations for iterates that come from polynomial and Hardy field of polynomial growth functions, for which the corresponding results in the measure theoretic setting are known.
In particular, we chose to state the following conjecture which is the topological analogue of [Reference Donoso, Koutsogiannis and Sun15, Theorem 1.4]: a natural extension of the linear case.
Conjecture 1.9. Let
$(X,S_1,\ldots ,S_k)$
be a minimal system,
$T_1,\ldots ,T_d \in \langle S_1,\ldots ,S_k\rangle $
and
$p\in \mathbb {Z}[t]$
. Then
$(T_1^{p(n)})_n,\ldots ,(T_d^{p(n)})_n$
are jointly transitive if and only if both of the following conditions are satisfied:
-
(i)
$((T_i^{-1}T_j)^{p(n)})_n$
is transitive (in the space X) for all
$1\leq i, j\leq d,\; i\neq j$
; and -
(ii) the sequence
$((T_1\times \cdots \times T_d)^{p(n)})_{n}$
is transitive (in the product space
$X^{d}$
).
1.3 Structure of the paper
In Section 2, we recall some notions from the theory of dynamical systems. In particular, we provide equivalent statements to joint transitivity (Lemma 2.1), and we list properties of dynamical cubes and regional proximal relations.
In Section 3, we first characterize (in Theorem 3.1) regional proximal relations for product transformations, and finally, we prove Theorem 1.6 by an inductive argument.
2 Background material and useful facts
Definitions and conventions. For any
$\mathbb {Z}^{k}$
-system
$(X,S_{1},\dots ,S_{k})$
and
$m=(m_{1},\dots ,m_{k})\in \mathbb {Z}^{k}$
, we write 
. So, we may write a
$\mathbb {Z}^{k}$
-system as
$(X,(S_{m})_{m\in \mathbb {Z}^{k}})$
whenever we do not need to stress the generators. With this convention, a
$\mathbb {Z}^k$
-system is minimal if for any point
$x\in X$
, its orbit
$\{ S_{m} x: m\in \mathbb {Z}^k\}$
is dense in X. We adopt a similar notation for subgroups of
$\mathbb {Z}^k$
. If
$G\subseteq \mathbb {Z}^k$
is a subgroup, then
$(X,(S_{m})_{m\in G})$
denotes the system given by the subaction of G.
We will use
$\rho $
to denote the metric on X, and slightly abusing notation, we will use
$\rho $
to denote the metric on the product space
$X^d$
as well, where
$\rho ((x_1,\ldots ,x_d),(x_1',\ldots ,x_d'))=\sup _{1\leq i\leq d} \rho (x_i,x_i')$
.
Let
$(X,S_{1},\dots ,S_{k})$
and
$(Y,R_{1},\dots ,R_{k})$
be two
$\mathbb {Z}^{k}$
-systems. We say that Y is a factor of X (or that X is an extension of Y) if there exists a continuous and onto map
$\pi \colon X\to Y$
(called the factor map from X to Y) such that
$\pi \circ S_{i}=R_{i}\circ \pi $
for all
$1\leq i\leq k$
. By slightly abusing the notation, we will sometimes use the same letter to denote the transformations that act on the space X and the factor
$Y.$
When
$\pi $
is a homeomorphism, we say that the systems are topologically conjugate.
There is a one-to-one correspondence between the factors and the closed invariant equivalence relations on X. Indeed, we can associate a factor map
$\pi \colon X\to Y$
with the relation
$R_{\pi }=\{(x,y): \pi (x)=\pi (y)\}$
, and conversely, given a closed invariant equivalence relation R, we can associate it with a factor map
$X\to X/R$
being the quotient map. A factor map
$\pi \colon X\to Y$
is almost one-to-one if there exists a
$G_{\delta }$
-dense subset
$\Omega $
of X such that for any
$x\in \Omega $
,
$\pi ^{-1}(\pi (x))=\{x\}$
. A factor map
$\pi \colon X\to Y$
is open if
$\pi (A)\subseteq Y$
is open whenever
$A\subseteq X$
is open. Note that the latter implies that for any
$(x,x')\in R_{\pi }$
, and
$\epsilon>0$
, there exists
$\delta>0$
such that if
$\rho (x,y)<\delta $
, then there exists
$y'\in X$
with
$\rho (x',y')<\epsilon $
and
$(y,y')\in R_{\pi }$
.
A system
$(X,S_1,\ldots ,S_k)$
is equicontinuous if the family of functions generated by
$S_1,\ldots ,S_k$
is equicontinuous. Any minimal equicontinuous
$\mathbb {Z}^k$
-system is topologically conjugate to a rotation on a compact abelian group (see [Reference Auslander2, Chapter 2]). The maximal equicontinuous factor of a
$\mathbb {Z}^k$
-system
$(X,S_1,\ldots ,S_k)$
, is the largest equicontinuous factor of it. That is, any equicontinuous factor of
$(X,S_1,\ldots ,S_k)$
is also a factor of the maximal one. The maximal equicontinuous factor of a minimal
$\mathbb {Z}^k$
-system is induced by the regionally proximal relation
$\mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(X)$
(see Section 2.2).
Let
$(X,S_1,\ldots ,S_k)$
be a
$\mathbb {Z}^k$
-system. A pair
$(x,y)$
is proximal if there exists a sequence
$(n_i)_i$
in
$\mathbb {Z}^k$
such that
$\rho (S_{n_i}x,S_{n_i}y)\to 0$
as i goes to infinity. The set of all proximal pairs is denoted by
$P(X)$
. It is well-known that
$P(X)\subseteq \mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(X)$
(see, for instance, [Reference Shao and Ye38]). A factor map
$\pi \colon X\to Y$
is proximal if
$R_{\pi }\subseteq P(X)$
. Any almost one-to-one factor map between minimal systems is proximal (see [Reference Auslander2, Chapter 11]).
2.1 Equivalent definitions for joint transitivity
The following lemma provides a couple of equivalent definitions for joint transitivity. We will use this lemma implicitly throughout this paper. Its proof is a direct generalization of [Reference Qiu37, Lemma 2.8] (see also [Reference Huang, Shao and Ye29, Lemma 2.4] and [Reference Moothathu36]).
Lemma 2.1. Let
$(X,S_{1},\dots ,S_{k})$
be a minimal
$\mathbb {Z}^k$
-system and
$(a_{1}(n))_n,\ldots ,(a_{d}(n))_n$
be sequences with values in
$\mathbb {Z}^k$
. The following are equivalent:
-
(i) There exists a dense
$G_{\delta }$
subset
$\Omega $
of X such that the set is dense in
$$ \begin{align*}\{(S_{a_{1}(n)}x,\dots,S_{a_{d}(n)}x)\colon n\in\mathbb{Z}\}\end{align*} $$
$X^{d}$
for every
$x\in \Omega $
.
-
(ii) There exists some
$x\in X$
such that the set is dense in
$$ \begin{align*}\{(S_{a_{1}(n)}x,\dots,S_{a_{d}(n)}x)\colon n\in\mathbb{Z}\}\end{align*} $$
$X^{d}$
.
