# Joint transitivity for linear iterates

Sebastián Donoso, Andreas Koutsogiannis and Wenbo Sun  Departamento de
Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de
Chile & IRL 2807 - CNRS, Beauchef 851, Santiago, Chile<EMAIL_ADDRESS>Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki,
54124, Greece<EMAIL_ADDRESS>Department of Mathematics, Virginia
Tech, 225 Stanger Street, Blacksburg, VA, 24061, USA<EMAIL_ADDRESS>
###### Abstract.

We establish sufficient and necessary conditions for the joint transitivity of
linear iterates in a minimal topological dynamical system with commuting
transformations. This result provides the first topological analogue of the
classical Berend and Bergelson joint ergodicity criterion in measure-
preserving systems.

###### Key words and phrases:

Joint transitivity, topological dynamics, transitivity, proximal relations.

###### 2020 Mathematics Subject Classification:

Primary: 37B05; Secondary: 37B02, 37B20

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.

##  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 non-zero 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.$111Throughout this paper, whenever a sequence is written as $(a(n))_{n}$
without specifying the range of $n$, it is understood as a
$\mathbb{Z}$-sequence $(a(n))_{n\in\mathbb{Z}}$.

######  Theorem 1.1 ([24]).

Let $(X,\mathcal{B},\mu,T)$ be a weakly mixing measure-preserving system.
Then, for any $d\in\mathbb{N},$ any distinct non-zero integers
$a_{1},\ldots,a_{d},$ and any $f_{1},\ldots,f_{d}\in L^{\infty}(\mu)$ we have

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}T^{a_{1}n}f_{1}\cdot\ldots\cdot
T^{a_{d}n}f_{d}=\int_{X}f_{1}\;d\mu\cdot\ldots\cdot\int_{X}f_{d}\;d\mu,$ (1)

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 [5], 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), also known as the
“expected limit,” developed to be a topic on its own (e.g., see [5, 7, 9, 12,
14, 15, 18, 19, 20, 21, 29, 30, 31, 32, 38] for various results on polynomial
and Hardy field, of polynomial growth, functions); 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 transformations222 Naturally, by this we mean
that $(X,\mathcal{B},\mu)$ is a standard probability space and
$T_{1},\ldots,T_{d}:X\to X$ are invertible measure-preserving transformations
with $T_{i}T_{j}=T_{j}T_{i}.$ 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

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}T_{1}^{a_{1}(n)}f_{1}\cdot\ldots\cdot
T_{d}^{a_{d}(n)}f_{d}=\int_{X}f_{1}\;d\mu\cdot\ldots\cdot\int_{X}f_{d}\;d\mu,$
(2)

where the convergence takes place in $L^{2}(\mu)$ (i.e., the
$L^{2}(\mu)$-limit of the left-hand side of (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 ([4]).

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 $\mu^{\otimes d}$).

There is a plethora of analogous joint ergodicity characterizations for
generalized linear functions [6], polynomial functions [12, 15, 22, 23], and
Hardy field functions [13, 16].

The objective of this article is to prove the corresponding to Theorem 1.3
result in the topological setting.

###  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 $S_{i}S_{j}=S_{j}S_{i}$. 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.

######  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

$\\{(T_{1}^{a_{1}(n)}x,\ldots,T_{d}^{a_{d}(n)}x):n\in\mathbb{Z}\\}$

is dense in $X^{d}$.333 In [28], the authors call the family
$(T^{n}_{1},\dots,T^{n}_{d})$ to be _$\Delta$ -transitive_. We use the term
“joint transitivity” to emphasize the direct parallelism of Theorem 1.3 to
Theorem 1.6 in the measure theoretic setting on “joint ergodicity.”

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 [25], 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 ([25]).

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 [28], 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 [8,
40].

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:

  1. $(i)$

$(X,T_{i}^{-1}T_{j})$ is transitive for all $1\leq i,j\leq d,\;i\neq j$; and

  2. $(ii)$

$(X^{d},T_{1}\times\cdots\times T_{d})$ 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 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 (strongly)
mixing system (see [34]).

We also want to emphasize that Theorem 1.6 fails without the minimality
assumption. Indeed, in [35], Moothathu showed that there exists a non-minimal,
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.

###  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
$S_{m}\coloneqq S_{1}^{m_{1}}\cdot\ldots\cdot S_{k}^{m_{k}}$. 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
$\rho((x_{1},\ldots,x_{d}),(x_{1}^{\prime},\ldots,x_{d}^{\prime}))=\sup_{1\leq
i\leq d}\rho(x_{i},x_{i}^{\prime})$.

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 $1\leq i\leq k$. 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 $x,x^{\prime}\in R_{\pi}$, and $\epsilon>0$, there
exists $\delta>0$ such that if $\rho(x,y)<\delta$, then there exists
$y^{\prime}\in X$ with $\rho(x^{\prime},y^{\prime})<\epsilon$ and
$(x^{\prime},y^{\prime})\in R_{\pi}$.

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 [2, 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 $\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(X)$
(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
$P(X)\subseteq\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(X)$ (see, for
instance, [37]). 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 [2, 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 [36, Lemma 2.8] (see also [28, Lemma 2.4]
and [35]).

######  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:

  1. $(i)$

There exists a dense $G_{\delta}$ subset $\Omega$ of $X$ such that the set

$\\{(S_{a_{1}(n)}x,\dots,S_{a_{d}(n)}x)\colon n\in\mathbb{Z}\\}$

is dense in $X^{d}$ for every $x\in\Omega$.

  2. $(ii)$

There exists some $x\in X$ such that the set

$\\{(S_{a_{1}(n)}x,\dots,S_{a_{d}(n)}x)\colon n\in\mathbb{Z}\\}$

is dense in $X^{d}$.

  3. $(iii)$

For every non-empty open subsets $U,V_{1},\dots,V_{d}$ of $X$, there is some
$n\in\mathbb{Z}$ such that

$U\cap S_{-a_{1}(n)}V_{1}\cap\dots\cap S_{-a_{d}(n)}V_{d}\neq\emptyset.$

###### 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

$\\{(S_{a_{1}(n)}x,\dots,S_{a_{d}(n)}x)\colon n\in\mathbb{Z}\\}$

is dense in $X^{d}$. Then, for any $m\in\mathbb{Z}^{k}$, the set

$X(x,m):=\\{(S_{a_{1}(n)+m}x,\dots,S_{a_{d}(n)+m}x)\colon n\in\mathbb{Z}\\}$

is also dense in $X^{d}$. Now, for any non-empty 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

$U\cap S_{-a_{1}(n)}V_{1}\cap\dots\cap S_{-a_{d}(n)}V_{d}$

contains the point $T_{m}x,$ hence it is non-empty.

