1 Introduction
In the process of studying the classification of topological dynamical systems, entropy as a conjugacy invariant plays an important role which divides them into two classes. For
$\mathbb {Z}$
-systems, the notion of uniformly positive entropy (u.p.e. for short) was introduced by Blanchard in [Reference Blanchard6] as an analogue in topological dynamics for the notion of a K-process in ergodic theory. He then naturally defined the notion of entropy pairs and used it to show that a u.p.e. system is disjoint from all minimal zero entropy systems [Reference Blanchard7]. Further research concerning u.p.e. systems and entropy pairs can be found in [Reference Blanchard, Glasner and Host8, Reference Blanchard, Host, Maass, Martinez and Rudolph9, Reference Glasner and Weiss13, Reference Glasner and Ye16, Reference Huang and Ye17, Reference Lemańczyk and Siemaszko27].
Recently, there has been a lot of significant progress in studying relative entropy via local relative entropy theory for
$\mathbb {Z}$
-systems. For a factor map between two
$\mathbb {Z}$
-systems, Glasner and Weiss [Reference Glasner and Weiss14] introduced the relative uniformly positive entropy (rel-u.p.e.) and the notion of relative topological Pinsker factor based on the idea of u.p.e. extensions. Later, Park and Siemaszko [Reference Park and Siemaszko30] interpreted another relative topological Pinsker factor, defined by Lemańczyk and Siemaszko [Reference Lemańczyk and Siemaszko27], using relative measure-theoretical entropy and discussed the relative product. In [Reference Huang, Ye and Zhang19], Huang, Ye and Zhang introduced the notions of relative entropy tuples in both topological and measure-theoretical settings. They showed that the finite product of rel-u.p.e. extensions has rel-u.p.e. if and only if the factors are fully supported (for definitions see §2.3). They also proved some classical results about the rel-u.p.e. extension. We will refer readers to [Reference Boyle, Fiebig and Fiebig10, Reference Downarowicz and Serafin11, Reference Huang, Ye and Zhang18, Reference Ledrappier and Walters26] for more results related to local relative entropy theory.
Bauer and Sigmund [Reference Bauer and Sigmund3] initiated a systematic study of the connections between dynamical properties of a
$\mathbb {Z}$
-system and its induced system (whose phase space consists of all Borel probability measures on the original space, for details see §2). A well-known result due to Glasner and Weiss [Reference Glasner and Weiss15] in 1995 reveals that if a system has zero topological entropy, then so does its induced system. Later, this connection was further developed by Kerr and Li in [Reference Kerr and Li23]. They obtained that a system is null if and only if its induced system is null. More research concerning relations of these systems was developed in [Reference Akin, Auslander and Nagar1, Reference Banks2, Reference Sharma and Nagar33, Reference Zhou and Qiao37]. Recently, Bernardes et al [Reference Bernardes, Darji and Vermersch4] proved that a
$\mathbb {Z}$
-system has u.p.e. if and only if its induced system does.
After Ornstein and Weiss’s pioneering work for amenable group actions in 1987 [Reference Ornstein and Weiss29], there have been many developments in the process of studying the amenable group action systems. We will refer the reader to the related papers [Reference Huang, Ye and Zhang20, Reference Lindenstrauss28, Reference Rudolph and Weiss31, Reference Ward and Zhang35, Reference Weiss36, Reference Zimmer38]. In this paper, we always assume that G is a countably infinite discrete amenable group. By a G-system
$(X,G)$
, we mean a compact metric space X together with G acting on X by homeomorphisms, that is, there exists a continuous map
$\Gamma :G\times X\to X$
, satisfying:
-
•
$\Gamma (e_{G},x)=x$
for every
$x\in X$
; -
•
$\Gamma (g,\Gamma (h,x))=\Gamma (gh,x)$
for each
$g,h\in G$
and
$x\in X$
.
We write
$\Gamma (g,x)$
as
$gx$
for every
$g\in G$
and
$x\in X$
.
Motivated by those works which were previously mentioned for
$\mathbb {Z}$
-systems and the local entropy theory developed for countable discrete amenable group action systems due to Huang, Ye and Zhang [Reference Huang, Ye and Zhang20], and Kerr and Li [Reference Kerr and Li24], the present paper aims to investigate the properties of the relative uniformly positive entropy (rel-u.p.e.) for an induced factor map of a factor map between two G-systems (see §2 for definitions).
More precisely, let
$(X,G)$
be a G-system,
$\mathcal {B}_X$
be the set of Borel subsets of X and
$\mathcal {M}(X)$
be the space of Borel probability measures on the compact metric space X endowed with the weak*-topology. Then the G-system
$(X,G)$
induces a system
$(\mathcal {M}(X),G)$
(see §2 for details). For any
$x\in X$
, let
$\delta _{x}$
denote the Dirac measure on x and
$$ \begin{align*}\mathcal{M}_n(X)=\bigg\{\frac{1}{n}\sum_{i=1}^n\delta_{x_i}: x_1, x_2,\ldots,x_n\in X \bigg\}\end{align*} $$
for each
$n\in \mathbb {N}$
. Then
$\mathcal {M}_n(X)$
is closed and invariant under G (that is,
$g\mathcal {M}_n(X)=\mathcal {M}_n(X)$
for every
$g\in G$
). Hence, we can consider the subsystems
$(\mathcal {M}_n(X),G)$
of
$(\mathcal {M}(X),G)$
for each
$n\in \mathbb {N}$
. For a factor map
$\pi :(X,G)\to (Y,G)$
between two G-systems, when
$\mathrm {supp}(Y)=Y$
(for definitions see §2.3), we have the following result.
Theorem 1.1. Let
$\pi : (X,G)\to (Y,G)$
be a factor map between two G-systems,
$\widetilde {\pi }: (\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$
be the factor map induced by
$\pi $
and
$\widetilde {\pi }_n: (\mathcal {M}_n(X),G) \ \to (\mathcal {M}_n(Y),G)$
be the restriction of
$\widetilde {\pi }$
on
$\mathcal {M}_n(X)$
. When
$\mathrm {supp}(Y)=Y$
, the following are equivalent:
-
(1)
$\pi $
has relative uniformly positive entropy; -
(2)
$\widetilde {\pi }_n$
has relative uniformly positive entropy for some
$n\in \mathbb {N}$
; -
(3)
$\widetilde {\pi }_n$
has relative uniformly positive entropy for every
$n\in \mathbb {N}$
; -
(4)
$\widetilde {\pi }$
has relative uniformly positive entropy.
Notice that when Y is a singleton, we obtain that
$(X,G)$
has u.p.e. if and only if the induced system
$(\mathcal {M}(X),G)$
has u.p.e. (when
$G=\mathbb {Z}$
, see [Reference Bernardes, Darji and Vermersch4, Theorem 4]).
We say a map
$\pi : X\to Y$
between two topological spaces is open if the images of open sets are open. Then we have the following result.
Theorem 1.2. Let
$\pi :X\to Y$
be a surjective continuous map between two compact metrizable spaces, and
$\widetilde {\pi }:\mathcal {M}(X)\to \mathcal {M}(Y)$
be the induced map of
$\pi $
. Then
$\pi $
is open if and only if
$\widetilde {\pi }$
is open.
This paper is organized as follows. In §2, we will list some basic notions and results needed in our argument. In §§3 and 4, we will give a proof of Theorem 1.1. Finally, we prove Theorem 1.2 in §5.