-
(iii) For every nonempty open subsets
$U,V_{1},\dots ,V_{d}$
of X, there is some
$n\in \mathbb {Z}$
such that
$$ \begin{align*}U \cap S_{-a_{1}(n)}V_{1}\cap \dots\cap S_{-a_{d}(n)}V_{d}\neq\emptyset.\end{align*} $$
Proof. The implication
$(i) \Rightarrow (ii)$
is obvious. We next prove that
$(ii)$
implies
$(iii)$
. To this end, let
$x\in X$
be such that the set
$$ \begin{align*}\{(S_{a_{1}(n)}x,\dots,S_{a_{d}(n)}x)\colon n\in\mathbb{Z}\}\end{align*} $$
is dense in
$X^{d}$
. Then, for any
$m\in \mathbb {Z}^{k}$
, the set
$$ \begin{align*}X(x,m):=\{(S_{a_{1}(n)+m}x,\dots,S_{a_{d}(n)+m}x)\colon n\in\mathbb{Z}\}\end{align*} $$
is also dense in
$X^{d}$
. Now, for any nonempty open subsets
$U,V_{1},\dots ,V_{d}$
of X, by the minimality of
$(X,S_{1},\dots ,S_{k})$
, we may find some
$m\in \mathbb {Z}^{k}$
such that
$S_{m}x\in U,$
and since
$X(x,m)$
is dense in
$X^{d}$
, there exists
$n\in \mathbb {Z}$
such that
$S_{a_{i}(n)+m}x\in V_{i}$
for all
$1\leq i\leq d$
. Therefore, the set
$$\begin{align*}U \cap S_{-a_{1}(n)}V_{1}\cap \dots\cap S_{-a_{d}(n)}V_{d}\end{align*}$$
contains the point
$T_{m}x$
; hence, it is nonempty.
It remains to show that
$(iii)$
implies
$(i)$
. Let
$\mathcal {F}$
be a countable basis of the topology of X and define
It follows from
$(iii)$
that
$\Omega $
is a dense
$G_{\delta }$
set. Moreover, the set
$$\begin{align*}\{(S_{a_{1}(n)}x,\dots,S_{a_{d}(n)}x)\colon n\in\mathbb{Z}\}\end{align*}$$
is dense in
$X^{d}$
for every
$x\in \Omega $
.
2.2 Dynamical cubes and regionally proximal relations
The notion of the regionally proximal relation was introduced by Ellis and Gottschalk [Reference Ellis and Gottschalk18] in the 1960s. Here, we will adapt a few notions from [Reference Donoso and Sun17] to our setting.
Let
$(X,S_1,\ldots ,S_k)$
be a system and
$(G_1,G_2)$
be a pair of subgroups of
$\mathbb {Z}^k$
. Recalling the definition of
$S_g$
for a
$g\in \mathbb {Z}^k$
from the beginning of Section 2, we define the space of
$(G_1,G_2)$
-cubes as
(We remark that such definitions were initially introduced in [Reference Donoso and Sun17], for the case where each
$G_i$
is generated by a single transformation.) Given
$x\in X$
, and
$(G_1,G_2)$
, let
When
$G_i$
is generated by a single element
$g_i$
, we write
$\boldsymbol{Q}_{G_i,G_i}(X)$
simply as
$\boldsymbol{Q}_{S_{g_i},S_{g_i}}(X)$
; a similar notation is used for
$\mathcal {F}_{G_i,G_i}(x)$
. Given a single subgroup G of
$\mathbb {Z}^k$
, we write
(this relation is called the prolongation relation in [Reference Auslander and Guerin3]). Note that if
$(X,(S_m)_{m\in G})$
is transitive (meaning that there exists a point x such that
$\{S_mx : m \in G \}$
is dense in X), then
$\mathbf {RP}_{G}(X)=X\times X$
. Similarly to [Reference Donoso and Sun17] (or [Reference Host, Kra and Maass27] for the case of a
$\mathbb {Z}$
-action), we define the relation
$\mathbf {RP}_{G_1,G_2}(X)$
as the set of points
$(x,y)\in X^2$
such that
$(x,y,y,y)\in \boldsymbol{Q}_{G_1,G_2}(X)$
. It should be noted that if
$(X,G)$
is minimal, then
$\mathbf {RP}_{G,G}(X)$
is nothing more than the classical regionally proximal relation (see [Reference Auslander2, Chapter 9] for more information on this relation).
We need the following lemma.
Lemma 2.2. Let
$(X,S_{1},\dots ,S_{k})$
be a
$\mathbb {Z}^k$
-system and
$G_{1},G_2$
be subgroups of
$\mathbb {Z}^k$
.
-
(i) Let
$\sigma \colon X^4\to X^4$
be the map with . Then
$\sigma (\boldsymbol{Q}_{G_1,G_2}(X))=\boldsymbol{Q}_{G_2,G_1}(X)$
. -
(ii) Consider the system
$(\mathbf {RP}_{G_1}(X),G_2^{\Delta })$
, where
$G_2^{\Delta }$
is the action given by
$g(x,y)=(gx,gy)$
, for all
$g\in G_2$
and
$(x,y)\in X^2.$
Footnote
4
Then
$\mathbf {RP}_{G_2^{\Delta }}(\mathbf {RP}_{G_1}(X))=\boldsymbol{Q}_{G_1,G_2}(X)$
. -
(iii) If H is a subgroup of
$\mathbb {Z}^k$
and
$\mathbf {RP}_{G_1}(X)=\mathbf {RP}_{G_2}(X)$
, then
$\boldsymbol{Q}_{G_1,H}(X)=\boldsymbol{Q}_{G_2,H}(X)$
. -
(iv) If
$G_1',G_2'$
and
$G_1,G_2$
are subgroups of
$\mathbb {Z}^k$
such that
$G_1'\subseteq G_1$
, and
$G_2'\subseteq G_2$
, then
$\boldsymbol{Q}_{G_1',G_2'}(X) \subseteq \boldsymbol{Q}_{G_1,G_2}(X).$
. Then 
Proof.
$(i)$
and
$(iv)$
follow directly from the definitions.
To show
$(ii)$
, first note that for all
$x \in X$
,
$g_1 \in G_1$
, and
$g_2\in G_2$
, the point
$(x,S_{g_1}x,S_{g_2}x,S_{g_1+g_2}x)$
belongs to
$\mathbf {RP}_{G_2^{\Delta }}(\mathbf {RP}_{G_1}(X))$
(here, we naturally identify this point with
$((x,S{g_1}x),(S_{g_2}x,S_{g_1+g_2}x))$
). Therefore,
$\boldsymbol{Q}_{G_1,G_2}(X)\subseteq \mathbf {RP}_{G_2^{\Delta }}(\mathbf {RP}_{G_1}(X))$
. For the converse inclusion, it suffices to show that for any
$(x,y)\in \mathbf {RP}_{G_1}(X)$
and any
$g_2\in G_2$
, we have
$(x,y,S_{g_2}x,S_{g_2}y)\in \boldsymbol{Q}_{G_1,G_2}(X)$
. Let
$\epsilon>0$
and choose
$0<\delta <\epsilon $
so that if
$z,z'\in X$
and
$\rho (z,z')<\delta $
, then
$\rho (S_{g_2}z,S_{g_2}z')<\epsilon $
. We can find
$x'\in X$
and
$g_1\in G_1$
such that
$\rho ((x',S_{g_1}x'),(x,y))<\delta $
. It follows that
$( x',S_{g_1}x',S_{g_2}x',S_{g_1+g_2}x')$
is at distance at most
$\epsilon $
of
$(x,y,S_{g_2}x,S_{g_2}y)$
. Since
$\epsilon>0$
is arbitrary, we conclude that
$(x,y,S_{g_2}x,S_{g_2}y)\in \boldsymbol{Q}_{G_1,G_2}(X)$
as desired.