It remains to show that $(iii)$ implies $(i)$. Let $\mathcal{F}$ be a
countable basis of the topology of $X$ and define

$\Omega\coloneqq\bigcap_{V_{1},\dots,V_{d}\in\mathcal{F}}\bigcup_{n\in\mathbb{Z}}S_{-a_{1}(n)}V_{1}\cap\dots\cap
S_{-a_{d}(n)}V_{d}.$

It follows from $(iii)$ that $\Omega$ is a dense $G_{\delta}$ set. Moreover,
the set

$\\{(S_{a_{1}(n)}x,\dots,S_{a_{d}(n)}x)\colon n\in\mathbb{Z}\\}$

is dense in $X^{d}$ for every $x\in\Omega$. ∎

###  2.2. Dynamical cubes and regionally proximal relations

Let $(X,S_{1},\ldots,S_{k})$ be a system and $(G_{1},G_{2})$ be a pair of
subgroups of $\mathbb{Z}^{k}$. Define the space of $(G_{1},G_{2})$-cubes as

$\bm{Q}_{G_{1},G_{2}}(X)\coloneqq\overline{\\{(x,S_{g_{1}}x,S_{g_{2}}x,S_{g_{1}+g_{2}}x):x\in
X,g_{1}\in G_{1},g_{2}\in G_{2}\\}}\subseteq X^{4}.$

(Remark that such definitions were initially introduced in [17], for the case
where each $G_{i}$ is generated by a single transformation.) Given $x\in X$,
and $(G_{1},G_{2})$, let

$\overline{\mathcal{F}_{G_{1},G_{2}}}(x)\coloneqq\overline{\\{(S_{g_{1}}x,S_{g_{2}}x,S_{g_{1}+g_{2}}x):g_{1}\in
G_{1},g_{2}\in G_{2}\\}}\subseteq X^{3}.$

When $G_{i}$ is generated by a single element $g_{i}$ we write
$\bm{Q}_{G_{i},G_{i}}(X)$ simply as $\bm{Q}_{S_{g_{i}},S_{g_{i}}}(X)$; a
similar notation is used for $\overline{F_{G_{i},G_{i}}}(x)$. Given a single
subgroup $G$ of $\mathbb{Z}^{k}$, we write

$\mathbb{RP}_{G}(X)\coloneqq\overline{\\{(x,S_{g}x):x\in X,g\in G\\}}\subseteq
X^{2}$

(this relation is coined the prolongation relation in [3]). Note that if
$(X,(S_{m})_{m\in G})$ is transitive (meaning that there exists a point $x$
such that $\\{S_{m}x:m\in G\\}$ is dense in $X$), then
$\mathbb{RP}_{G}(X)=X\times X$. Similarly to [17] (or [26] for the case of a
$\mathbb{Z}$-action), we define the relation $\mathbb{RP}_{G_{1},G_{2}}(X)$ as
the set of points $(x,y)\in X^{2}$ such that
$(x,y,y,y)\in\bm{Q}_{G_{1},G_{2}}(X)$. It should be noted that if $(X,G)$ is
minimal, then $\mathbb{RP}_{G,G}(X)$ is nothing more than the classical
regionally proximal relation (see [2, 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 $\mathbb{Z}^{k}$.

  1. $(i)$

Let $\sigma\colon X^{4}\to X^{4}$ be the map with
$\sigma(a,b,c,d)\coloneqq(a,c,b,d)$. Then
$\sigma(\bm{Q}_{G_{1},G_{2}}(X))=\bm{Q}_{G_{2},G_{1}}(X)$.

  2. $(ii)$

Consider the system $(\mathbb{RP}_{G_{1}}(X),G_{2}^{(2)})$, where
$G_{2}^{(2)}$ is the action given by $g(x,y)=(gx,gy)$, for all $g\in G_{2}$
and $(x,y)\in X^{2}.$444Note that $\mathbb{RP}_{G_{1}}(X)$ is invariant under
this action since $G_{1}$ and $G_{2}$ commute. Then
$\mathbb{RP}_{G_{2}^{(2)}}(\mathbb{RP}_{G_{1}}(X))=\bm{Q}_{G_{1},G_{2}}(X)$.

  3. $(iii)$

If $H$ is a subgroup of $\mathbb{Z}^{k}$, and
$\mathbb{RP}_{G_{1}}(X)=\mathbb{RP}_{G_{2}}(X)$, then
$\bm{Q}_{G_{1},H}(X)=\bm{Q}_{G_{2},H}(X)$.

  4. $(iv)$

If $G_{1}^{\prime},G_{2}^{\prime}$ and $G_{1},G_{2}$ are subgroups of
$\mathbb{Z}^{k}$ such that $G_{1}^{\prime}\subseteq G_{1}$, and
$G_{2}^{\prime}\subseteq G_{2}$, then
$\bm{Q}_{G_{1}^{\prime},G_{2}^{\prime}}(X)$
$\subseteq\bm{Q}_{G_{1},G_{2}}(X).$

###### 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 $\mathbb{RP}_{G_{2}^{(2)}}(\mathbb{RP}_{G_{1}}(X))$ (here we
naturally identify this point with
$((x,S{g_{1}}x),(S_{g_{2}}x,S_{g_{1}+g_{2}}x))$). Therefore,
$\bm{Q}_{G_{1},G_{2}}(X)\subseteq\mathbb{RP}_{G_{2}^{(2)}}(\mathbb{RP}_{G_{1}}(X))$.
For the converse inclusion, it suffices to show that for any
$(x,y)\in\mathbb{RP}_{G_{1}}(X)$ and any $g_{2}\in G_{2}$, we have
$(x,y,S_{g_{2}}x,S_{g_{2}}y)\in\bm{Q}_{G_{1},G_{2}}(X)$. Let $\epsilon>0$ and
choose $0<\delta<\epsilon$ so that if $z,z^{\prime}\in X$ and
$\rho(z,z^{\prime})<\delta$, then
$\rho(S_{g_{2}}z,S_{g_{2}}z^{\prime})<\epsilon$. We can find $x^{\prime}\in X$
and $g_{1}\in G_{1}$ such that
$\rho((x^{\prime},S_{g_{1}}x^{\prime}),(x,y))<\delta$. It follows that
$(x^{\prime},S_{g_{1}}x^{\prime},S_{g_{2}}x^{\prime},S_{g_{1}+g_{2}}x^{\prime})$
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
$(x,y,S_{g_{2}}x,S_{g_{2}}y)\in\bm{Q}_{G_{1},G_{2}}(X)$, 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
$\bm{Q}_{G,H}(X)=\bm{Q}_{\mathbb{Z}^{k},H}(X)$. In particular,
$\mathbb{RP}_{G,G}(X)=\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(X)$.

###### Proof.

Note that by Lemma 2.2 $(iv)$, the inclusion
$\bm{Q}_{G,H}(X)\subseteq\bm{Q}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(X)$ always
holds. In addition, if $G$ is transitive, since $\mathbb{RP}_{G}(X)=X\times
X=\mathbb{RP}_{\mathbb{Z}^{k}}(X)$, Lemma 2.2 $(iii)$ implies that
$\bm{Q}_{G,H}(X)=\bm{Q}_{\mathbb{Z}^{k},H}(X)$. Using $(i)$ and $(iii)$ of
Lemma 2.2, we get $\bm{Q}_{G,G}(X)=\bm{Q}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(X),$
from where
$\mathbb{RP}_{G,G}(X)=\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(X)$. ∎

We remark that Corollary 2.3 could also be deduced by using a proof similar to
that of [17, Lemma 6.13].

The following theorem is classical (see, for instance, [2]).

######  Theorem 2.4.

Let $(X,S_{1},\ldots,S_{k})$ be a minimal system. Then
$\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(X)$ is an equivalence relation,
the system $X/\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}$ is the maximal
equicontinuous factor of $X$, and this factor is topologically conjugate to a
rotation on a compact abelian group.

###  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 [10, Chapter VI, Section 3] or [2,
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)

${\tilde{X}}$${X}$${\tilde{Y}}$${Y}$$\scriptstyle{\tilde{\sigma}}$$\scriptstyle{\tilde{\pi}}$$\scriptstyle{\pi}$$\scriptstyle{\tilde{\tau}}$

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 estimates in product
spaces (see [15, Lemma 5.2] or [12, Lemma 3.4] for analogous statements in the
measurable setting).