2 Preliminaries
In this section, we recall some basic notation and results which will be used repeatedly in our paper. Denote by
$\mathbb {N}$
and
$\mathbb {R}$
the set of natural numbers and real numbers, respectively. For
$n\in \mathbb {N}$
, we write
$[n]$
for
$\{1,2, \ldots ,n\}$
.
2.1 Amenable group
We say a countably infinite discrete group G is amenable if there always exists an invariant Borel probability measure when it acts on any compact metric space. In the case where G is a countably infinite discrete group, amenability is equivalent to the existence of a F
$\phi $
lner sequence: a sequence of non-empty finite subsets
$\{F_n\}_{n=1}^{\infty }$
of G such that
$$ \begin{align*}\lim_{n\to\infty}\frac{\vert F_n\Delta gF_n\vert}{\vert F_n\vert}=0\end{align*} $$
for all
$g\in G$
. One should refer to Ornstein and Weiss’ paper [Reference Ornstein and Weiss29] for more details about an amenable group. In this paper, we always assume that G is a countably infinite discrete amenable group and denote by
$\mathcal {F}(G)$
the collection of non-empty finite subsets of G. The following result is well known (see [Reference Kerr and Li25, Theorem 4.48]).
Theorem 2.1. Let
$\phi $
be a real-valued function on
$\mathcal {F}(G)$
satisfying:
-
(1)
$\phi (Fs)=\phi (F)$
for all
$F\in \mathcal {F}(G)$
and
$s\in G$
; and -
(2)
$\phi (F)\leq ({1}/{k})\sum _{E\in \mathcal {E}}\phi (E)$
for every
$k\in \mathbb {N}$
,
$F\in \mathcal {F}(G)$
and finite collection
$\mathcal {E}\subseteq \mathcal {F}(G)$
with
$\bigcup _{E\in \mathcal {E}}E\subseteq F$
and
$\sum _{E\in \mathcal {E}}1_E\geq k1_F$
.
Then
${\phi (F)}/{\vert F\vert }$
converges to a limit as F becomes more and more invariant and this limit is equal to
$\inf _{F}{\phi (F)}/{\vert F\vert }$
, where F ranges over all non-empty finite subsets of G.
2.2 Induced systems
Assume that X is a compact metric space. Let
$\mathcal {B}_X$
be the collection of Borel subsets of X,
$C(X)$
be the space of continuous maps from X to
$\mathbb R$
endowed with the supremum norm
$\|\cdot \|_{\infty }$
and
$\mathcal {M}(X)$
be the set of Borel probability measures on X endowed with the weak
$^\ast $
-topology, which is the smallest topology making the map
$$ \begin{align*}D_g:\mathcal{M}(X)\to\mathbb{R},\quad \mu\mapsto\int_Xg\,d\mu\end{align*} $$
continuous for every
$g\in C(X)$
, and the topology basis of weak
$^\ast $
-topology consists of the following sets:
$$ \begin{align} \mathbb{V}(\mu;f_1,\ldots,f_k;\epsilon):= \bigg\{\nu\in\mathcal{M}(X): \bigg\vert\int_Xf_i\,d\mu-\int_Xf_i\,d\nu\bigg\vert<\epsilon\ \text{for all } i\in[k]\bigg\}, \end{align} $$
where
$\mu \in \mathcal {M}(X)$
,
$k\geq 1$
,
$\epsilon>0$
and
$f_i:X\to \mathbb {R}$
are continuous functions for
$i\in [k]$
. The Prohorov metric on
$\mathcal {M}(X)$
,
$$ \begin{align*} d_P(\mu,\nu):=\mathrm{inf}\{\delta>0:\mu(A)\leq\nu(A^{\delta})+\delta\ \text{and}\ \nu(A)\leq \mu(A^{\delta})+\delta\ \text{for all}\ A\in\mathcal{B}_X\}, \end{align*} $$
where
$A^{\delta }=\{x\in X: d(x,A)<\delta \}$
, is compatible with the weak*-topology. We will refer the readers to the books [Reference Billingsley5, Reference Dudley12, Reference Kechris22] for the knowledge of space
$\mathcal {M}(X)$
. Moreover,
$$ \begin{align*}d_P(\mu,\nu)= \mathrm{inf}\{\delta>0:\mu(A)\leq\nu(A^{\delta})+\delta\ \text{for all}\ A\in\mathcal{B}_X\}\end{align*} $$
(see [Reference Billingsley5, p. 72]). Proposition 2.2 describes a basis for the weak*-topology on
$\mathcal {M}(X)$
due to Bernardes et al (see [Reference Bernardes, Darji and Vermersch4, Lemma 1]).
Proposition 2.2. The set of the form
$$ \begin{align*}\mathbb{W}(U_1,U_2,\ldots,U_k:\eta_1,\eta_2,\ldots,\eta_k):=\{\mu\in\mathcal{M}(X): \ \mu(U_i)>\eta_i \text{ for } i\in[k]\},\end{align*} $$
where
$k\geq 1$
,
$U_1,U_2,\ldots ,U_k$
are non-empty disjoint open sets in X and
$\eta _1,\eta _2,\ldots ,\eta _k$
are positive real numbers with
$\eta _1+\eta _2+\cdots +\eta _k<1$
, form a basis for the weak*-topology on
$\mathcal {M}(X)$
.
A G-system
$(X,G)$
induces a system
$(\mathcal {M}(X),G)$
, where
$g: \mathcal {M}(X)\to \mathcal {M}(X)$
is defined by
$(g\mu )(A):=\mu (g^{-1}A)$
for every
$g\in G$
,
$\mu \in \mathcal {M}(X)$
and
$A\in \mathcal {B}_X$
. We call
$(\mathcal {M}(X),G)$
the induced system of
$(X,G)$
.
Let
$(X,G)$
and
$(Y,G)$
be two G-systems. A continuous map
$\pi :(X,G)\to (Y,G)$
is called a factor map between
$(X,G)$
and
$(Y,G)$
if it is onto and
$\pi \circ g=g\circ \pi $
for every
$g\in G$
. Here,
$\pi $
can induce a factor map
$\widetilde {\pi }:(\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$
by
$$ \begin{align*}(\widetilde{\pi}\mu)(B)=\mu(\pi^{-1}B)\end{align*} $$
for every
$\mu \in \mathcal {M}(X)$
and
$B\in \mathcal {B}_Y$
. For every
$n\in \mathbb {N}$
, we denote
$$ \begin{align*}\widetilde{\pi}_n:=\tilde{\pi}\vert_{\mathcal{M}_n(X)}: \mathcal{M}_n(X)\to\mathcal{M}_n(Y)\end{align*} $$
by the restriction of
$\tilde {\pi }$
on
$\mathcal {M}_n(X)$
. Note that
$\widetilde {\pi }_n$
is also a factor map for each
$n\in \mathbb {N}$
.
2.3 Support
Let
$(X,G)$
be a G-system,
$(\mathcal {M}(X),G)$
be the induced G-system of
$(X,G)$
. We denote by
$\mathcal {M}(X,G)$
the set of all G-invariant measures. For
$\mu \in \mathcal {M}(X)$
, we denote by
$\mathrm {supp}(\mu )$
the support of
$\mu $
, that is, the smallest closed subset
$W\subseteq X$
such that
$\mu (W)=1$
. We denote by
$\mathrm {supp}(X,G)$
the support of
$(X, G)$
, that is,
$$ \begin{align*}\mathrm{supp}(X, G) =\bigcup_{\mu\in\mathcal{M}(X,G)}\mathrm{supp}(\mu).\end{align*} $$
Here,
$(X,G)$
is called fully supported if there is an invariant measure
$\mu \in \mathcal {M}(X,G)$
with full support (that is,
$\mathrm {supp}(\mu )=X$
), equivalently,
$\mathrm {supp}(X, G) = X$
.