$(iii)$
follows immediately from
$(ii)$
.
Corollary 2.3. Let
$(X,S_1,\ldots ,S_k)$
be a
$\mathbb {Z}^k$
-system and let
$G\subseteq \mathbb {Z}^k$
be a subgroup such that
$(X,(S_m)_{m\in G})$
is transitive. Then, for any subgroup H of
$\mathbb {Z}^k$
, we have
$\boldsymbol{Q}_{G,H}(X)=\boldsymbol{Q}_{\mathbb {Z}^k,H}(X)$
. In particular,
$\mathbf {RP}_{G,G}(X)=\mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(X)$
.
Proof. Note that by Lemma 2.2
$(iv)$
, the inclusion
$\boldsymbol{Q}_{G,H}(X)\subseteq \boldsymbol{Q}_{\mathbb {Z}^k,\mathbb {Z}^k}(X)$
always holds. In addition, if G is transitive, since
$\mathbf {RP}_{G}(X)=X\times X=\mathbf {RP}_{\mathbb {Z}^k}(X)$
, Lemma 2.2
$(iii)$
implies that
$\boldsymbol{Q}_{G,H}(X)=\boldsymbol{Q}_{\mathbb {Z}^k,H}(X)$
. Using
$(i)$
and
$(iii)$
of Lemma 2.2, we get
$\boldsymbol{Q}_{G,G}(X)=\boldsymbol{Q}_{\mathbb {Z}^k,\mathbb {Z}^k}(X),$
from where
$\mathbf {RP}_{G,G}(X)=\mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(X)$
.
We remark that Corollary 2.3 could also be deduced by using a proof similar to that of [Reference Donoso and Sun17, Lemma 6.13].
Veech [Reference Veech40] proved that the regionally proximal relation is an equivalence relation for a minimal system and an abelian action. The first part of the following, now classical, theorem can be found, for example, in [Reference Auslander2, Chapter 9], while the second one can be found in [Reference Shao and Ye38].
Theorem 2.4. Let
$(X,S_1,\ldots ,S_k)$
be a minimal system. Then
$\mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(X)$
is an equivalence relation, the system
$X/\mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}$
is the maximal equicontinuous factor of X, and this factor is topologically conjugate to a rotation on a compact abelian group.
Furthermore, if
$\pi \colon X\to Y$
is a factor map between minimal
$\mathbb {Z}^k$
-systems, then
$\pi \times \pi (\mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(X))=\mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(Y)$
.
2.3 The O-diagram
The following is a classical theorem in the structural theory of topological dynamical systems and will be very useful for our purposes. We state a version for
$\mathbb {Z}^k$
, giving only the information we need for our work. We note that this theorem is valid for general group actions. For further details, the interested reader may consult [Reference de Vries10, Chapter VI, Section 3] or [Reference Auslander2, Chapter 14].
Theorem 2.5. Let
$(X,S_1,\ldots ,S_k)$
and
$(Y, R_1,\ldots ,R_k)$
be two
$\mathbb {Z}^k$
-minimal systems and
$\pi \colon X \to Y$
a factor map. Then there exist two
$\mathbb {Z}^k$
-minimal systems
$(\tilde {X},\tilde {S}_1,\ldots ,\tilde {S}_k)$
and
$(\tilde {Y}, \tilde {R}_1,\ldots ,\tilde {R}_k)$
and factor maps
$\tilde {\pi }\colon \tilde {X}\to X$
,
$\tilde {\sigma }\colon \tilde {X}\to X$
,
$\tilde {\tau }\colon \tilde {Y}\to Y$
such that the following diagram (which is called the O-diagram)
is commutative,
$\tilde {\sigma }$
and
$\tilde {\tau }$
are almost one-to-one, and
$\tilde {\pi }$
is open.
Theorem 2.5 says that, modulo almost one-to-one extensions, we may assume that the factor map is open.
3 The proof of the main result
3.1 A characterization for the regional proximal relation for product transformations
The following is the main tool we use in the proof of Theorem 1.6 and can be interpreted as a topological analogue of seminorm control in product spaces (see [Reference Donoso, Koutsogiannis and Sun15, Lemma 5.2] or [Reference Donoso, Ferré Moragues, Koutsogiannis and Sun12, Lemma 3.4] for analogous statements in the measurable setting).
Theorem 3.1. Let
$(X,S_1,\dots ,S_k)$
be a minimal
$\mathbb {Z}^k$
-system and
$T_1,\ldots ,T_d \in \langle S_1,\ldots , S_k\rangle $
. Let
$(Y_i,S_1,\ldots ,S_k)$
,
$1\leq i\leq d$
be factors of X, such that for all i, the factor map
$\pi _i\colon X\to Y_i$
is open, and
$R_{\pi _i}\subseteq \mathbf {RP}_{T_i,T_i}(X)$
. For all
$1\leq i\leq d$
, let
$(x_i,y_i)\in R_{\pi _i}$
. Then
$$\begin{align*}((x_1,\dots,x_d),(y_1,\dots,y_d))\in \mathbf{RP}_{T_1\times\dots\times T_d}(X^{d}). \end{align*}$$
Roughly speaking, Theorem 3.1 can be interpreted as saying that
$(X^{d}/\mathbf {RP}_{T_1\times \dots \times T_d}(X^{d}),T_{1}\times \dots \times T_{d})$
is a factor of
$(X/\mathbf {RP}_{T_1,T_1}(X)\times \dots \times X/\mathbf {RP}_{T_d,T_d}(X),T_{1}\times \dots \times T_{d})$
(assuming that all the quotient spaces are well defined).
The main ingredient in proving Theorem 3.1 is to show that if
$(x_{i},y_{i})\in \mathbf {RP}_{T_i,T_i}(X)$
for some fixed
$1\leq i\leq d$
, then we may find a common time n such that a neighborhood of
$x_{j}$
returns to itself under
$T_{j}^{n}$
for all j, while a neighborhood of
$x_{i}$
visits a neighborhood of
$y_{i}$
under
$T_{i}^{n}$
. This is done in Lemma 3.3. We then use Lemma 3.3 repeatedly to move the point
$(x_{1},\dots ,x_{d})$
to
$(y_{1},\dots ,y_{d})$
by changing one coordinate at a time.
In order to prove Lemma 3.3, we need the following lemma which shows that the set of visiting times of x to a neighborhood of y under
$T^{n}$
, when
$(x,y,x) \in {\overline {\mathcal {F}_{T,T}}}(x)$
, contains the difference set of an infinite sequence. The proof can be deduced from the proof of [Reference Huang, Shao and Ye28, Theorem 7.3.2] or by using arguments from [Reference Veech40]. We give a proof for completeness.
Lemma 3.2. Let
$(X,T)$
be a
$\mathbb {Z}$
-system and let
$x,y\in X$
be such that
$(x,y,x) \in {\overline {\mathcal {F}_{T,T}}}(x)$
. Then for any open neighborhood U of y, there is a sequence
$(a_i)_{i\in \mathbb {N}}\subseteq \mathbb {Z}$
of integers taking infinitely many values such that the set
$\{n\in \mathbb {Z} : T^nx\in U\}$
contains
$\{a_j-a_i:\;j>i\}$
.