######  Theorem 3.1.

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
$(Y_{i},S_{1},\ldots,S_{k})$, $1\leq i\leq d$ be factors of $X$, such that for
all $i$, the factor map $\pi_{i}\colon X\to Y_{i}$ is open, and
$R_{\pi_{i}}\subseteq\mathbb{RP}_{T_{i},T_{i}}(X)$. For all $1\leq i\leq d$,
let $(x_{i},y_{i})\in R_{\pi_{i}}$. Then

$((x_{1},\dots,x_{d}),(y_{1},\dots,y_{d}))\in\mathbb{RP}_{T_{1}\times\dots\times
T_{d}}(X^{d}).$

Interpreting appropriately the coordinates, this can be stated as
“$\mathbb{RP}_{T_{1},T_{1}}(X)\times\dots\times\mathbb{RP}_{T_{d},T_{d}}(X)\subseteq\mathbb{RP}_{T_{1}\times\dots\times
T_{d}}(X^{d})$.”

We need some lemmas before proving Theorem 3.1. The following lemma can be
deduced from the proof of [27, Theorem 7.3.2], or using [39]. 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 infinitely many values such that the set
$\\{n\in\mathbb{Z}:T^{n}x\in U\\}$ 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 $\rho(z,z^{\prime})<\delta_{2}$ implies that
$\rho(T^{a}z,T^{a}z^{\prime})<\epsilon_{2}$ 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:

$\rho(T^{n_{2}+n_{1}}x,x)<\epsilon_{1}+\epsilon_{2},\
\rho(T^{n_{2}+n_{1}+m_{1}}x,x)<\epsilon_{1}+\epsilon_{2},\
\rho(T^{n_{2}+m_{2}+n_{1}}x,x)<\epsilon_{1}+\epsilon_{2},\\\
\rho(T^{n_{2}+m_{2}+n_{1}+m_{1}}x,x)<\epsilon_{1}+\epsilon_{2},\
\rho(T^{n_{2}+m_{1}}x,y)<\epsilon_{1}+\epsilon_{2},\;\;\text{and}\;\;\rho(T^{n_{2}+m_{2}+m_{1}}x,y)<\epsilon_{1}+\epsilon_{2}.$

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 $\rho(T^{a}x,x)<\sum_{t=1}^{l}\epsilon_{j_{t}}$,
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
$\rho(z,z^{\prime})<\delta_{i+1}$ implies that
$\rho(T^{a}z,T^{a}z^{\prime})<\epsilon_{i+1}$ 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
$\rho(T^{a}x,x)<\sum_{t=1}^{l}\epsilon_{j_{t}}$, 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 $\rho(T^{a}x,x)<\sum_{t=1}^{l}\epsilon_{j_{t}}$, 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 $R_{\pi}\subseteq\mathbb{RP}_{T_{i},T_{i}}(X)$ 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 $1\leq j\leq d,j\neq
i$.

###### Proof.

The set $\Omega_{i}$ of $\tilde{x}\in X$ such that the set
$\\{(a,b,c):(\tilde{x},a,b,c)\in\bm{Q}_{T_{i},T_{i}}(X)\\}$ equals
$\overline{\mathcal{F}_{T_{i},T_{i}}}(\tilde{x})$ is a dense $G_{\delta}$-set
of points (see, for instance, [25, Lemma 4.5]). Since $\pi$ is open, we can
find $\tilde{x}\in U_{x_{i}}\cap\Omega_{i}$ and $\tilde{y}\in
U_{x_{i}^{\prime}}$ with $(\tilde{x},\tilde{y})\in R_{\pi}$. Because
$R_{\pi}\subseteq\mathbb{RP}_{T_{i},T_{i}}(X)$, we have
$(\tilde{x},\tilde{y},\tilde{x})\in\overline{\mathcal{F}_{T_{i},T_{i}}}(\tilde{x})$
and by Lemma 3.2, the set $\\{n\in\mathbb{Z}:T_{i}^{n}\tilde{x}\in
U_{x_{i}^{\prime}}\\}$ contains a set of the form
$\\{a_{j}-a_{j}^{\prime}:\;j>j^{\prime}\\}$ 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
$\\{n\in\mathbb{Z}:T_{i}^{n}U_{x_{i}}\cap U_{x_{i}^{\prime}}\neq\emptyset\\}$.
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
$U=U_{1}\times\cdots\times U_{d}$. 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 non-trivially every set of the form
$\\{a_{j}-a_{j}^{\prime}:\;j>j^{\prime}\\},$ which takes infinitely many
values. This implies that it has non-empty intersection with
$\\{n\in\mathbb{Z}:T_{i}^{n}U_{x_{i}}\cap U_{x_{i}^{\prime}}\neq\emptyset\\}$,
as desired. ∎

###### Proof of Theorem 3.1.

Fix $\epsilon>0$ and set $\epsilon_{d}\coloneqq\epsilon$. 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 $(x_{r+1},y_{r+1})\in
R_{\pi_{r+1}}$, 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, 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
$\delta^{\prime}\coloneqq\min\\{\epsilon_{r+1}/2,\epsilon_{r+1}-\epsilon_{r}\\}$.
Take $0<\delta<\delta^{\prime}$ such that for all $x,y\in X$, if
$\rho(x,y)<\delta$, then $\rho(T^{n}x,T^{n}y)<\delta^{\prime}$ ($n$ is the one
from Property $r$ above). By Lemma 3.3, there exist
$z^{\prime}_{1},\dots,z^{\prime}_{d}\in X$ and $n^{\prime}\in\mathbb{Z}$ such
that

  * •

$\rho(z_{i},z^{\prime}_{i})<\delta$ for all $1\leq i\leq d,i\neq r+1$;

  * •

$\rho(z_{i},T_{i}^{n^{\prime}}z^{\prime}_{i})<\delta$ for all $1\leq i\leq
d,i\neq r+1$;

  * •

$\rho(x^{\prime}_{r+1},z^{\prime}_{r+1})<\delta$;

  * •

$\rho(z_{r+1},T_{r+1}^{n^{\prime}}z^{\prime}_{r+1})<\delta$.

Then for all $1\leq i\leq r$,

$\rho(x_{i},z^{\prime}_{i})\leq\rho(x_{i},z_{i})+\rho(z_{i},z^{\prime}_{i})<\epsilon_{r}+\delta\leq\epsilon_{r+1}.$

For all $r+2\leq i\leq d$,

$\rho(y_{i},z^{\prime}_{i})\leq\rho(y_{i},z_{i})+\rho(z_{i},z^{\prime}_{i})<\epsilon_{r}+\delta\leq\epsilon_{r+1}.$

Moreover,

$\rho(x_{r+1},z^{\prime}_{r+1})\leq\rho(x_{r+1},x^{\prime}_{r+1})+\rho(x^{\prime}_{r+1},z^{\prime}_{r+1})<\epsilon_{r+1}/2+\delta\leq\epsilon_{r+1}.$

On the other hand, for all $1\leq i\leq d,i\neq r+1$, since
$\rho(z_{i},T_{i}^{n^{\prime}}z_{i})<\delta$, we have
$\rho(T_{i}^{n}z_{i},T_{i}^{n+n^{\prime}}z_{i})<\delta^{\prime}$ and so

$\rho(y_{i},T_{i}^{n+n^{\prime}}z_{i})\leq\rho(y_{i},T_{i}^{n}z_{i})+\rho(T_{i}^{n}z_{i},T_{i}^{n+n^{\prime}}z_{i})<\epsilon_{r}+\delta^{\prime}\leq\epsilon_{r+1}.$