2.4 Relative uniformly positive topological entropy
For a given G-system
$(X,G)$
, a cover of X is a family of Borel subsets of X, whose union is X. Denote the set of finite covers by
$\mathcal {C}_X$
. For
$n\in \mathbb N$
and
$\mathcal {U}_1,\mathcal {U}_2,\ldots ,\mathcal {U}_n\in \mathcal {C}_X$
, we denote
$$ \begin{align*}\bigvee_{i=1}^{n}\mathcal{U}_i=\{A_1\cap A_1\cap\cdots\cap A_n: A_i\in\mathcal{U}_i,\ i\in[n]\}.\end{align*} $$
Let
$\pi :(X,G)\to (Y,G)$
be a factor map between two G-systems and
$\mathcal {U}\in \mathcal {C}_X$
. For any non-empty subset E of X, let
$N(\mathcal {U},E)$
be the minimum among the cardinalities of the subsets of
$\mathcal {U}$
which cover E, and define
$$ \begin{align*}N(\mathcal{U}|\pi)=\sup_{y\in Y}N(\mathcal{U},\pi^{-1}(y)).\end{align*} $$
The topological conditional entropy of
$\mathcal {U}$
with respect to
$\pi $
is defined by
$$ \begin{align*}h_{\mathrm{top}}(\mathcal{U},G|\pi)=\lim_{n\to\infty}\frac{1}{\vert F_n\vert}\log N(\mathcal{U}_{F_n}|\pi),\end{align*} $$
where
$\mathcal {U}_{F_{n}}=\bigvee _{g\in F_{n}}g^{-1}\mathcal {U}$
and
$\{F_n\}_{n=1}^{\infty }$
is a F
$\phi $
lner sequence of G. It is well known that
$h_{\mathrm {top}}(\mathcal {U},G\vert \pi )$
is well defined and is independent of the choice of the F
$\phi $
lner sequences of G.
Let
$\pi :(X,G)\to (Y,G)$
be a factor map between G-systems. Here,
$\mathcal {U} = \{ U_{1},\ldots ,U_{n}\}\in \mathcal {C}_X$
is said to be non-dense-on-
$\pi $
-fibre if there is
$y\in Y$
such that
$\pi ^{-1}(y)$
is not contained in any element of
$\overline {\mathcal {U}}$
which consists of the closures of elements of
$\mathcal {U}$
in X. Clearly, if an open cover
$\mathcal {U}=\{U_{1},U_{2}\}$
is non-dense-on-
$\pi $
-fibre, then
$\pi (U_{1})\cap \pi (U_{2})\neq \emptyset $
. We say
$(X,G)$
or
$\pi $
has relative uniformly positive entropy (rel-u.p.e. for short) if for any non-dense-on-
$\pi $
-fibre open cover
$\mathcal {U}$
of X with two elements, we have
$h_{\mathrm{top}}(\mathcal {U},G|\pi )>0$
.
For
$n\in \mathbb {N}$
and G-systems
$(Z_i,G)$
,
$i\in [n]$
, we set
$$ \begin{align*}\prod_{i\in[n]}Z_i=\{(z_1,z_2,\ldots,z_n): z_i\in Z_i;\ i\in[n]\}\end{align*} $$
and
$$ \begin{align*}g(z_1,z_2,\ldots,z_n)=(gz_1,gz_2,\ldots,gz_n)\end{align*} $$
for every
$g\in G$
and
$z_i\in Z_i$
for
$i\in [n]$
. Clearly,
$(\prod _{i\in [n]}Z_i,G)$
is also a G-system. When
$Z_i=Z$
for all
$i\in [n]$
, we write
$\prod _{i\in [n]}Z_i$
as
$Z^{(n)}$
. Let
$\pi _i:(X_i,G)\to (Y_i,G)$
be factor maps between G-systems for
$i\in [n]$
. Then
$\{\pi _i\}_{i\in [n]}$
induce a factor map
$$ \begin{align*}\prod_{i\in[n]}\pi_i: \bigg(\prod_{i\in[n]}X_i,G\bigg) \to \bigg(\prod_{i\in[n]}Y_i,G\bigg)\end{align*} $$
by
$$ \begin{align*}\prod_{i\in[n]}\pi_i(x_1,x_2,\ldots,x_n)=(\pi_1 x_1,\pi_2 x_2,\ldots,\pi_n x_n)\end{align*} $$
for every
$(x_1,x_2,\ldots ,x_n)\in \prod _{i\in [n]}X_i$
. When
$\pi _i=\pi $
for all
$i\in [n]$
, we write
$\prod _{i\in [n]}\pi _i$
as
$\pi ^{(n)}$
. In [Reference Huang, Ye and Zhang19], Huang, Ye and Zhang showed that the finite product of rel-u.p.e. factor maps between
$\mathbb {Z}$
-systems has rel-u.p.e. It also holds for G-systems.
Theorem 2.3. Let
$\pi _i: (X_i,G)\to (Y_i,G)$
be a factor map between two G-systems and
$\mathrm {supp}(Y_i)=Y_i$
for
$i=1,2$
. Then
$\pi _1$
and
$\pi _2$
have rel-u.p.e. if and only if
$\pi _1\times \pi _2: (X_1\times X_2,G)\to (Y_1\times Y_2,G)$
has rel-u.p.e.
We will give a proof of Theorem 2.3 in Appendix A (see Theorem A.5).
3
$\pi $
has rel-u.p.e. if and only if
$\widetilde {\pi }_n$
has rel-u.p.e
Let X be a compact metric space and
$\rho _X$
be a compatible metric for X. We denote
$B_{\rho _X}(x,\delta )=\{y\in X: \rho _X(x,y)<\delta \}$
for
$x\in X$
and
$\delta>0$
, and denote
$$ \begin{align*}\Delta(X)=\{(x,x): x\in X\}.\end{align*} $$
For
$(x_1,x_2)\in X\times X\backslash \Delta (X)$
and
$\mathcal {U}=\{U_1,U_2\}\in \mathcal {C}_X$
, we say
$\mathcal {U}$
is an admissible cover of X with respect to
$(x_1,x_2)$
if for any
$i\in [2]$
, one has
$\{x_1,x_2\}\nsubseteq \overline {U_i}$
. Let
$\pi :(X,G)\to (Y,G)$
be a factor map between two G-systems. Here,
$(x_1,x_2)\in X\times X\backslash \Delta (X)$
is called an entropy pair relevant to
$\pi $
if for any admissible cover
$\mathcal {U}$
with respect to
$(x_1,x_2)$
, we have
$h_{\mathrm{top}}(\mathcal {U},G|\pi )>0$
. Denote by
$E(X,G|\pi )$
the set of all entropy pairs relevant to
$\pi $
. Let
$$ \begin{align*}R_{\pi}=\{(x_1,x_2)\in X\times X :\pi(x_1)=\pi(x_2)\}.\end{align*} $$
It is easy to see that
$E(X,G|\pi )\subseteq R_{\pi }\setminus \Delta (X)$
, and
$\pi $
has rel-u.p.e. if and only if
$E(X,G|\pi )=R_{\pi }\setminus \Delta (X)$
.