Proof. Let
$\epsilon>0$
be such that
$B(y,\epsilon )\subseteq U$
. For
$i\in \mathbb {N}$
, set
$\epsilon _i=\epsilon /2^i$
. Construct a sequence
$(\delta _i)_{i\in \mathbb {N}}$
, with
$0<\delta _i<\epsilon _i$
and a sequence
$(m_i,n_i)_{i\in \mathbb {N}}$
in
$\mathbb {Z}\times \mathbb {Z}$
as follows: Let
$n_1,m_1$
be such that
$\rho (T^{n_1}x,x)<\epsilon _1$
,
$\rho (T^{m_1}x,y)<\epsilon _1,$
and
$\rho (T^{n_1+m_1}x,x)<\epsilon _1$
. Pick
$0<\delta _2<\epsilon _2$
such that
$\rho (z,z')<\delta _2$
implies that
$\rho (T^a z,T^a z')<\epsilon _2$
for all
$|a|\leq |n_1|+|m_1|$
. Take
$n_2$
,
$m_2$
such that
$\rho (T^{n_2}x,x)< \delta _2$
,
$\rho (T^{m_2}x,y)< \delta _2$
and
$\rho (T^{n_2+m_2}x,x)<\delta _2$
. (We highlight here that the numbers
$n_2, m_2$
can be taken to be arbitrarily large.) Note that the definition of
$\delta _2$
implies that
$$ \begin{align*} &\rho(T^{n_2+n_1}x,x)< \epsilon_1+\epsilon_2, \ \rho(T^{n_2+n_1+m_1}x,x) < \epsilon_1+\epsilon_2, \ \rho(T^{n_2+m_2+n_1}x,x)<\epsilon_1+\epsilon_2, \\& \qquad\rho(T^{n_2+m_2+n_1+m_1}x,x)< \epsilon_1+\epsilon_2, \ \rho(T^{n_2+m_1}x,y) < \epsilon_1+\epsilon_2, \;\; \text{and}\;\; \rho(T^{n_2+m_2+m_1}x,y) < \epsilon_1+\epsilon_2. \end{align*} $$
So, if we set
$R_1=\{n_1,n_1+m_1\}$
,
$P_1=\{m_1\}$
,
$R_2=\{n_2,n_2+m_2\}$
, we have that
$\rho (T^{a+b}x,x)< \epsilon _1+\epsilon _2$
for all
$a\in R_2$
and
$b\in R_1$
, and
$\rho (T^{a+c}x,y)<\epsilon _1+\epsilon _2$
for all
$a\in R_2$
and
$c\in P_1$
(here, R stands for ‘return’ and P for ‘passage’).
The idea of the proof is that return times associated with large indices are compatible with return times and passages associated with smaller indices. More precisely, inductively, suppose that we have defined
$\delta _i$
,
$m_i$
and
$n_i$
for all
$1\leq i\leq l$
for some
$l\in \mathbb {N}$
, and for the set
$R_i=\{n_i,n_i+m_i\}$
and
$P_i=\{m_i\}$
, we have that if
$a=r_{j_1}+\dots +r_{j_l}$
, with
$r_{j_k}\in R_{j_k}$
,
$j_1<\ldots < j_l$
, then
$\rho (T^ax,x) <\sum _{t=1}^{l} \epsilon _{j_t} $
, and
$\rho (T^{a+c}x,y)< \epsilon _k + \sum _{t=1}^{l} \epsilon _{j_t} $
if
$c\in P_k$
,
$k<j_1$
.
Let
$0<\delta _{i+1}<\epsilon _{i+1}$
be such that
$\rho (z,z')<\delta _{i+1}$
implies that
$\rho (T^a z,T^a z')<\epsilon _{i+1}$
for all
$|a|\leq |n_1|+|m_1|+\cdots + |n_i|+|m_i|$
. Then choose
$n_{i+1}$
and
$m_{i+1}$
such that
$\rho (T^{n_{i+1}}x,x)< \delta _{i+1}$
,
$\rho (T^{n_{i+1}+m_{i+1}}x,x)< \delta _{i+1},$
and
$\rho (T^{m_{i+1}}x,y)< \delta _{i+1}$
, and set
$R_{i+1}=\{n_{i+1}, n_{i+1}+m_{i+1}\}$
,
$P_{i+1}=\{m_{i+1}\}$
.
We claim that if
$a=r_{j_1}+r_{j_2}+\dots + r_{j_l}$
, with
$r_{j_k}\in R_{j_k}$
,
$j_1<\ldots < j_l\leq i+1$
, then
$\rho (T^ax,x)< \sum _{t=1}^{l} \epsilon _{j_t} $
, and
$\rho (T^{a+c}x,y)< \epsilon _k + \sum _{t=1}^{l} \epsilon _{j_t} $
if
$c\in P_k$
,
$k<j_1$
.
We only need to check the case
$j_l=i+1$
. Assume that
$r_{i+1}=n_{i+1}$
(the case
$r_{i+1}=n_{i+1}+m_{i+1}$
is identical). Since
$\rho (T^{n_{i+1}}x,x)< \delta _{i+1}$
, and
$|a-n_{i+1}|\leq |n_1|+|m_1|+\dots +|n_i|+|m_i|$
, we get
$\rho (T^{a}x,T^{a-n_{i+1}}x)< \epsilon _{i+1}$
. By induction,
$\rho (T^{a-n_{i+1}}x,x)< \sum _{t=1}^{l-1}\epsilon _{j_t}$
, so the triangle inequality implies that
$\rho (T^ax,x)<\sum _{t=1}^{l}\epsilon _{j_t}$
, as desired. The estimate of
$\rho (T^{a+c}x,y)$
follows in a similar way.
Now set
$a_i=n_{i+1}+\sum _{j=1}^i (n_j+m_j)$
. Note that we may further require the sequence
$(a_i)_{i\in \mathbb {N}}$
to take infinitely many values (by choosing the
$n_i$
’s and
$m_i$
’s to go to infinity) and
$a_{i+l}-a_i =n_{i+l+1} + \sum _{j=i+2}^{i+l} (n_{j}+m_j) +m_{i+1}$
. Hence, we can rewrite this as
$a_{i+l}-a_i=r_{i+l+1}+\sum _{j=i+2}^{i+l} r_j + c$
, where
$r_j\in R_j$
and
$c\in P_{i+1}$
. It follows from the construction of the sequence
$\{n_i,m_i:i \in \mathbb {N}\}$
that
$\rho (T^{a_{i+l}-a_i}x,y)<\sum _{t=i+1}^{i+l+1}\epsilon _i< \epsilon $
, which implies that
$T^{a_{i+l}-a_i}x\in U,$
as was to be shown.
Lemma 3.3. Let
$(X,S_1,\dots ,S_k)$
be a minimal
$\mathbb {Z}^k$
-system,
$T_1,\ldots ,T_d\in \langle S_1,\ldots , S_k\rangle $
and
$\pi \colon X\to Y$
an open factor map with
$R_{\pi }\subseteq \mathbf {RP}_{T_i,T_i}(X)$
for some
$1\leq i\leq d$
. Let
$x_{1},\dots ,x_{d},x^{\prime }_{i}\in X$
with
$(x_{i},x^{\prime }_{i})\in R_{\pi }$
, and
$U_{z}$
be a neighborhood of z for
$z=x_{1},\dots ,x_{d},x^{\prime }_{i}$
. There exist
$n\in \mathbb {Z}$
such that
$T_{i}^{n}U_{x_{i}}\cap U_{x^{\prime }_{i}}\neq \emptyset $
and
$T_{j}^{n}U_{x_{j}}\cap U_{x_{j}}\neq \emptyset $
for all
$1\leq j\leq d$
.