Finally, since $\rho(z_{r+1},T_{r+1}^{n^{\prime}}z_{r+1})<\delta$, we have
that $\rho(T_{r+1}^{n}z_{r+1},T_{r+1}^{n+n^{\prime}}z_{r+1})<\delta^{\prime}$
and so

$\rho(y_{r+1},T_{r+1}^{n+n^{\prime}}z_{r+1})\leq\rho(y_{r+1},T_{r+1}^{n}z_{r+1})+\rho(T_{r+1}^{n}z_{r+1},T_{r+1}^{n+n^{\prime}}z_{r+1})<\epsilon_{r}+\delta^{\prime}\leq\epsilon_{r+1}.$

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

$\rho((x_{1},\dots,x_{d}),(z_{1},\dots,z_{d}))<\epsilon\text{ and
}\rho((y_{1},\dots,y_{d}),(T_{1}^{n}z_{1},\dots,T_{d}^{n}z_{d}))<\epsilon.$

Since $\epsilon$ is arbitrary, we have that
$((x_{1},\dots,x_{d}),(y_{1},\dots,y_{d}))\in\mathbb{RP}_{T_{1}\times\dots\times
T_{d}}(X^{d})$. ∎

As a consequence of Theorem 3.1, we have:

######  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
$Y=X/\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(X)$.

###### 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
$(\tilde{X},T_{1}),\dots,(\tilde{X},T_{d})$ and
$(\tilde{Y}^{d},T_{1}\times\dots\times T_{d})$ are also transitive, because
this property is preserved under almost one-to-one extensions (see [1]). We
now show that $(\tilde{X}^{d},T_{1}\times\dots\times T_{d})$ 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 $\tilde{X}/\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(\tilde{X})$ and
$X/\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(X)$ are conjugate, so we have
that $R_{\tilde{\pi}}$ is a subset of $\mathbb{RP}_{T_{i},T_{i}}(\tilde{X})$
for all $i$. 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 $X/\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(X)$. It suffices
to show that
$R_{q}\subseteq\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(\tilde{X})$. Let
$\tilde{x},\tilde{x}^{\prime}$ with $q(\tilde{x})=q(\tilde{x}^{\prime})$. Then
$(\tilde{\sigma}(\tilde{x}),\tilde{\sigma}(\tilde{x}^{\prime}))\in\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(X)$.
We can find
$(\tilde{y},\tilde{y}^{\prime})\in\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(\tilde{X})$
such that
$(\tilde{\sigma}(\tilde{y}),\tilde{\sigma}(\tilde{y}^{\prime}))=(\tilde{\sigma}(\tilde{x}),\tilde{\sigma}(\tilde{x}^{\prime}))$.
It follows that
$(\tilde{x},\tilde{y}),(\tilde{x}^{\prime},\tilde{y}^{\prime})\in
P(\tilde{X})$ (the proximal relation on $\tilde{X}$). Since
$P(\tilde{X})\subseteq\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(\tilde{X})$,
and this is an equivalence relation, we conclude
$(\tilde{x},\tilde{x}^{\prime})\in\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(\tilde{X})$.555
This should be a well-known result; we chose to present its short proof (which
simplifies the one of [11, Lemma 5.3] that covers the $\mathbb{Z}$ case and
almost one-to-one extensions) for completeness.

Let $U,V$ be nonempty open subsets of $\tilde{X}^{d}$. Then
$\tilde{\pi}^{\times d}(U)$ and $\tilde{\pi}^{\times d}(V)$ are nonempty open
sets, where $\tilde{\pi}^{\times d}\coloneqq\pi\times\cdots\times\pi$
($d$-times). Since $(\tilde{Y}^{d},S_{1}\times\dots\times S_{d})$ is
transitive, there exist $(x_{1},\dots,x_{d})\in U$ and $n\in\mathbb{Z}$ such
that $\tilde{\pi}^{\times
d}(T_{1}^{n}x_{1},\ldots,T_{d}^{n}x_{d})\in\tilde{\pi}^{\times d}(V)$. That
is, there exists $(x^{\prime}_{1},\dots,x^{\prime}_{d})\in V$ such that
$(T_{i}^{n}x_{i},x^{\prime}_{i})\in R_{\tilde{\pi}}(\tilde{X})$ 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
$\delta>0$ so that $\rho(a,b)<\delta$ implies
$\rho(T_{i}^{-n}a,T_{i}^{-n}b)<\epsilon$ for $1\leq i\leq d$. By Theorem 3.1,
as $R_{\tilde{\pi}}(\tilde{X})\subseteq\mathbb{RP}_{T_{i},T_{i}}(\tilde{X})$,
we obtain
$((T_{1}^{n}x_{1},\dots,T_{d}^{n}x_{d}),(x^{\prime}_{1},\dots,x^{\prime}_{d}))\in\mathbb{RP}_{T_{1}\times\dots\times
T_{d}}(\tilde{X}^{d})$. Therefore, there exist
$(y_{1},\dots,y_{d})\in\tilde{X}^{d}$ and $m\in\mathbb{Z}$ such that
$\rho((y_{1},\ldots,y_{d}),$ $(T_{1}^{n}x_{1},\dots,T_{d}^{n}x_{d}))<\delta$
and
$\rho((T_{1}^{m}y_{1},\dots,T_{d}^{m}y_{d}),(x_{1}^{\prime},\ldots,x_{d}^{\prime}))<\delta$.
It follows that $(T_{1}^{-n}y_{1},\dots,$ $T_{d}^{-n}y_{d})\in U$ and
$(T_{1}^{m+n}T_{1}^{-n}y_{1},\dots,T_{d}^{m+n}T_{d}^{-n}y_{d})\in V$. Using
Lemma 2.1, we conclude that $(\tilde{X}^{d},T_{1}\times\dots\times T_{d})$ 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

$V_{0}\cap T^{-n}_{1}V_{1}\cap\dots\cap T^{-n}_{d}V_{d}\neq\emptyset.$ (3)

To show $(i)$, pick any $i\neq j,$ and let $U_{0},U_{1}$ be non-empty 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) 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 $R_{2}\coloneqq
T_{2}T_{1}^{-1},\ldots,R_{d}\coloneqq T_{d}T_{1}^{-1}$. 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
$\mathbb{RP}_{R_{2},R_{2}}(X)=\ldots=\mathbb{RP}_{R_{d},R_{d}}(X)=\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(X)$,
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
$Y=X/\mathbb{RP}_{\mathbb{Z}^{k},\mathbb{Z}^{k}}(X)$. By Theorem 2.4 we have
that $(Y,S_{1},\ldots,S_{k})$ is a rotation on a compact abelian group, so we
may write $T_{i}(y)=y+\alpha_{i}$ for $\alpha_{i}\in Y$ for $1\leq i\leq d$.
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, hence minimal. (This
holds because rotations are distal, and in this class, transitivity and
minimality are equivalent conditions–e.g., see [2, 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
$\rho(y+n(\alpha_{i}-\alpha_{1}),y_{i})<2\epsilon$ for all $2\leq i\leq d$,
and since $y,y_{2},\ldots,y_{d}$ are arbitrary, we get that
$(Y^{d-1},R_{2}\times\cdots\times R_{d})$ is minimal. Proposition 3.4 allows
us to conclude. ∎

The following lemma is a generalization of [8, Lemma 2.9] (see also [33, 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 non-empty
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 non-empty open subset $\tilde{V}_{i}$ of
$V_{i}$ such that

$T_{i}^{-n_{j}}R^{-1}_{j}\tilde{V}_{i}\subseteq V_{i}\text{ for all }1\leq
j\leq N,1\leq i\leq d.$