The concept of dynamical independence is introduced in [Reference Kerr and Li24, Definition 2.1]. Now we consider its relative version. Let
$\pi : (X,G)\to (Y,G)$
be a factor map between two G-systems. For any
$n\in \mathbb {N}$
and a tuple
$\mathcal {V}=(V_1,V_2,\ldots ,V_n)$
of subsets of X, we say
$J\subseteq G$
is an independence set of
$\mathcal {V}$
with respect to
$\pi $
if for every non-empty finite subset
$I\subset J$
, there exists
$y\in Y$
such that
$$ \begin{align*}\pi^{-1}(y)\cap \bigcap_{g\in I}g^{-1}V_{\sigma(g)}\neq\emptyset\end{align*} $$
holds for every
$\sigma \in [n]^{I}$
. We denote by
$\mathcal {P}_{\mathcal {V}}^{\pi }$
the set of all independence sets of
$\mathcal {V}$
with respect to
$\pi $
.
Remark 3.1. For every
$n\in \mathbb {N}$
and a tuple
$\mathcal {V}=(V_1,V_2,\ldots ,V_n)$
of subsets of X, if we set
$$ \begin{align*}\mathcal{I}_{\mathcal{V}}: \mathcal{F}(G)\to \mathbb{R}; \quad \mathcal{I}_{\mathcal{V}}(F):=\max_{I\subseteq F,I\in \mathcal{P}_{\mathcal{V}}^{\pi}}\vert I\vert,\end{align*} $$
then by Theorem 2.1,
$\mathcal {I}_{\mathcal {V}}(F)/\vert F\vert $
converges as F becomes increasingly more invariant and this limit is equal to
$\inf _{F}({\mathcal {I}_{\mathcal {V}}(F)}/{\vert F\vert })$
, where F ranges over
$\mathcal {F}(G)$
. When this limit is positive, we say
$\mathcal {V}$
is independent with respect to
$\pi $
.
The next lemma follows [Reference Kerr and Li24, Lemma 3.4] (see also [Reference Huang and Ye17, Theorem 7.4]).
Lemma 3.2. Let
$\pi : (X,G)\to (Y,G)$
be a factor map between two G-systems, and
$V_1, V_2$
be two disjoint subsets of X. If we set
$\mathcal {U}=\{X\setminus V_1, X\setminus V_2\}$
, then
$h_{\mathrm {top}}(\mathcal {U},G|\pi )>0$
if and only if
$\{V_1,V_2\}$
is independent with respect to
$\pi $
.
Let
$\pi : (X,G)\to (Y,G)$
be a factor map between two G-systems. For any
$(x_1,x_2)\in X\times X\backslash \Delta (X)$
, disjoint open subsets
$V_1, V_2$
of X with
$x_i\in V_i$
for
$i\in [2]$
,
$\mathcal {V}=\{X\setminus V_1, X\setminus V_2\}$
is an admissible cover of X with respect to
$(x_1,x_2)$
. Then by Lemma 3.2, we immediately have the following corollary.
Corollary 3.3. Let
$\pi : (X,G)\to (Y,G)$
be a factor map between two G-systems and
$(x_1,x_2)\in X\times X\backslash \Delta (X)$
. Then
$(x_1,x_2)\in E(X,G|\pi )$
if and only if for any disjoint open subsets
$V_1, V_2$
of X with
$x_i\in V_i$
for
$i=1,2$
,
$\{V_1,V_2\}$
is independent with respect to
$\pi $
.
We note that for any two non-empty finite sets H, W, if
$H\subseteq W$
and
$S\subset \{1,2\}^{W}$
, one has
$$ \begin{align} \vert S\vert_H\vert\geq\frac{\vert S\vert}{2^{\vert W\vert-\vert H\vert}}, \end{align} $$
where
$S\vert _H$
is the restriction of S on H, that is,
$$ \begin{align*} S\vert_H=\{\sigma\in\{1,2\}^H:\text{there exists}\ \sigma'\in S\ \text{such that}\ \sigma(h)=\sigma'(h)\ \text{for all}\ h\in H\}. \end{align*} $$
The following consequence of Karpovsky and Milman’s generalization of the Sauer–Perles–Shelah lemma [Reference Karpovsky and Milman21, Reference Sauer32, Reference Shelah34] is well known, and one can also refer to [Reference Kerr and Li24, Lemma 3.5].
Lemma 3.4. Given
$k\geq 2$
and
$\unicode{x3bb}>1$
, there exists a constant
$c>0$
such that for all
$n\in \mathbb {N}$
, if
$S\subseteq [k]^{[n]}$
satisfies
$\vert S\vert \geq ((k-1)\unicode{x3bb} )^n$
, then there is an
$I\subseteq [n]$
with
$\vert I\vert \geq cn$
and
$S\vert _I=[k]^I$
.
Theorem 1.1 follows from Theorems 3.5, 4.2 and 4.3.
Theorem 3.5. Let
$n\in \mathbb {N}$
,
$\pi : (X,G)\to (Y,G)$
be a factor map between two G-systems,
$\widetilde {\pi }: (\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$
be the factor map induced by
$\pi $
and
$\widetilde {\pi }_n: \ (\mathcal {M}_n(X),G) \ \to (\mathcal {M}_n(Y),G)$
be the restriction of
$\widetilde {\pi }$
on
$\mathcal {M}_n(X)$
. When
$\mathrm {supp}(Y) \ =Y$
, the following are equivalent:
-
(1)
$\pi $
has rel-u.p.e.; -
(2)
$\widetilde {\pi }_n$
has rel-u.p.e. for some
$n\in \mathbb {N}$
; -
(3)
$\widetilde {\pi }_n$
has rel-u.p.e. for every
$n\in \mathbb {N}$
.
Proof.
$(3)\Rightarrow (2)$
is trivial. We will prove
$(1)\Rightarrow (3)$
and
$(2)\Rightarrow (1)$
.
$(1)\Rightarrow (3)$
. Assume that
$\pi $
has rel-u.p.e. For every fixed
$1\le n< \infty $
, to obtain that
$\widetilde {\pi }_n$
has rel-u.p.e., it is sufficient to prove that
$E(\mathcal {M}_n(X),G|\widetilde {\pi }_n)\supseteq R_{\widetilde {\pi }_n}(\mathcal {M}_{n}(X),G) \ \backslash \Delta (\mathcal {M}_n(X))$
.
Let
$(\mu _1,\mu _2)\in R_{\widetilde {\pi }_n}(\mathcal {M}_n(X), \ G)\backslash \Delta (\mathcal {M}_n(X))$
, and
$\widetilde {V}_1$
and
$\widetilde {V}_2$
be two disjoint open subsets of
$\mathcal {M}_n(X)$
with
$\mu _i\in \widetilde {V}_i$
for
$i\in [2]$
. By Corollary 3.3, we shall show that
$\{\widetilde {V}_1,\widetilde {V}_2\}$
is independent with respect to
$\widetilde {\pi }_n$
.