Proof. The set
$\Omega _i$
of
$\tilde {x}\in X$
such that the set
$\{(a,b,c):(\tilde {x},a,b,c)\in \boldsymbol{Q}_{T_i,T_i}(X) \}$
equals
$\overline {\mathcal {F}_{T_{i},T_i}}(\tilde {x})$
is a dense
$G_{\delta }$
set of points (see, for instance, [Reference Glasner26, Lemma 4.5]). Since
$\pi $
is open, we can find
$\tilde {x}\in U_{x_i}\cap \Omega _{i}$
and
$\tilde {y}\in U_{x_i'}$
with
$(\tilde {x},\tilde {y})\in R_{\pi }$
. Because
$R_{\pi }\subseteq \mathbf {RP}_{T_i,T_i}(X)$
, we have that
$(\tilde {x},\tilde {x},\tilde {y},\tilde {x})\in \boldsymbol{Q}_{T_i,T_i}(X)$
, and since
$\tilde {x}\in \Omega $
, we obtain
$(\tilde {x},\tilde {y},\tilde {x})\in \overline {\mathcal {F}_{T_{i},T_i}}(\tilde {x})$
. By Lemma 3.2, the set
$\{ n\in \mathbb {Z}: T_i^n \tilde {x} \in U_{x_i'}\}$
contains a set of the form
$\{a_r-a_{r'}:\;r>r'\}$
for some
$\mathbb {Z}$
-valued sequence
$(a_i)_{i\in \mathbb {N}}$
taking infinitely many values. In particular, the same is true for the set
$\{n\in \mathbb {Z}: T_i^n U_{x_i}\cap U_{x_i'}\neq \emptyset \}$
. Let
$\mu $
be a
$\mathbb {Z}^k$
-invariant measure on X. By the minimality of
$\langle S_1, \ldots , S_k\rangle $
,
$\mu $
has full support. Consider the product system
$(X_1\times \cdots \times X_d,\mathcal {B}(X)^{\otimes d},\mu ^{\otimes d},T_1\times \cdots \times T_d)$
and
$U=U_{x_1}\times \cdots \times U_{x_d}$
. Then
$\mu ^{\otimes d}(U)>0,$
and so by the proof of the Poincaré recurrence theorem, the set
$\{n\in \mathbb {Z} : \mu ^{\otimes d}(U\cap (T_1\times \cdots \times T_d)^{-n}U)>0\}$
must intersect nontrivially every infinite set of the form
$\{a_r-a_{r'}:\;r>r'\}$
. This implies that it has nonempty intersection with
$\{n\in \mathbb {Z}: T_i^n U_{x_i}\cap U_{x_i'}\neq \emptyset \}.$
Picking now an
$n\in \mathbb {Z}$
in the intersection, we get that
$T_{i}^{n}U_{x_{i}}\cap U_{x^{\prime }_{i}}\neq \emptyset $
and
$T_{j}^{n}U_{x_{j}}\cap U_{x_{j}}\neq \emptyset $
for all
$1\leq j\leq d,$
as desired.
Proof of Theorem 3.1.
Our goal here is to find, for every
$\epsilon> 0,$
a point
$(z_1, \ldots , z_d)$
and an integer n so that
$(x_1, \ldots , x_d)$
is close to
$(z_1, \ldots , z_d)$
and
$(y_1, \ldots , y_d)$
is close to
$(T^n z_1, \ldots , T^n z_d).$
We do so by induction on the coordinates. To this end, fix
$\epsilon>0$
and set 
. Suppose that we have constructed
$\epsilon _{r+1},\dots ,\epsilon _{d}>0$
for some
$1\leq r\leq d-1$
. We let
$0<\epsilon _{r}<\epsilon _{r+1}/2$
to be a number such that for any
$z_{r+1}\in X$
with
$\rho (y_{r+1},z_{r+1})<\epsilon _{r}$
, there exists
$x^{\prime }_{r+1}\in X$
with
$\rho (x^{\prime }_{r+1},x_{r+1})<\epsilon _{r+1}/2$
such that
$(x^{\prime }_{r+1},z_{r+1})\in R_{\pi _{r+1}}$
. The existence of such
$\epsilon _{r}$
follows from the assumptions that
$(x_{r+1},y_{r+1})\in R_{\pi _{r+1}}$
, and that the map
$X\mapsto Y_{{r+1}}=X/R_{\pi _{r+1}}$
is open.
For
$1\leq r\leq d$
, we say that Property r holds if there exist
$z_{1},\dots ,z_{d}\in X$
and
$n\in \mathbb {Z}$
such that
-
•
$\rho (x_{i},z_{i})<\epsilon _{r}$
for all
$1\leq i\leq r$
; -
•
$\rho (y_{i},z_{i})<\epsilon _{r}$
for all
$r+1\leq i\leq d$
; -
•
$\rho (y_{i},T_{i}^{n}z_{i})<\epsilon _{r}$
for all
$1\leq i\leq d$
.
By Lemma 3.3, setting
$x_j=y_j, 1\leq j\leq d, x_1'=x_1,$
and
$U_z=B(z,\epsilon _1)$
for
$z=y_1, \ldots , y_d, x_1,$
for
$-n$
, we have that Property 1 holds. Now suppose that Property r holds for some
$1\leq r\leq d-1$
. Since
$(x_{r+1},y_{r+1})\in R_{\pi _{r+1}}$
and
$\rho (y_{r+1},z_{r+1})<\epsilon _{r}$
, by the construction of
$\epsilon _{r}$
, there exists
$x^{\prime }_{r+1}\in X$
with
$\rho (x^{\prime }_{r+1},x_{r+1})<\epsilon _{r+1}/2$
such that
$(x^{\prime }_{r+1},z_{r+1})\in R_{\pi _{r+1}}$
. Let 
. Take
$0<\delta <\delta '$
such that for all
$x,y\in X$
, if
$\rho (x,y)<\delta $
, then
$\rho (T^{n}x,T^{n}y)<\delta '$
(n is the one from Property r above). By Lemma 3.3, there exist
$z^{\prime }_{1},\dots ,z^{\prime }_{d}\in X$
and
$n'\in \mathbb {Z}$
such that
-
•
$\rho (z_{i},z^{\prime }_{i})<\delta $
for all
$1\leq i\leq d, i\neq r+1$
; -
•
$\rho (z_{i},T_{i}^{n'}z^{\prime }_{i})<\delta $
for all
$1\leq i\leq d, i\neq r+1$
; -
•
$\rho (x^{\prime }_{r+1},z^{\prime }_{r+1})<\delta $
; -
•
$\rho (z_{r+1},T_{r+1}^{n'}z^{\prime }_{r+1})<\delta $
.