###### 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 $\tilde{V}_{i}=V^{(1)}_{i}\coloneqq T_{i}^{-n_{1}}R^{-1}_{1}V_{i}\cap
V_{i}$. 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 non-empty open sets $V_{i}\supseteq
V^{(1)}_{i}\supseteq\ldots\supseteq V^{(N-1)}_{i}$ such that

$T_{i}^{-n_{j}}R^{-1}_{j}V^{(m)}_{i}\subseteq V_{i}\text{ for all }1\leq j\leq
m,1\leq m\leq N-1,1\leq i\leq d.$

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

$V_{i}\cap T_{i}^{-n_{N}}R^{-1}_{N}V_{i}^{(N-1)}=V_{i}\cap
T_{i}^{-n_{N-1}}U_{i}\neq\emptyset.$

Let

$V_{i}^{(N)}:=V_{i}^{(N-1)}\cap(T_{i}^{-n_{N}}R^{-1}_{N})^{-1}V_{i}.$

Then $V_{i}^{(N)}\subseteq V_{i}^{(N-1)}$ is an non-empty 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

$T_{i}^{-n_{j}}R^{-1}_{j}V^{(N)}_{i}\subseteq V_{i}\text{ for all }1\leq j\leq
N-1,1\leq i\leq d.$

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 $(X^{d-1},T_{2}T_{1}^{-1}\times\dots\times
T_{d}T_{1}^{-1})$ is transitive. On the other hand, 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 $m=(m_{1},\dots,m_{k})\in\mathbb{Z}^{k}$, let
$S_{m}B:=S_{1}^{m_{1}}\cdot\ldots\cdot S_{k}^{m_{k}}B$. Let
$U,V_{1},\dots,V_{d}$ be open and nonempty. We wish to show that there exists
$n\in\mathbb{Z}$ such that

$U\cap T_{1}^{-n}V_{1}\cap\dots\cap T^{-n}_{d}V_{d}\neq\emptyset.$

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 non-empty open sets
$\tilde{V}_{1},\dots,\tilde{V}_{d}$ and for all $r\in F$ some
$n_{r}\in\mathbb{Z}$, such that

$(T_{1}\times\dots\times T_{d})^{n_{r}}(S_{k}\times\dots\times
S_{k}(\tilde{V}_{1}\times\dots\times\tilde{V}_{d}))\subseteq
V_{1}\times\dots\times V_{d}$

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

$\tilde{V}_{1}\cap(T_{2}T_{1}^{-1})^{-m}\tilde{V}_{2}\cap\dots\cap(T_{d}T_{1}^{-1})^{-m}\tilde{V}_{d}\neq\emptyset.$

Take $x\in\tilde{V}_{1}$ such that $T_{i}^{m}T_{1}^{-m}x\in\tilde{V}_{i}$ 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 $z\in U\cap T_{1}^{-(m+n_{r})}V_{1}\cap\dots\cap
T_{d}^{-(m+n_{r})}V_{d}$. ∎

We close this article with the following widely open problem. 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 3.7.

Obtain the analogous to Theorem 1.6 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.

Acknowledgments. 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.

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# Theory of Huge Thermoelectric Effect Based on Magnon Drag Mechanism:
Application to Thin-Film Heusler Alloy

Hiroyasu Matsuura<EMAIL_ADDRESS>Department of Physics,
University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  Masao Ogata
Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo
113-0033, Japan  Trans-scale Quantum Science Institute, University of Tokyo,
Bunkyo-ku, Tokyo 113-0033, Japan  Takao Mori International Center for
Materials, Nanoarchitectonics (WPI-MANA), National Institute for Materials
Science (NIMS), Tsukuba, Japan  Graduate School of Pure and Applied Sciences,
University of Tsukuba, Tennodai 1-1-1, Tsukuba 305-8671, Japan  Ernst Bauer
Institute of Solid State Physics, Technische Universität Wien, Vienna, Austria

###### Abstract

To understand the unexpectedly high thermoelectric performance observed in the
thin-film Heusler alloy Fe2V0.8W0.2Al, we study the magnon drag effect,
generated by the tungsten based impurity band, as a possible source of this
enhancement, in analogy to the phonon drag observed in FeSb2. Assuming that
the thin-film Heusler alloy has a conduction band integrating with the
impurity band, originated by the tungsten substitution, we derive the
electrical conductivity $L_{11}$ based on the self-consistent t-matrix
approximation and the thermoelectric conductivity $L_{12}$ due to magnon drag,
based on the linear response theory, and estimate the temperature dependent
electrical resistivity, Seebeck coefficient and power factor. Finally, we
compare the theoretical results with the experimental results of the thin-film
Heusler alloy to show that the origin of the exceptional thermoelectric
properties is likely to be due to the magnon drag related with the tungsten-
based impurity band.

Introduction.— Thermoelectric materials have attracted much attention because
they can directly convert thermal energy to electric energy [2, 1, 3].
Especially, the development of thermoelectric materials, utilizing magnetism,
has been in the focus, and many materials with high thermoelectric performance
have been found [4, 5, 6, 7, 8]. The efficiency of the thermoelectric
conversion is expressed by the figure of merit, $ZT$, defined by $ZT\equiv
S^{2}\sigma T/\kappa$ where $S$, $\sigma$, $T$ and $\kappa$ are the Seebeck
coefficient, electrical conductivity, temperature and thermal conductivity,
respectively. However, it is well known that $ZT$ is usually much lower than
unity, because it is difficult to control these physical quantities
independently.

Recently, it was found that a thin-film Heusler alloy, Fe2V0.8W0.2Al, shows a
huge $ZT$ ($ZT\sim$ 5 ) at $T\sim$ 350 K, deriving from a huge power factor
defined as $PF\equiv S^{2}\sigma$ [9]. The origin of these huge $ZT$ and $PF$
is expected to be related to the anomalous temperature dependence of the
electrical resistivity and the Seebeck coefficient, because the electrical
resistivity changes from a metallic behavior to a semiconducting behavior at
$T\sim 350$ K, and the Seebeck coefficient has a peak structure with a huge
value ($S\sim-500$ $\mu$V/K) around this temperature.

In a previous study, on the basis of the first principles calculation, the
origin of this huge Seebeck coefficient was suggested to be a result of the
large mobility due to many Weyl points and a large logarithmic energy
derivative of the electronic density of states near the Fermi energy [9]. On
the other hand, it was also claimed [10] that the crystal structure assumed in
ref. [9] is different from the experimental one. Then, it was reported that a
new alloy model suggested in ref. [10] gives only rise to a Seebeck
coefficient $S\sim 30$ $\mu$V/K at $T\sim 400$ K , which is much smaller than
the experimental value. However, the actual alloy structures of Fe2V0.8W0.2Al
have not been fully explored both theoretically and experimentally.
Furthermore, a contribution of magnetism related to the thin-film Heusler
alloy [9] to the Seebeck coefficient has not yet been taken into account. In
addition to recent experimental reports, revealing an enhancement of the
Seebeck coefficient of various systems through magnetic interactions [4, 6,
8], it has recently been experimentally demonstrated that spin fluctuation
enhances the Seebeck coefficient of a doped itinerant ferromagnetic Fe2VAl
system [5].

The temperature dependencies of the electrical resistivity and Seebeck
coefficient observed in this thin-film Heusler alloy are very similar to those
in FeSb2: FeSb2 shows a huge Seebeck coefficient at low temperatures ($T\sim
10$ K ), and at the same temperature, the electrical resistivity changes its
temperature dependence to the semiconducting behavior as the temperature
decreases [11]. The origin of this huge Seebeck effect observed in FeSb2 has
been suggested being caused by a phonon drag, in which acoustic phonons couple
with large effective mass electrons in an impurity band [12, 13, 14]. From the
analogy with FeSb2, the origin of huge Seebeck effect observed in the thin-
film Heusler alloy is supposed to be magnon drag related in the context of an
impurity band and the conduction band with a large effective electron masses.