For
$i\in [2]$
and
$j\in [n]$
, there exist points
$x_j^i\in X$
such that
$\mu _i=({1}/{n})\sum _{j=1}^n\delta _{x_j^i}$
. We note that the map
$\Phi : X^{(n)}\to \mathcal {M}(X)$
, defined by
$$ \begin{align*}\Phi(z_1,z_2,\ldots,z_n)=\frac{1}{n}\sum_{i=1}^{n}\delta_{z_i}\end{align*} $$
is continuous. Thus, for every
$i\in [2]$
and
$j\in [n]$
, there exists open neighbourhoods
$V_j^i$
of
$x_j^i$
such that
$$ \begin{align*}\mu_i\in\bigg\{\frac{1}{n}\sum_{j=1}^{n}\delta_{z_j}: z_j\in V_j^i, j\in[n]\bigg\}\subseteq \widetilde{V}_i.\end{align*} $$
Since
$\widetilde {V}_1{\kern-1pt}\cap{\kern-1pt} \widetilde {V}_2 {\kern-1pt}={\kern-1pt}\emptyset $
, if we set
$W_i{\kern-1pt}={\kern-1pt}V^i_1{\kern-1pt}\times{\kern-1pt} V_2^i{\kern-1pt}\times \cdots \times{\kern-1pt} V_n^i$
for
$i{\kern-1pt}={\kern-1pt}1,2$
, one has
$W_1{\kern-1pt}\cap{\kern-1pt} W_2{\kern-1pt}={\kern-1pt}\emptyset $
. Without loss of generality, we can assume that
$\pi (x_j^1)=\pi (x_j^2)$
for all
$j\in [n]$
since
$\widetilde {\pi }_n(\mu _1)=\widetilde {\pi }_n(\mu _2)$
. Let
$\omega _i=(x_1^i,x_2^i,\ldots ,x_n^i)\in W_i$
for
$i=1,2$
. Then
$$ \begin{align*}(\omega_1,\omega_2)\in R_{\pi^{(n)}}\setminus\Delta(X^{(n)})=E(X^{(n)},G\vert\pi^{(n)})\end{align*} $$
as
$\pi ^{(n)}$
has rel-u.p.e. by Theorem 2.3. Thus,
$\{W_1,W_2\}$
is independent with respect to
$\pi ^{(n)}$
. We note that
$\mathcal {P}^{\pi ^{(n)}}_{\{W_1,W_2\}}\subseteq \mathcal {P}^{\widetilde {\pi }_n}_{\{\widetilde {V_1},\widetilde {V_2}\}}$
. This implies
$\{\widetilde {V_1},\widetilde {V_2}\}$
is independent with respect to
$\widetilde {\pi }_n$
.
$(2)\Rightarrow (1)$
. We assume that
$\widetilde {\pi }_n$
has rel-u.p.e. for some positive integer
$1\le n< \infty $
. In the following, we prove that
$R_{\pi }\setminus \Delta (X)\subseteq E(X,G\vert \pi )$
. Let
$(x_1,x_2)\in R_{\pi }\setminus \Delta (X)$
,
$V_1$
and
$V_2$
be two disjoint open subsets of X with
$x_i\in V_i$
,
$i=1,2$
. By Corollary 3.3, we only need to show that
$\{V_1,V_2\}$
is independent with respect to
$\pi $
.
We set
$$ \begin{align*}\widetilde{V_i}=\bigg\{\mu\in\mathcal{M}_n(X): \mu(V_i)>1-\frac{1}{2n}\bigg\}\end{align*} $$
for
$i=1,2$
. Clearly,
$\widetilde {V_1}$
and
$\widetilde {V_2}$
are disjoint open subsets of
$\mathcal {M}_n(X)$
with
$\delta _{x_i}\in \widetilde {V_i}$
for
$i=1,2$
. Since
$\widetilde {\pi }_n$
has rel-u.p.e., and
$(\delta _{x_1},\delta _{x_2})\in R_{\widetilde {\pi }_n}\setminus \Delta (\mathcal {M}_n(X))=E(\mathcal {M}_n(X),G\vert \widetilde {\pi }_n)$
,
$\{\widetilde {V_1},\widetilde {V_2}\}$
is independent with respect to
$\widetilde {\pi }_n$
. Then there exists a constant
$c>0$
, such that for every fixed
$F\in \mathcal {F}(G)$
, there exist
$I\subseteq F$
with
$\vert I\vert>c\vert F\vert $
and
$\nu =({1}/{n})\sum _{i=1}^{n}\delta _{y_i}\in \mathcal {M}_n(Y)$
for some
$y_i\in Y$
such that
$$ \begin{align*}A_{\sigma}:=\widetilde{\pi}_n^{-1}(\nu)\cap \bigcap_{g\in I}g^{-1}\widetilde{V}_{\sigma(g)}\neq\emptyset\end{align*} $$
for every
$\sigma \in \{1,2\}^{I}$
.
For every
$\sigma \in \{1,2\}^{I}$
and
$\mu _{\sigma }=({1}/{n})\sum _{i=1}^{n}\delta _{z_i^{\sigma }}\in A_{\sigma }$
, we can assume
$\pi (z_i^{\sigma })=y_i$
for
$i\in [n]$
. Moreover, for every
$g\in I$
, one has
$g\mu _{\sigma }=({1}/{n})\sum _{i=1}^n\delta _{gz_i^{\sigma }}\in \widetilde {V}_{\sigma (g)}$
. That is,
$$ \begin{align*}\frac{1}{n}\sum_{i=1}^n\delta_{gz_i^{\sigma}}(V_{\sigma(g)})>1-\frac{1}{2n},\end{align*} $$
which implies
$gz_{i}^{\sigma }\in V_{\sigma (g)}$
for every
$i\in [n]$
. In particular,
$$ \begin{align*}z_{1}^{\sigma}\in \pi^{-1}(y_1)\cap\bigcap_{g\in I}g^{-1}V_{\sigma(g)}\end{align*} $$
for every
$\sigma \in \{1,2\}^I$
. Thus,
$\{V_1,V_2\}$
is independent with respect to
$\pi $
. This ends our proof.
4
$\pi $
is rel-u.p.e. if and only if
$\widetilde {\pi }$
is rel-u.p.e
In this section, we will prove
$\pi $
is rel-u.p.e. if and only if
$\widetilde {\pi }$
is rel-u.p.e. We need the following lemma.
Lemma 4.1. Let
$\pi : X\to Y$
be a continuous surjective map between two compact metric spaces,
$\widetilde {\pi }: \mathcal {M}(X)\to \mathcal {M}(Y)$
be the map induced by
$\pi $
and
$\widetilde {\pi }_n:\mathcal {M}_n(X)\to \mathcal {M}_n(Y)$
be the restriction of
$\widetilde {\pi }$
on
$\mathcal {M}_n(X)$
. Then
$\bigcup _{n\in \mathbb {N}}R_{\widetilde {\pi }_n}$
is dense in
$R_{\widetilde {\pi }}$
.