Then for all
$1\leq i\leq r$
,
$$ \begin{align*}\rho(x_{i},z^{\prime}_{i})\leq \rho(x_{i},z_{i})+\rho(z_{i},z^{\prime}_{i})<\epsilon_{r}+\delta\leq\epsilon_{r+1}.\end{align*} $$
For all
$r+2\leq i\leq d$
,
$$ \begin{align*}\rho(y_{i},z^{\prime}_{i})\leq \rho(y_{i},z_{i})+\rho(z_{i},z^{\prime}_{i})<\epsilon_{r}+\delta\leq\epsilon_{r+1}.\end{align*} $$
Moreover,
$$ \begin{align*}\rho(x_{r+1},z^{\prime}_{r+1})\leq \rho(x_{r+1},x^{\prime}_{r+1})+\rho(x^{\prime}_{r+1},z^{\prime}_{r+1})<\epsilon_{r+1}/2+\delta\leq\epsilon_{r+1}.\end{align*} $$
However, for all
$1\leq i\leq d, i\neq r+1$
, since
$\rho (z_{i},T_{i}^{n'}z_{i})<\delta $
, we have
$\rho (T_{i}^{n}z_{i},T_{i}^{n+n'}z_{i})<\delta '$
and so
$$ \begin{align*}\rho(y_{i},T_{i}^{n+n'}z_{i})\leq \rho(y_{i},T_{i}^{n}z_{i})+\rho(T_{i}^{n}z_{i},T_{i}^{n+n'}z_{i})<\epsilon_{r}+\delta'\leq\epsilon_{r+1}.\end{align*} $$
Finally, since
$\rho (z_{r+1},T_{r+1}^{n'}z_{r+1})<\delta $
, we have that
$\rho (T_{r+1}^{n}z_{r+1},T_{r+1}^{n+n'}z_{r+1})<\delta '$
and so
$$ \begin{align*}\rho(y_{r+1},T_{r+1}^{n+n'}z_{r+1})\leq \rho(y_{r+1},T_{r+1}^{n}z_{r+1})+\rho(T_{r+1}^{n}z_{r+1},T_{r+1}^{n+n'}z_{r+1})<\epsilon_{r}+\delta'\leq\epsilon_{r+1}.\end{align*} $$
In conclusion, we have that that Property
$r+1$
holds.
So it follows from induction that Property d holds, which means that there exist
$(z_{1},\dots ,z_{d})\in X^{d}$
and
$n\in \mathbb {Z}$
such that
$$ \begin{align*}\rho((x_{1},\dots,x_{d}),(z_{1},\dots,z_{d}))< \epsilon \text{ and } \rho((y_{1},\dots,y_{d}),(T_{1}^{n}z_{1},\dots,T_{d}^{n}z_{d}))<\epsilon.\end{align*} $$
Since
$\epsilon $
is arbitrary, we have that
$((x_{1},\dots ,x_{d}),(y_{1},\dots ,y_{d}))\in \mathbf {RP}_{T_{1}\times \dots \times T_{d}}(X^{d})$
.
As a consequence of Theorem 3.1, we have the following:
Proposition 3.4. Let
$(X,S_{1},\dots ,S_{k})$
be a minimal
$\mathbb {Z}^{k}$
-system and
$T_1,\ldots ,T_d\in \langle S_1,\ldots ,S_k\rangle $
. Suppose that
$(X,T_{1}),\dots ,(X,T_{d})$
are transitive. Then
$(X^{d},T_{1}\times \dots \times T_{d})$
is transitive if and only if
$(Y^{d},T_{1}\times \dots \times T_{d})$
is transitive, where
$Y=X/\mathbf {RP}_{\mathbb {Z}^{k},\mathbb {Z}^{k}}(X)$
.
Proof. The ‘only if’ part is straightforward. Now assume that
$(Y^d,T_1\times \dots \times T_d)$
is transitive. By the O-diagram (Theorem 2.5), we may consider almost one-to-one extensions
$\tilde {X}$
,
$\tilde {Y}$
of X and Y, respectively, such that the projection
$\tilde {\pi }\colon \tilde {X}\to \tilde {Y}$
is open. Note that
$(\tilde {X},\tilde {T}_1),\dots ,(\tilde {X},\tilde {T}_d)$
and
$(\tilde {Y}^d,\tilde {T}_1\times \dots \times \tilde {T}_d)$
are also transitive because this property is preserved under almost one-to-one extensions (see [Reference Akin and Glasner1]). We now show that
$(\tilde {X}^d,\tilde {T}_1\times \dots \times \tilde {T}_d)$
is transitive, which implies that
$(X^d,T_1\times \dots \times T_d)$
is transitive.
Note that since
$\tilde {X}$
is an almost one-to-one extension of X, we have that
$\tilde {X}/\mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(\tilde {X})$
and
$X/\mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(X)$
are conjugate, so we have that
$R_{\tilde {\pi }}$
is a subset of
$\mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(\tilde {X})$
. To see this, by only assuming that
$\tilde {\sigma }$
is proximal (which covers the almost one-to-one case), let q be the projection from
$\tilde {X}$
to
$X/\mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(X)$
. It suffices to show that
$R_{q}\subseteq \mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(\tilde {X})$
. Let
$\tilde {x},\tilde {x}'\in \tilde {X}$
with
$q(\tilde {x})=q(\tilde {x}')$
. Then
$(\tilde {\sigma }(\tilde {x}),\tilde {\sigma }(\tilde {x}'))\in \mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(X)$
. By the second part of Theorem 2.4, we can find
$(\tilde {y}, \tilde {y}')\in \mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(\tilde {X})$
such that
$ (\tilde {\sigma }(\tilde {y}),\tilde {\sigma }(\tilde {y}'))=(\tilde {\sigma }(\tilde {x}),\tilde {\sigma }(\tilde {x}'))$
. It follows that
$(\tilde {x},\tilde {y}), (\tilde {x}',\tilde {y}') \in P(\tilde {X})$
(the proximal relation on
$\tilde {X}$
). Since
$P(\tilde {X})\subseteq \mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(\tilde {X})$
, and this is an equivalence relation, we conclude
$(\tilde {x},\tilde {x}')\in \mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(\tilde {X})$
.Footnote
5
Let
$U,V$
be nonempty open subsets of
$\tilde {X}^{d}$
. Our goal is to find
$m\in \mathbb {N}$
such that
$U\cap (\tilde {T}_{1}\times \dots \times \tilde {T}_{d})^{-m} V\neq \emptyset $
. Then
$\tilde {\pi }^{\times d}(U)$
and
$\tilde {\pi }^{\times d}(V)$
are nonempty open sets, where 
(d-times). Since
$(\tilde {Y}^d,\tilde {T}_1\times \dots \times \tilde {T}_d)$
is transitive, there exist
$(x_{1},\dots ,x_{d})\in U$
and
$n\in \mathbb {Z}$
such that
$\tilde {\pi }^{\times d}(\tilde {T}_1^nx_1,\ldots ,\tilde {T}_d^nx_{d})\in \tilde {\pi }^{\times d}(V)$
. That is, there exists
$(x^{\prime }_{1},\dots ,x^{\prime }_{d})\in V$
such that
$(\tilde {T}_i^nx_{i},x^{\prime }_{i})\in R_{\tilde {\pi }}$
for all
$1\leq i\leq d$
. Let
$\epsilon>0$
be such that
$B((x_{1},\dots ,x_{d}),\epsilon )\subseteq U$
and
$B((x^{\prime }_{1},\dots ,x^{\prime }_{d}),\epsilon )\subseteq V$
. Take
$0<\delta <\epsilon $
so that
$\rho (a,b)<\delta $
implies
$\rho (\tilde {T}_i^{-n}a,\tilde {T}_i^{-n} b)<\epsilon $
for
$1\leq i\leq d$
. Thanks to the transitivity of
$\tilde {T}_i$
, using Corollary 2.3, we get that
$\mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(\tilde {X})= \mathbf {RP}_{\tilde {T}_i,\tilde {T}_i}(\tilde {X})$
and thus
$R_{\tilde {\pi }}\subseteq \mathbf {RP}_{\tilde {T}_i,\tilde {T}_i}(\tilde {X})$
for all
$1\leq i\leq d$
. By Theorem 3.1, we obtain
$((\tilde {T}_1^nx_{1},\dots ,\tilde {T}_d^nx_{d}),(x^{\prime }_{1},\dots ,x^{\prime }_{d}))\in \mathbf {RP}_{\tilde {T}_1\times \dots \times \tilde {T}_d}(\tilde {X}^d)$
. Therefore, there exist
$(y_{1},\dots ,y_{d})\in \tilde {X}^d$
and
$m\in \mathbb {Z}$
such that
$\rho ((y_1,\ldots ,y_d), (\tilde {T}_1^nx_{1},\dots ,\tilde {T}_d^nx_{d}))<\delta $
and
$\rho ((\tilde {T}_1^my_{1},\dots ,\tilde {T}_d^my_{d}),(x_1',\ldots ,x_d'))<\delta $
. It follows that
$(\tilde {T}_1^{-n}y_{1},\dots , \tilde {T}_d^{-n}y_{d})\in U$
and
$(\tilde {T}_1^{m+n}\tilde {T}_1^{-n}y_{1},\dots ,\tilde {T}_d^{m+n}\tilde {T}_d^{-n}y_{d})\in V$
. Therefore,
$U\cap (\tilde {T}_{1}\times \dots \times \tilde {T}_{d})^{-(m+n)}V\neq \emptyset $
. We conclude that
$(\tilde {X}^{d},\tilde {T}_{1}\times \dots \times \tilde {T}_{d})$
is transitive.