The contribution of the magnon drag to the Seebeck effect has been studied
experimentally [15, 16, 17, 18] and theoretically [19, 20, 21, 22, 23] from
the 1960s. However, it appears that the magnon drag, related with an impurity
band such as for the present alloy, is not sufficiently understood.

In this letter, we study the magnon drag effect with an impurity state to
clarify the origin of the huge Seebeck coeffcient and $PF$ observed in the
thin-film Heusler alloy. Firstly, since the electronic state of the thin-film
Heusler alloy has not been entirely understood yet, we assume an electronic
state from the view point of a dimensional reduction. Extending the phonon
drag theory studied in FeSb2 [14] to the thin-film Heusler alloy, we study the
temperature dependence of the electrical resistivity, Seebeck coefficient, and
PF related with such an impurity state. We then compare the obtained
theoretical results with experimental results to understand the origin of huge
thermoelectric effect observed in the thin-film Heusler alloy.

Schematic picture of electronic states.— Firstly, we deduce the electronic
state of the thin-film Heusler alloy based on the electronic state of bulk
Fe2VAl. Figure 1(a) shows a schematic picture of the electronic state of the
bulk Fe2VAl near the Fermi level. It is found [9, 24, 25] that this electronic
state is a typical semimetallic state. In the thin-film, it is expected that
the bandwidth decreases due to the dimensional reduction. Therefore, we
suggest that a band gap appears by the lower dimension in the thin-film Fe2VAl
(Fig.1(b)). When vanadium (V) is replaced by tungsten (W) in this thin-film,
it is natural to expect that impurity states appear near the bottom of the
conduction band, because the energy level of 5d electrons in W is lower than
the 3d energy level in V.

Figure 1:  Schematic picture on the electronic state of a) bulk Fe2VAl, b)
thin-film Fe2VAl, and c) thin film Fe2(V,W)Al.

Figure 1(c) shows a schematic picture of the electronic state of the thin-film
Heusler alloy substituted by W. In this letter, we study the electrical and
thermal transports on the basis of this electronic state shown in Fig.1(c).

Model Hamiltonian and Formulation of Electric and Thermal transports.— To
study the magnon drag based on the electronic state shown in Fig. 1(c), we use
a following model Hamiltonian [21, 22, 23, 14].

$\displaystyle H=H_{\rm 0}+H_{\rm W}+H_{\rm mag}+H_{\rm e-mag},$ (1)

where $H_{\rm 0}$, $H_{\rm W}$, $H_{\rm mag}$ and $H_{\rm e-mag}$ are
Hamiltonians for a ferromagnetic conduction band, W sites, a ferromagnetic
magnon, and an electron-magnon interaction, respectively. These Hamiltonians
are given as $H_{0}=\sum_{{\bf k},\sigma}(\epsilon_{{\bf k}\sigma}-\mu)c_{{\bf
k}\sigma}^{\dagger}c_{{\bf k}\sigma}$, $H_{W}=V_{0}\sum_{\langle
i\rangle}c_{i\sigma}^{\dagger}c_{i\sigma}$, $H_{mag}=\sum_{{\bf
q}}\hbar\omega_{\bf q}b_{\bf q}^{\dagger}b_{\bf q}$, and
$H_{e-mag}=\frac{I}{\sqrt{V}}\sum_{{\bf k},{\bf q}}\bigr{[}b_{\bf
q}^{\dagger}c_{{\bf k}\uparrow}^{\dagger}c_{{\bf k}+{\bf q}\downarrow}+b_{\bf
q}c_{{\bf k}+{\bf q}\downarrow}^{\dagger}c_{{\bf k}\uparrow}\bigr{]}$, where
$c_{{\bf k}\sigma}$ or $c_{i\sigma}$ ($c_{{\bf k}\sigma}^{\dagger}$ or
$c_{i\sigma}^{\dagger}$) is an annihilation (creation) operator of an electron
with the wave number ${\bf k}$ on the $i$-th site and spin
$\sigma=\uparrow\downarrow$; $b_{\bf q}$ ($b_{\bf q}^{\dagger}$) is an
annihilation (creation) operator of a magnon with wave vector ${\bf q}$.
$\epsilon_{{\bf k}\sigma}$ is the energy dispersion in the ferromagnetic
state, $\mu$ is a chemical potential, $V_{0}$ is the strength of a random
impurity potential, $\langle i\rangle$ is the position of impurities, and
$\hbar\omega_{\bf q}$ is the energy dispersion of ferromagnetic magnons given
by $\hbar\omega_{\bf q}=Dq^{2}$, where $D$ is the spin wave stiffness
constant. Finally, $I=J\sqrt{V}$ is the strength of the electron-magnon
interaction, where $J$ and $V$ are the coupling constant between electron and
magnon, and the volume of unit cell, respectively. In this letter, we use the
following simple energy dispersion: $\epsilon_{{\bf
k}\uparrow}=\frac{\hbar^{2}k^{2}}{2m^{*}}-\Delta,$ and $\epsilon_{{\bf
k}\downarrow}=\frac{\hbar^{2}k^{2}}{2m^{*}}$, where $m^{*}$ is the effective
mass of conduction electrons, $\Delta$ is the energy difference between up
spin and down spin electrons to express the ferromagnetic state, which
corresponds to d orbitals of Iron (Fe) in FeV0.8W0.2Al [9]. We assume that
$\Delta$ is independent of temperature for simplicity. Because the Fermi
energy is located near the bottom of the conduction band or in the impurity
band as shown in Fig. 1(c), the valence band is neglected although it will
contribute at high temperatures.

To treat the random potential of the W site, we use a self-consistent
$t$-matrix approximation [26, 28, 27, 14, 29]. As discussed in Ref [14], we
define the retarded Green’s function of electron with spin $\sigma$ as

$\displaystyle G^{R}_{\sigma}(k,\epsilon)=\frac{1}{\epsilon-\epsilon_{{\bf
k}\sigma}-\Sigma^{R}_{\sigma}(\epsilon)},$ (2)

where by the self-consistent t-matrix approximation, a retarded self-energy,
$\Sigma^{R}_{\sigma}(\epsilon)$, is given as
$\Sigma^{R}_{\sigma}(\epsilon)=\frac{n_{i}V_{0}}{1-\frac{V_{0}}{V}\sum_{\bf
k}^{\prime}G^{R}_{\sigma}(k^{\prime},\epsilon)}.$ Here, $n_{i}$ is the
concentration of W sites. The density of state (DOS) is obtained by
$D_{\sigma}(\epsilon)=D_{0}{\rm Im}[y_{\sigma}],$ where
$D_{0}=\frac{\sqrt{(m^{*})^{3}\epsilon_{B}}}{\sqrt{2}\pi^{2}\hbar^{3}},$ and
$y_{\sigma}$ is determined by solving the cubic equation:
$y_{\sigma}^{3}-2y_{\sigma}+\bigr{(}1+\frac{\epsilon+\Delta\delta_{\sigma,\uparrow}}{\epsilon_{B}}\bigr{)}y_{\sigma}-\nu=0$
[14]. Here, $\nu\equiv 2\pi
n_{i}\hbar^{3}/\sqrt{2(m^{*})^{3}\epsilon_{B}^{3}}$. We assumed that
$\epsilon_{B}$ ($\epsilon_{B}+\Delta$) is the binding energy of a single W
impurity for down spin (up spin) as a first step. It should be noted that the
first principles calculation shows no spin splitting in 5d orbitals of W [9].