Proof. Fix compatible metrics
$\rho _X$
for X and
$\rho _Y$
for Y. Let
$(\mu _1,\mu _2)\in R_{\widetilde {\pi }}$
. Without loss of generality, we can assume
$\mu _1\neq \mu _2$
. For any two disjoint open subsets
$\widetilde {V}_1$
,
$\widetilde {V}_2$
of
$\mathcal {M}(X)$
with
$\mu _i\in \widetilde {V}_i$
for
$i\in [2]$
, by (2.1), there exist a constant
$r>0$
small enough, integers
$L_1$
and
$L_2$
,
$f_1,\ldots ,f_{L_1}\in C(X)$
and
$g_1,\ldots ,g_{L_2}\in C(X)$
such that
$$ \begin{align*}\mu_1\in\widetilde{W}_1:=\bigg\{\mu\in\mathcal{M}(X):\bigg\vert\int_X f_i\,d\mu-\int_X f_i\,d\mu_1\bigg\vert<r,i\in [L_1]\bigg\}\subseteq\widetilde{V_1}\end{align*} $$
and
$$ \begin{align*}\mu_2\in\widetilde{W}_2:=\bigg\{\mu\in\mathcal{M}(X):\bigg\vert\int_X g_j\,d\mu-\int_X g_j\,d\mu_2\bigg\vert<r,j\in[L_2]\bigg\}\subseteq\widetilde{V_2}.\end{align*} $$
It is sufficient to prove that
$(\widetilde {W}_1\times \widetilde {W}_2)\cap R_{\widetilde {\pi }_N}\neq \emptyset $
for some
$N\in \mathbb {N}$
.
Without loss of generality, we can assume
$\Vert f_i\Vert \leq 1$
and
$\Vert g_j\Vert \leq 1$
for
$i\in [L_1]$
and
$j\in [L_2]$
. Moreover, since
$f_i,g_j\in C(X)$
for
$i\in [L_1]$
and
$j\in [L_2]$
, there exists
$\varepsilon>0$
such that for any
$x,z\in X$
with
$\rho _X(x,z)<\varepsilon $
, one has
$$ \begin{align} \begin{array}{l}\vert f_i(x)-f_i(z)\vert<\dfrac{r}{2}\quad \text{for every } i\in [L_1], \\[6pt] \vert g_j(x)-g_j(z)\vert<\dfrac{r}{2}\quad \text{for every } j\in[L_2]. \end{array}\end{align} $$
For every
$y\in Y$
, since
$\pi $
is continuous, one can find an open neighbourhood
$V_y\subseteq Y$
such that
$$ \begin{align*}\pi^{-1}(y)\subseteq\pi^{-1}(V_y)\subseteq\overline{\pi^{-1}(V_y)}\subseteq(\pi^{-1}(y))^{{\varepsilon}/{2}}, \end{align*} $$
where
$(\pi ^{-1}(y))^{{\varepsilon }/{2}}=\{x\in X: \rho _X(x,\{\pi ^{-1}(y)\})<{\varepsilon }/{2}\}$
. Moreover, since Y is compact, there exist
$K\in \mathbb {N}$
and pairwise different points
$y_1,\ldots ,y_K$
of Y such that
$Y=\bigcup _{i=1}^{K}V_{y_i}$
. Then one can find
$t>0$
such that
$y_i\in B_{\rho _Y}(y_i,t)\subset V_{y_i}$
for any
$i\in [K]$
and
$\{B_{\rho _Y}(y_1,t),\ldots ,B_{\rho _Y}(y_K,t)\}$
are pairwise disjoint. We set
$$ \begin{align*}W_1= V_{y_1}\setminus\bigcup_{i=2}^{K}B_{\rho_Y}(y_i,t) \quad \text{and}\quad W_i= V_{y_i}\setminus\bigg(\bigcup_{j=1}^{i-1}V_{y_j}\cup\bigcup_{j=i+1}^{K}B_{\rho_Y}(y_j,t)\bigg) \end{align*} $$
for
$i=2,\ldots ,K$
. Then
$\{W_1,\ldots ,W_K\}$
is a partition of Y and
$y_i\in W_i\subseteq V_{y_i}$
for
$i\in [K]$
. Moreover,
$\{\pi ^{-1}(W_1),\ldots ,\pi ^{-1}(W_K)\}$
is a partition of X which satisfies
$$ \begin{align*} \pi^{-1}(y_i)\subseteq\pi^{-1}(W_i)\subseteq \overline{\pi^{-1}(V_{y_i})}\subseteq(\pi^{-1}(y_i))^{{\varepsilon}/{2}}\end{align*} $$
for every
$i\in [K]$
. Then for every
$i\in [K]$
, there exist
$P_i\in \mathbb {N}$
and pairwise different
$x_1^i,x_2^i,\ldots ,x_{P_i}^i\in \pi ^{-1}(y_i)$
, such that
$\{x_j^i: j\in [P_i]\}$
is a
${\varepsilon }/{2}$
-net of
$\pi ^{-1}(W_i)$
. Then one can choose Borel subsets
$A_j^i$
of X for
$i\in [K]$
and
$j\in [P_i]$
, such that:
-
(i)
$\mathrm{diam}(A_j^i)<\varepsilon $
for every
$i\in [K],j\in [P_i]$
; -
(ii)
$x_j^i\in A_j^i$
for every
$i\in [K], j\in [P_i]$
; -
(iii)
$\{A_j^i: j\in [P_i]\} $
is a partition of
$\pi ^{-1}(W_i)$
for every
$i\in [K]$
.
For every
$i\in [K]$
,
$j\in [P_i]$
, we set
$a_{ij}=\mu _{1}(A_j^{i})$
and
$b_{ij}=\mu _{2}(A_j^{i})$
. Since
$\widetilde {\pi }(\mu _1)=\widetilde {\pi }(\mu _2)$
, we have
$$ \begin{align*}\sum_{j=1}^{P_i}a_{ij}=\sum_{j=1}^{P_i}\mu_{1}(A_j^{i})=\mu_{1}(\pi^{-1}(W_i))=\mu_{2}(\pi^{-1}(W_i))=\sum_{j=1}^{P_i}\mu_{2}(A_j^{i})=\sum_{j=1}^{P_i}b_{ij}\end{align*} $$
for
$i\in [K]$
. Then for any
$i\in [K]$
and
$j\in [P_i]$
, there exist integers
$q_{ij}$
,
$\tilde {q}_{ij}$
,
$Q_i$
and
$N\in \mathbb {N}$
large enough satisfying the following conditions:
-
(i*)
${q_{ij}}/{N}\leq a_{ij}<({q_{ij}+1}/{N})$
; -
(ii*)
${\tilde {q}_{ij}}/{N}\leq b_{ij}<{\tilde {q}_{ij}+1}/{N}$
; -
(iii*)
${Q_i}/{N}\leq \sum _{j=1}^{P_i}a_{ij}=\sum _{j=1}^{P_i}b_{ij}<({Q_i+1})/{N}$
.
Now, we choose an
$x_0\in X$
arbitrarily and set
$$ \begin{align*}\widetilde{\mu}_1=\frac{1}{N}\bigg(\sum_{i=1}^{K}\bigg(\sum_{j=1}^{P_i-1}q_{ij}\delta_{x_{j}^{i}}+\bigg(Q_i-\sum_{j=1}^{P_i-1}q_{ij}\bigg)\delta_{x_{P_i}^{i}}\bigg)\bigg)+\frac{N-\sum_{i=1}^{K}Q_{i}}{N}\delta_{x_0}\end{align*} $$
and
$$ \begin{align*}\widetilde{\mu}_2=\frac{1}{N}\bigg(\sum_{i=1}^{K}\bigg(\sum_{j=1}^{P_i-1}\tilde{q}_{ij}\delta_{x_{j}^{i}}+\bigg(Q_i-\sum_{j=1}^{P_i-1}\tilde{q}_{ij}\bigg)\delta_{x_{P_i}^{i}}\bigg)\bigg)+\frac{N-\sum_{i=1}^{K}Q_{i}}{N}\delta_{x_0}.\end{align*} $$
It is clear that
$(\widetilde {\mu }_1,\widetilde {\mu }_2)\in R_{\widetilde {\pi }_N}$
. Now we shall show that
$\widetilde {\mu }_i\in \widetilde {W_i}$
for
$i\in [2]$
.