3.2 The proof of Theorem 1.6
In this last subsection we prove Theorem 1.6. We start with its forward direction, which is almost straightforward.
Proof of the forward direction of Theorem 1.6.
We use Lemma 2.1 implicitly throughout the proof. Assume that
$(T_1^n)_n,\ldots ,(T_d^n)_n$
are jointly transitive. Equivalently, for all
$V_{0},\dots ,V_{d}$
nonempty and open subsets of
$X,$
there exists
$n\in \mathbb {Z}$
such that
$$ \begin{align} V_{0}\cap T^{-n}_{1}V_{1}\cap\dots\cap T^{-n}_{d}V_{d}\neq \emptyset. \end{align} $$
To show
$(i)$
, pick any
$i\neq j,$
and let
$U_0, U_1$
be nonempty opens sets. Setting
$V_i=U_0, V_j=U_1$
and
$V_k=X$
for all
$k\in \{0,\ldots ,d\}\setminus \{i,j\},$
it follows from (3.1) that
$T_i^{-n}V_i\cap T_j^{-n}V_j\neq \emptyset $
, or
$U_0\cap (T_i^{-1}T_j)^{-n} U_1\neq \emptyset $
. To show
$(ii)$
, pick any point x for which
$\{(T_1^n x, \ldots , T_d^n x):\;n\in \mathbb {Z}\}$
is dense in
$X.$
Then the point
$(x,\ldots ,x)$
is a transitive point for
$T_1\times \dots \times T_d$
.
It remains to show the inverse direction of Theorem 1.6. To this end, we first need a couple of statements. In particular, the first one will allow us to run an inductive argument.
Proposition 3.5. Let
$(X,S_{1},\ldots ,S_{k})$
be a minimal
$\mathbb {Z}^{k}$
-system,
$T_1,\ldots ,T_d\in \langle S_1,\ldots ,S_k\rangle $
, and 
. If
$(X,R_{2}), \ldots , (X,R_{d})$
and
$(X^{d},T_{1}\times \dots \times T_{d})$
are transitive, then
$(X^{d-1},R_{2}\times \dots \times R_{d})$
is transitive.
Proof. Since
$(X,R_{2}),\ldots , (X,R_{d})$
are transitive, by Corollary 2.3, we have
$\mathbf {RP}_{R_{2},R_{2}}(X)=\ldots =\mathbf {RP}_{R_{d},R_{d}}(X)=\mathbf {RP}_{\mathbb {Z}^k,\mathbb {Z}^k}(X)$
, which is an equivalence relation since
$(X,S_{1},\ldots ,S_{k})$
is minimal. By Proposition 3.4, it suffices to show that
$(Y^{d-1},R_{2}\times \dots \times R_{d})$
is transitive, where
$Y=X/\mathbf {RP}_{\mathbb {Z}^{k},\mathbb {Z}^{k}}(X)$
. By Theorem 2.4, we have that
$(Y,S_1,\ldots ,S_k)$
is a rotation on a compact abelian group, and so we may write
$T_i(y)=y+\alpha _i$
for
$\alpha _i\in Y$
for
$1\leq i \leq d$
. We can choose a metric
$\rho $
on Y that is compatible with its topology, such that
$S_1,\ldots ,S_k$
act as isometries on Y. Since
$(X^{d},T_{1}\times \dots \times T_{d})$
is transitive, we get that
$(Y^d,T_1\times \dots \times T_d)$
is transitive, and hence minimal. (This holds because rotations are distal, and in this class transitivity and minimality are equivalent conditions – for example, see [Reference Auslander2, Chapters 2 and 5].)
Take
$y\in Y$
and
$(y_2,\ldots ,y_d)\in Y^{d-1}$
. Since
$(Y^d,T_1\times \cdots \times T_d)$
is minimal, given
$\epsilon>0$
, there exists n such that
$\rho (y+n\alpha _1,y)<\epsilon $
, and
$\rho (y+n\alpha _i, y_i)<\epsilon $
, for all
$2\leq i \leq d$
. It follows that
$$ \begin{align*}\rho(y+n(\alpha_i-\alpha_1) , y_i)\leq \rho(y+n(\alpha_i-\alpha_1) , y+n\alpha_i)+\rho(y+n\alpha_i, y_i)=\rho(y, y+n\alpha_1)+\rho(y+n\alpha_i, y_i)<2\epsilon\end{align*} $$
for all
$2\leq i \leq d$
. As
$\epsilon>0$
is arbitrary, we get that
$(y_2,\ldots ,y_d)$
belongs to the orbit closure of
$(y,\ldots ,y)\in X^{d-1}$
under
$R_2\times \cdots \times R_d$
. Since
$y_2,\ldots ,y_d$
are arbitrary, this orbit closure is all of
$Y^{d-1}$
. We get that
$(Y^{d-1},R_2\times \cdots \times R_d)$
is minimal, as it is the orbit closure of a point in an equicontinuous system. Proposition 3.4 allows us to conclude.
The following lemma is a generalization of [Reference Cao and Shao8, Lemma 2.9] (see also [Reference Kwietniak and Oprocha34, Lemma 3]).