The Fermi energy ($E_{F}$) and the temperature dependence of the chemical
potential are determined self-consistently by
$\sum_{\sigma}\int_{-\infty}^{\infty}f(\epsilon)D_{\sigma}(\epsilon)d\epsilon=\sum_{\sigma}\int_{-\infty}^{E_{F}}D_{\sigma}(\epsilon)d\epsilon=n_{i},$
where $f(\epsilon)$ is the Fermi distribution function defined by
$f(\epsilon)=1/(e^{\beta(\epsilon-\mu)}+1)$.

The electrical current (${\bf J}_{e}$) and the heat current due to electrons
(${\bf J}_{Q}^{\it ele}$), and the heat current due to ferromagnetic magnons
(${\bf J}_{Q}^{\it mag}$) are defined as ${\bf J}_{e}=e\sum_{{\bf
k}\sigma}v_{{\bf k},\sigma}c_{{\bf k},\sigma}^{\dagger}c_{{\bf k},\sigma}$,
${\bf J}_{Q}^{\it ele}=\sum_{{\bf k}\sigma}(\epsilon_{{\bf
k}\sigma}-\mu)v_{{\bf k},\sigma}c_{{\bf k},\sigma}^{\dagger}c_{{\bf
k},\sigma}$, and ${\bf J}_{Q}^{\it mag}=\sum_{\bf q}\hbar\omega_{\bf
q}\frac{\partial\omega_{\bf q}}{\partial q_{x}}b_{\bf q}^{\dagger}b_{\bf q}$,
where $v_{{\bf k},\sigma}=\frac{1}{\hbar}\frac{\partial\epsilon_{{\bf
k},\sigma}}{\partial k_{x}}$ and $e$ is the electron charge ($e<0$).

Under an electric field ${\bf E}$ and temperature gradient $\nabla T$, the
electrical current density ${\bf j}$ is described in the linear response
theory as ${\bf j}=L_{11}{\bf
E}+L_{12}\bigr{(}-\frac{{\bf\nabla}T}{T}\bigr{)}$, where $L_{11}$ and $L_{12}$
are electrical conductivity and thermoelectric conductivity, respectively
[30]. These coefficients are calculated from the correlation function between
the electrical currents, and that between the electrical and heat currents
derived by Kubo and Luttinger [31, 32, 33]:

$\displaystyle L_{ij}=\lim_{\omega\rightarrow
0}\frac{\Phi_{ij}(\omega+i\delta)-\Phi_{ij}(0)}{i\omega+i\delta},$ (3)

where $\omega$ is a frequency of the external field. In the present case,
$L_{12}$ contains two components due to $J_{Q}^{\it ele}$ and $J_{Q}^{\it
mag}$, which we refer to $L_{12}^{\it ele}$ and $L_{12}^{\it drag}$,
respectively.

The transport coefficient $L_{11}$ due to the electrical currents and
$L_{12}^{\it ele}$ owing to the electrical current and the heat current due to
electrons are [33]

$\displaystyle L_{11}$ $\displaystyle=$ $\displaystyle\int
d\epsilon\bigr{(}-\frac{\partial
f(\epsilon)}{\partial\epsilon}\bigr{)}\sigma(\epsilon),$ (4) $\displaystyle
L_{12}^{\it ele}$ $\displaystyle=$ $\displaystyle\frac{1}{e}\int
d\epsilon\bigr{(}-\frac{\partial
f(\epsilon)}{\partial\epsilon}\bigr{)}(\epsilon-\mu)\sigma(\epsilon),$ (5)

where $\sigma(\epsilon)$ is the function of electrical conductivity, depending
on $\epsilon$. The relaxation time of electrons is included in
$\sigma(\epsilon)$. When we use a Green’s function, which is obtained in Eq.
(2), $\sigma(\epsilon)$ is given by

$\displaystyle\sigma(\epsilon)=\sum_{\sigma}\frac{e^{2}\sqrt{m^{*}}}{12\pi^{2}\hbar^{2}}\frac{(\sqrt{x_{\sigma}^{2}+\Gamma_{\sigma}(\epsilon)^{2}}+x_{\sigma})^{\frac{3}{2}}}{\Gamma_{\sigma}(\epsilon)},$
(6)

where $x_{\sigma}=\epsilon+\Delta\delta_{\sigma,\uparrow}-{\rm
Re}\Sigma_{\sigma}^{\rm R}(\epsilon)$ and $\Gamma_{\sigma}(\epsilon)=-{\rm
Im}\Sigma_{\sigma}^{\rm R}(\epsilon)$, respectively. It has to be noted that
we consider only the effect of the random potential, given by the self-
consistent t-matrix approximation, and neglect the effect of relaxation due to
the electron-magnon interaction in the calculation of the electrical
conductivity $L_{11}$.

Next, we study the correlation function between the electrical current and the
heat current of magnons defined as $\Phi_{12}(\tau)=\frac{1}{V}\langle
T_{\tau}[{\bf j}_{e}(\tau){\bf j}_{Q}^{\it mag}(0)]\rangle$, where $\tau$ is
an imaginary time and $T_{\tau}$ denotes the imaginary time ordering operator
[22, 23]. By the second order perturbation on the exchange interaction based
on the Green’s function of electrons, Eq. (2), the correlation function due to
the magnon drag is obtained as

$\displaystyle\Phi_{12}^{\rm
drag}(\omega)=i\omega\frac{I^{2}e(m^{*})^{2}}{48\pi^{3}\hbar^{5}\Gamma_{\rm
mag}(T)}\int_{0}^{\epsilon^{\rm
cut}_{q}}d\epsilon_{q}\frac{\beta\epsilon_{q}e^{\beta\epsilon_{q}}}{(e^{\beta\epsilon_{q}}-1)^{2}}$
$\displaystyle\times\int
dxf(x)\biggr{[}\frac{L_{1}^{+}}{\Gamma_{\downarrow}(x+\epsilon_{q})}-\frac{L_{2}^{+}}{\Gamma_{\downarrow}(x)}-\frac{L_{1}^{-}}{\Gamma_{\uparrow}(x)}+\frac{L_{2}^{-}}{\Gamma_{\uparrow}(x-\epsilon_{q})}\biggr{]},$
(7)

where $\Gamma_{\rm mag}(T)$ and $\epsilon^{\rm cut}_{q}$ are the temperature
dependent magnon relaxation rate, and an energy cutoff of magnons,
respectively; $L_{1}^{\pm}=\epsilon_{q}\pm\alpha\epsilon_{q}-\Delta-{\rm
Re}\Sigma_{\downarrow}^{({\rm R})}(x+\epsilon_{q})+{\rm
Re}\Sigma_{\uparrow}^{({\rm R})}(x)$ and
$L_{2}^{\pm}=\epsilon_{q}\pm\alpha\epsilon_{q}-\Delta+{\rm
Re}\Sigma_{\uparrow}^{({\rm R})}(x-\epsilon_{q})-{\rm
Re}\Sigma_{\downarrow}^{({\rm R})}(x)$ where
$\alpha=\frac{\hbar^{2}}{2m^{*}D}$. In the supplemental material, we show the
derivation of Eq. (7) in detail. Using Eq. (3), the thermoelectric
conductivity due to the magnon drag, $L_{12}^{\rm drag}$, is obtained. The
vetex corrections, which are neglected in this letter for simplicity, have
been discussed in ref. [23].