In fact, for any
$\ell \in [L_1]$
, one has
$$ \begin{align} \bigg\vert\int f_{\ell}\,d\mu_1-\int f_{\ell}\,d\tilde{\mu}_1\bigg\vert&= \bigg\vert\sum_{i=1}^{K}\sum_{j=1}^{P_i}\int_{A_{j}^{i}}f_l\,d\mu_1-\frac{1}{N}\sum_{i=1}^{K}\bigg(\sum_{j=1}^{P_i-1}q_{ij}f_l(x_{j}^{i})\nonumber\\ &\quad+\bigg(Q_i-\sum_{j=1}^{P_i-1}q_{ij}\bigg)f_l(x_{P_i}^{i})\bigg)-\frac{N-\sum_{i=1}^{K}Q_{i}}{N}f_l(x_0)\bigg\vert\nonumber\\ &\leq \bigg\vert\sum_{i=1}^{K}\sum_{j=1}^{P_i}\int_{A_{j}^{i}}f_l\,d\mu_1-\frac{1}{N}\sum_{i=1}^{K}\sum_{j=1}^{P_i}q_{ij}f_l(x_{j}^{i})\bigg\vert\nonumber\\ &\quad+\bigg\vert\frac{1}{N}\sum_{i=1}^{K}(Q_i-\sum_{j=1}^{P_i}q_{ij})f_l(x_{P_i}^{i})\bigg\vert+\bigg\vert\frac{N-\sum_{i=1}^{K}Q_{i}}{N}f_l(x_0)\bigg\vert. \end{align} $$
Since
$\mathrm {diam}(A_j^{i})<\varepsilon $
for
$i\in [K]$
and
$j\in [P_i]$
, by (4.1) and
$(\text {i}^*)$
, we have
$$ \begin{align} &\bigg\vert\sum_{i=1}^{K}\sum_{j=1}^{P_i}\int_{A_{j}^{i}}f_l(x)\,d\mu_1-\frac{1}{N}\sum_{i=1}^{K}\sum_{j=1}^{P_i}q_{ij}f_l(x_{j}^{i})\bigg\vert\nonumber\\ &\quad\leq\sum_{i=1}^{K}\sum_{j=1}^{P_i}\int_{A_{j}^{i}}\vert f_l(x)-f_l(x_{j}^{i})\vert\, d\mu_1+\sum_{i=1}^{K}\sum_{j=1}^{P_i}\bigg(a_{ij}-\frac{q_{ij}}{N}\bigg)\vert f_l(x_{j}^{i})\vert\nonumber\\ &\quad\leq\frac{r}{2}+\frac{\sum_{i=1}^{K}P_i}{N}. \end{align} $$
By
$(\text {i}^*)$
and
$(\text {iii}^*)$
, one has
$$ \begin{align} \sum_{i=1}^{K}\bigg\vert\frac{Q_{i}}{N}-\frac{1}{N}\sum_{j=1}^{P_i}q_{ij}\bigg\vert &\leq\sum_{i=1}^{K}\bigg\vert\frac{Q_{i}}{N}-\sum_{j=1}^{P_i}a_{ij}\bigg\vert+\sum_{i=1}^{K}\bigg\vert\sum_{j=1}^{P_i}a_{ij}-\frac{1}{N}\sum_{j=1}^{P_i}q_{ij}\bigg\vert\nonumber \\ &\leq\frac{K}{N}+\frac{\sum_{i=1}^{K}P_i}{N} \end{align} $$
and
$$ \begin{align*} \bigg\vert\frac{N-\sum_{i=1}^{K}Q_{i}}{N}\bigg\vert =\bigg\vert\sum_{i=1}^{K}\sum_{j=1}^{P_i}a_{ij}-\frac{1}{N}\sum_{i=1}^{K}Q_{i}\bigg\vert\leq\sum_{i=1}^{K}\bigg\vert\sum_{j=1}^{P_i}a_{ij}-\frac{Q_{i}}{N}\bigg\vert\leq\frac{K}{N}. \end{align*} $$
When N is large enough such that
${K}/{N}+{\sum _{i=1}^{K}P_i}/{N}\leq {r}/{6},$
by (4.2), (4.3) and (4.4), we have
$\widetilde {\mu }_1\in \widetilde {W_1}$
. Similarly, we can prove that
$\widetilde {\mu }_2\in \widetilde {W_2}$
. This ends our proof.
Theorem 4.2. Let
$\pi :(X,G)\to (Y,G)$
be a factor map between two G-systems with
$\mathrm {supp}(Y)=Y$
and
$\widetilde {\pi }:(\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$
be the induced map of
$\pi $
. Suppose
$\pi $
has rel-u.p.e., then
$\widetilde {\pi }$
also has rel-u.p.e.
Proof. Assume that
$\pi $
has rel-u.p.e. To show
$\widetilde {\pi }$
has rel-u.p.e., it suffices to prove that
$R_{\widetilde {\pi }}\setminus \Delta (\mathcal {M}(X))\subseteq E(\mathcal {M}(X),G\vert \widetilde {\pi })$
. Let
$(\mu _1,\mu _2)\in R_{\widetilde {\pi }}\setminus \Delta (\mathcal {M}(X))$
and
$\widetilde {V_1}$
,
$\widetilde {V_2}$
be two disjoint open subsets of
$\mathcal {M}(X)$
with
$\mu _i\in \widetilde {V}_i$
for
$i\in [2]$
. By Lemma 4.1, there exist
$n\in \mathbb {N}$
and
$(\mu _1',\mu _2')\in R_{\widetilde {\pi }_n}\cap (\widetilde {V_1}\times \widetilde {V_2})$
. Notice that, since
$\pi $
has rel-u.p.e., by Theorem 3.5,
$\widetilde {\pi }_n$
has rel-u.p.e. Then
$\{\widetilde {V_1}\cap \mathcal {M}_n(X),\widetilde {V_2}\cap \mathcal {M}_n(X)\}$
is independent with respect to
$\widetilde {\pi }_n$
, which implies
$\{\widetilde {V_1},\widetilde {V_2}\}$
is independent with respect to
$\widetilde {\pi }$
. This ends our proof.
We note that for any non-empty finite subsets A, H of
$\mathbb {N}$
with
$A\subseteq H$
and
$S\subseteq \{1,2\}^H$
, one can find
$S_0\subset S$
with
$\vert S_0\vert \geq {\vert S\vert }/{ 2^{\vert H\vert -\vert A\vert }}$
such that for every
$\sigma _1\neq \sigma _2\in S_0$
, there exists
$a\in A$
with
$$ \begin{align} \sigma_1(a)\neq\sigma_2(a). \end{align} $$
In fact, if we let
$\mathcal {W}=S\vert _A$
, then
$\vert \mathcal {W}\vert \geq {\vert S\vert }/{2^{\vert H\vert -\vert A\vert }}$
. For each
$w\in \mathcal {W}$
, there exists
$\sigma _w\in S$
such that
$\sigma _w\vert _A=w$
. Put
$S_0:=\{\sigma _w: w\in \mathcal {W}\}\subseteq S$
. Then
$\vert S_0\vert =\vert \mathcal {W}\vert \geq {\vert S\vert }/{2^{\vert H\vert -\vert A\vert }}$
, and for every
$\sigma _1\neq \sigma _2\in S_0$
, one has
$\sigma _1\vert _A\neq \sigma _2\vert _A$
.