Lemma 3.6. Let
$(X,S_{1},\dots ,S_{k})$
be a
$\mathbb {Z}^{k}$
-system and
$T_1,\ldots ,T_d\in \langle S_1,\ldots ,S_k\rangle $
. Let
$(R_{j})_{1\leq j\leq N}$
be a finite sequence of continuous maps from X to X. Assume that
$(X^{d},T_{1}\times \dots \times T_{d})$
is transitive. Then for all nonempty open sets
$V_{1},\dots ,V_{d}$
, there exists
$n_{j}\in \mathbb {Z}, 1\leq j\leq N$
, and for each
$1\leq i\leq d$
a nonempty open subset
$\tilde {V}_{i}$
of
$V_{i}$
such that
$$ \begin{align*}T_{i}^{-n_{j}}R^{-1}_{j}\tilde{V}_{i}\subseteq V_{i} \text{ for all } 1\leq j\leq N, 1\leq i\leq d.\end{align*} $$
Proof. We use induction on N. Since
$(X^{d},T_{1}\times \dots \times T_{d})$
is transitive, there exists
$n_{1}\in \mathbb {Z}$
such that
$T_{i}^{-n_{1}}R^{-1}_{1}V_{i}\cap V_{i}\neq \emptyset $
for all
$1\leq i\leq d$
. Set 
. This completes the proof for the case
$N=1$
.
Now assume that for some
$N\geq 2$
, we have constructed
$n_{1},\dots ,n_{N-1}\in \mathbb {N}$
with
$n_{1}<\ldots <n_{N-1}$
, and for each
$1\leq i\leq d$
a sequence of nonempty open sets
$V_{i}\supseteq V^{(1)}_{i}\supseteq \ldots \supseteq V^{(N-1)}_{i}$
such that
$$ \begin{align*}T_{i}^{-n_{j}}R^{-1}_{j}V^{(m)}_{i}\subseteq V_{i} \text{ for all } 1\leq j\leq m, 1\leq m\leq N-1, 1\leq i\leq d.\end{align*} $$
Let
$U_{i}:=R^{-1}_{N}V_{i}^{(N-1)}$
. Since
$(X^{d},T_{1}\times \dots \times T_{d})$
is transitive, there exists
$n_{N}\in \mathbb {N}$
with
$n_{N}>n_{N-1}$
such that
$T_{i}^{-n_{N}}U_{i}\cap V_{i}\neq \emptyset $
for all
$1\leq i\leq d$
. This implies that
$$ \begin{align*}V_i\cap T_{i}^{-n_{N}}R^{-1}_{N}V_{i}^{(N-1)}=V_i\cap T_{i}^{-n_{N-1}}U_{i}\neq \emptyset.\end{align*} $$
Let
$$ \begin{align*}V_{i}^{(N)}:=V_{i}^{(N-1)}\cap (T_{i}^{-n_{N}}R^{-1}_{N})^{-1}V_{i}.\end{align*} $$
Then
$V_{i}^{(N)}\subseteq V_{i}^{(N-1)}$
is a nonempty open set and
$T_{i}^{-n_{N}}R^{-1}_{N}V^{(N)}_{i}\subseteq V_{i}$
. Since
$V_{i}^{(N)}\subseteq V_{i}^{(N-1)}$
, we also have that
$$ \begin{align*}T_{i}^{-n_{j}}R^{-1}_{j}V^{(N)}_{i}\subseteq V_{i} \text{ for all } 1\leq j\leq N-1, 1\leq i\leq d.\end{align*} $$
This completes the induction step, and we are done by setting
$\tilde {V}_{i}:=V_{i}^{(N)}$
.
Proof of the inverse direction of Theorem 1.6.
There is nothing to prove when
$d=1$
. Now we assume that Theorem 1.6 holds for
$d-1$
for some
$d\geq 2$
, and we prove it for d. By Proposition 3.5, conditions (i) and (ii) imply that
$(X^{d-1},T_2T_1^{-1}\times \dots \times T_{d}T_{1}^{-1})$
is transitive. However, we have that
$(T_{i}T_{1}^{-1})^{-1}(T_{j}T_{1}^{-1})=T_{i}^{-1}T_{j}$
is transitive for all
$1\leq i,j\leq d, i\neq j$
. So, by induction hypothesis, we have that
$((T_{2}T_{1}^{-1})^{n})_n,\ldots ,((T_{d}T_{1}^{-1})^{n})_n$
are jointly transitive.
For
$m=(m_1,\dots ,m_k)\in \mathbb {Z}^k$
, recall that
$S_{m}=S_1^{m_1}\cdot \ldots \cdot S_k^{m_k}$
. Let
$U,V_{1},\dots ,V_{d}$
be open and nonempty. We wish to show that there exists
$n\in \mathbb {Z}$
such that
$$ \begin{align*}U\cap T_{1}^{-n}V_{1}\cap\dots\cap T^{-n}_{d}V_{d}\neq \emptyset.\end{align*} $$
Since
$(X,S_{1},\dots ,S_{k})$
is minimal, there exists a finite set
$F\subseteq \mathbb {Z}^k$
such that
$X=\bigcup _{r\in F} S_{r}U$
. By assumption (ii) and Lemma 3.6, there exist nonempty open sets
$\tilde {V}_1,\dots ,\tilde {V}_d$
and for all
$r\in F$
, some
$n_r\in \mathbb {Z}$
, such that
$$\begin{align*}(T_{1}\times\dots \times T_{d})^{n_{r}} (S_{r}\times\dots\times S_{r} (\tilde{V}_1\times\dots\times \tilde{V}_d))\subseteq V_1 \times\dots\times V_d\end{align*}$$
for all
$r\in F$
. Since
$(T_{2}T_{1}^{-1})^{n},\dots ,(T_{d}T_{1}^{-1})^{n}$
are jointly transitive, we can find
$m=m(F)\in \mathbb {Z}$
such that
$$\begin{align*}\tilde{V}_1 \cap (T_{2}T_1^{-1})^{-m}\tilde{V}_2\cap\dots\cap (T_{d}T_1^{-1})^{-m}\tilde{V}_d \neq \emptyset. \end{align*}$$
Take
$x\in \tilde {V}_1$
such that
$T_i^mT_1^{-m}x\in \tilde {V}_i$
for all
$2\leq i\leq d$
, and write
$y=T_1^{-m}x$
. Let
$r\in F$
be such that
$z:=S_{r}y\in U$
. Then
$T_{i}^{m+n_r}z=T_i^{m+n_r}S_{r}y=T_i^{n_r}S_{r}(T_{i}T_{1}^{-1})^{m}x\in T_i^{n_r}S_{r}(\tilde {V}_{i})\subseteq V_{i}$
for all
$1\leq i\leq d$
. It follows that
$z\in U\cap T_1^{-(m+n_r)}V_1\cap \dots \cap T_d^{-(m+n_r)}V_d$
.
Acknowledgements
After sharing our findings with D. Charamaras, F. K. Richter and K. Tsinas, we were informed that, by using elementary methods, they had obtained a special case of Theorem 1.6. Given the distinct nature of the methods, the two groups have decided not to publish their results together. Thanks go to the anonymous referees for suggestions that improved the initial version of the paper. Thanks also go to Bryna Kra for correcting some of the history around regionally proximal relations.
Competing interest
The authors have no competing interest to declare.
Funding statement
The first author was partially funded by Centro de Modelamiento Matemático (CMM) FB210005, BASAL funds for centers of excellence from ANID-Chile and ANID/Fondecyt/ 1241346. The second author was partially supported by the ‘Excellence in Research’ program of the Special Account for Research Funds AUTh (Code 10316). The third author was partially supported by the NSF Grant DMS-2247331.
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