Numerical Results.— Figure 2(a) shows the density of states for
$\tilde{\nu}\equiv\nu/\nu_{0}=1.1$, $2$ and $4$. We set
$\Delta/\epsilon_{B}=0.5$. For $\tilde{\nu}=2$ and $4$, the impurity band
hybridizes the conduction band naturally, while for $\tilde{\nu}=1.1$, the
impurity band only slightly touches with the conduction band. The Fermi energy
is located in $E_{F}/\epsilon_{B}\simeq-1.20$ for $\tilde{\nu}=1.1$ , $-1.16$
for $\tilde{\nu}=2$, and $-1.07$ for $\tilde{\nu}=4.0$. It should be noted
that the chemical potential does not show a drastic temperature dependence.

Figure 2: (a) Densitiy of states (DOS), (b) electrical resistivity, (c)
Seebeck coefficient due to the magnon drag ($S_{\rm drag}$) and (d) the power
factor due to the magnon drag (PFdrag) for $\tilde{\nu}\equiv\nu/\nu_{0}=1.1$,
$2$ and $4$ and $\Delta/\epsilon_{B}=0.5$. Inset of (b) Seebeck coefficient
due to the heat current of electron ($S_{\rm ele}$).

Figure 2(b) shows the temperature dependent electrical resistivity
($\rho=1/L_{11}$) for $\tilde{\nu}=1.1$, $2$ and $4$. Here,
$\rho_{0}=12\pi^{2}\hbar^{2}/\gamma e^{2}\sqrt{m^{*}\epsilon_{B}}$. It has to
be noted that we introduce the dimensionless phenomenological parameter
$\gamma$ to consider additional contributions of the valleys and other
unspecified processes to the electrical conductivity. As shown in Fig. 2(b),
the resistivity increases gradually, as the temperature decreases from high
temperatures, while around $k_{B}T/\epsilon_{B}\simeq 0.5$, the resistivity
drastically increases; the resistivity becomes constant at low temperatures.
This behavior is a result of the impurity band. We also conclude that the
constant resistivity value at low temperatures depends on the impurity
concentration.

Next, let us discuss the Seebeck coefficient due to the heat current of
electrons, i.e. $S_{\rm ele}=L_{12}^{\rm ele}/TL_{11}$ and the Seebeck
coefficient due to the magnon drag, i.e. $S_{\rm drag}=L_{12}^{\rm
drag}/TL_{11}$. The inset of Figure 2(b) shows the temperature dependent
Seebeck coefficients $S_{\rm ele}$ for $\tilde{\nu}=1.1$, $2$ and $4$. As the
impurity concentration decreases, the Seebeck coefficient increases, while the
Seebeck coefficient does not show a peak structure. Figure 2(c) shows the
temperature dependent term $S_{\rm drag}$, for $\tilde{\nu}=1.1$, $2$ and $4$
and $\alpha=1.0$. We assume a temperature dependent magnon relaxation rate,
$\Gamma_{\rm mag}=\bigr{(}\hbar/2\tau_{0}\bigr{)}T$, where $\tau_{0}$ is a
constant. The factor $S_{0}$ is defined by
$S_{0}=I^{2}(m^{*})^{3/2}k_{B}^{2}\tau_{0}/2e\pi\hbar^{4}\sqrt{\epsilon_{B}}$.
Note that the Seebeck coefficient does not depend on $\gamma$. As shown in
Fig.2(c), the Seebeck coefficient increases as the impurity concentration
decreases. We also find that a peak structure of the temperature dependent
Seebeck coefficient appears around $k_{B}T/\epsilon_{B}\simeq 0.5$ for
$\nu/\nu_{0}=1.1$, $0.6$ for $\nu/\nu_{0}=2$ and $0.7$ for $\nu/\nu_{0}=4$.
Figs. 2(d) show the temperature dependent power factor due to the magnon drag,
$PF_{\rm drag}$, where we define $PF_{0}=S_{0}^{2}/\rho_{0}$. The $PF_{\rm
drag}$ traces closely the temperature dependent Seebeck coefficient as shown
in Figs. 2(c) regarding several impurity concentrations, while we find that
the peak temperature of $PF_{\rm drag}$ is higher than that of Seebeck
coefficient, because of the distinct decrease of the electrical resistivity.
It should be noted that the temperature dependences of $S_{\rm drag}$ and
$PF_{\rm drag}$ are insensitive to $\alpha$, while these values strongly
depend on $\alpha$ (See the supplemental material).

Discussion: Comparison with experiments.— Here, we compare the obtained
theoretical results with the experimental results of the thin-film Heusler
alloy. Since there are no experimental data on theoretical parameters, we have
chosen a set of the reasonable values: $\epsilon_{B}/k_{B}=300$ K,
$m^{*}/m_{0}=10$, $J/k_{B}=1000$ K, $V=10^{-27}$ $m^{3}$, and $\gamma=10$. It
should be noted that the large effective mass is due to the large density of
states of the conduction band as shown from the first principles DFT
calculations [9]; then, the impurity concentration ($n_{i}$) is of the order
of $10^{27}$ $m^{-3}$ for $\tilde{\nu}=1\sim 4$, which is consistent with the
concentration of W in Fe2VAl. Here, we set the life time of magnon as
$\tau=\tau_{0}/T\sim 10^{-14}$ s at $T=300$ K. This value is reasonable for a
ferromagnetic metal [34].

Using these parameters, $\nu/\nu_{0}=4$ and $\alpha=1.0$, the temperature
dependent electrical resistivity and Seebeck coefficient due to the magnon
drag, as well as the power factor are displayed in Fig. 3.

Figure 3: Temperature dependences of electrical resistivity, Seebeck
coefficient due to the magnon drag, and power factor (PF) in the realistic
parameters.

We find that the electrical resistivity attains $\rho\simeq$ 1000
$\mu\Omega$cm at $T\sim 300$ K; we also find that the Seebeck coefficient due
to the magnon drag exhibits a peak structure, with $S_{\rm max}\sim-500$
$\mu$V/K at $T\sim$ 300 K. The power factor reaches $PF\sim$ 60 mW/m K2 around
$T\sim 400$ K. Since these theoretical results are similar to the experimental
results, we presume that the origin of the huge Seebeck coefficient and the
large $PF$ observed experimentally for the thin-film Heusler alloy is likely
due to a magnon drag, related to the tungsten-based impurity band.

Finally, we comment on the life time of magnons. In this letter, we used a
simple temperature dependent life time of magnons. However, the life time is
expected to be very complicated in a real material, because it is derived from
many kinds of scattering mechanisms such as impurity scattering, magnon-
electron, magnon-magnon, magnon-phonon interactions related with (without) the
Umklapp process, and so on. The understanding of these microscopic mechanisms
for the life time of magnon is a future problem.

Conclusion.— We studied the origin of the large Seebeck coefficient and
unprecedented large PF observed in the thin-film Heusler alloy FeV0.8W0.2Al on
the basis of the linear response theory. Assuming that this thin-film alloy
has a conduction band integrating with the impurity band originated from the W
substitution, and by extending the microscopic phonon drag theory observed in
FeSb2, we derived $L_{11}$ based on the self-consistent t-matrix approximation
and $L_{12}$ due to the magnon drag. As a result, we found that the
theoretical results of the Seebeck coefficient and PF are in agreement with
the experimental ones. Therefore, we concluded that the origin of these
striking thermoelectric properties is likely due to the magnon drag related
with the W-based impurity band.

Acknowledgments.— This work is supported by Grants-in-Aid for Scientific
Research from the Japan Society for the Promotion of Science (No. JP18H01162,
No. JP18K03482, and No. JP20K03802), and JST-Mirai Program Grant (No.
JPMJMI19A1).

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