Theorem 4.3. Let
$\pi :(X,G)\to (Y,G)$
be a factor map between two G-systems and
$\widetilde {\pi }:(\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$
be the induced map of
$\pi $
. If
$\widetilde {\pi }$
has rel-u.p.e., then so does
$\pi $
.
Proof. Assume that
$\widetilde {\pi }$
has rel-u.p.e. To show
$\pi $
has rel-u.p.e., we shall show that
$R_{\pi }\setminus \Delta (X)\subseteq E(X,G\vert \pi )$
. Let
$(x_1,x_2)\in R_{\pi }\setminus \Delta (X)$
,
$V_1,V_2$
be two non-empty disjoint open subsets of X with
$x_i\in V_i$
for
$i\in [2]$
. By Corollary 3.3, it is sufficient to show that
$(V_1,V_2)$
is independent with respect to
$\pi $
.
Take
$\epsilon \in (0,\tfrac 12)$
with
$$ \begin{align} 2^{1-\epsilon^2}\cdot (1-\epsilon^2)^{(1-\epsilon^2)}\cdot(\epsilon^2)^{(\epsilon^2)}>1. \end{align} $$
We set
$$ \begin{align} &\widetilde{V_i}=\{\mu\in\mathcal{M}(X): \mu(V_i)> 1-\epsilon^{4}\} \end{align} $$
for
$i\in [2]$
. Clearly,
$\delta _{x_i}\in \widetilde {V}_i$
. Since
$(\delta _{x_1},\delta _{x_2})\in R_{\widetilde {\pi }}$
and
$\widetilde {\pi }$
has rel-u.p.e.,
$(\widetilde {V}_1,\widetilde {V}_2)$
is independent with respect to
$\widetilde {\pi }$
. That is, there exists
$c>0$
such that for every
$F\in \mathcal {F}(G)$
, there exists an independence set
$E\subseteq F$
of
$(\widetilde {V}_1,\widetilde {V}_2)$
with respect to
$\widetilde {\pi }$
with
$\vert E\vert>c\vert F\vert $
.
Fix an
$F\in \mathcal {F}(G)$
and an independence set
$E\subseteq F$
of
$(\widetilde {V}_1,\widetilde {V}_2)$
with respect to
$\widetilde {\pi }$
with
$\vert E\vert>c\vert F\vert $
. Then there exists
$\nu \in \mathcal {M}(Y)$
, such that for every
$\sigma \in \{1,2\}^E$
,
$$ \begin{align} \boldsymbol{\widetilde{V}}_{\sigma}:=\bigg(\bigcap_{g\in E}g^{-1}\widetilde{V}_{\sigma(g)}\bigg)\cap\widetilde{\pi}^{-1}(\nu)\neq\emptyset. \end{align} $$
For every
$\sigma \in \{1,2\}^{E}$
, we take
$\mu _{\sigma }\in \boldsymbol {\widetilde {V}}_{\sigma }$
. Then
$\mu _{\sigma }\in g^{-1}\widetilde {V}_{\sigma (g)}$
for every
$g\in E$
and
$\sigma \in \{1,2\}^{E}$
, which implies
$\mu _{\sigma }(g^{-1}V_{\sigma })>1-\epsilon ^4$
for every
$g\in E$
and
$\sigma \in \{1,2\}^E$
. Thus,
$$ \begin{align*}\int_{X}\frac{1}{\vert E\vert}\sum_{g\in E}1_{g^{-1}V_{\sigma(g)}}(x)\,d{ \mu_{\sigma}}=\frac{1}{|E|}\sum_{g\in E}\mu_{\sigma}(g^{-1}V_{\sigma(g)})> 1-\epsilon^{4}\end{align*} $$
and
$\mu _{\sigma }(\widetilde {X}_{\sigma })> 1-\epsilon ^{2}$
for every
$\sigma \in \{1,2\}^{E}$
, where
$$ \begin{align} \widetilde{X}_{\sigma}=\bigg\{x\in X: \frac{1}{|E|}\sum_{g\in E}1_{g^{-1}V_{\sigma(g)}}(x)> 1-\epsilon^{2}\bigg\}. \end{align} $$
By the inner regular of measure, we can find a closed subset
$$ \begin{align} X_{\sigma}\subseteq\widetilde{X}_{\sigma} \quad \text{with} \quad \mu_{\sigma}(X_{\sigma})> 1-\epsilon^{2} \end{align} $$
for every
$\sigma \in \{1,2\}^{E}$
. Since
$\pi $
is continuous, for every
$\sigma \in \{1,2\}^E$
, we have
$$ \begin{align} Y_{\sigma}:=\pi(X_{\sigma}) \end{align} $$
is a closed subset of Y and
$$ \begin{align*}\nu(Y_{\sigma})=\widetilde{\pi}\mu_{\sigma}(Y_{\sigma})\geq\mu_{\sigma}(X_{\sigma})> 1-\epsilon^{2}.\end{align*} $$
Then
$$ \begin{align*}\int_{Y}\frac{1}{2^{|E|}}\sum_{\sigma\in \{1,2\}^{E}}1_{Y_{\sigma}}(y)\,d \nu> 1-\epsilon^{2}.\end{align*} $$
Put
$$ \begin{align} Y^{'}:=\bigg\{y\in Y: \frac{1}{2^{|E|}}\sum_{\sigma\in \{1,2\}^{E}}1_{Y_{\sigma}}(y)> 1-\epsilon\bigg\}, \end{align} $$
then
$\nu (Y^{'})> 1-\epsilon >\tfrac 12$
.
Now, we fix a point
$y_0\in Y^{'}$
and set
$$ \begin{align} \mathcal{E}:=\{\sigma\in\{1,2\}^{E}: y_0\in Y_{\sigma}\}. \end{align} $$
Then
$|\mathcal {E}|>(1-\epsilon )\cdot 2^{\vert E\vert }$
by (4.12). For any
$\sigma \in \mathcal {E}$
, by (4.13), (4.11), (4.10) and (4.9), there is
$x_{\sigma }\in X_{\sigma }$
with
$$ \begin{align*}\frac{1}{|E|}\sum_{g\in E}1_{g^{-1}V_{\sigma(g)}}(x_{\sigma})>1-\epsilon^2\end{align*} $$
such that
$\pi (x_{\sigma })=y_0$
. For every
$\sigma \in \mathcal {E}$
, we set
$$ \begin{align*}A(\sigma)=\{g\in E: x_{\sigma}\in g^{-1}V_{\sigma(g)}\},\end{align*} $$
then
$\vert A(\sigma )\vert>(1-\epsilon ^2)\vert E\vert $
. Now we define
$$ \begin{align*}\Omega:=\{H\subseteq E:\vert H\vert=\lfloor (1-\epsilon^2)\cdot\vert E\vert\rfloor\}\end{align*} $$
and
$$ \begin{align*}\mathcal{Q} (H)=\{\sigma\in\mathcal{E}:H\subseteq\{g\in E:gx_{\sigma}\in V_{\sigma(g)}\}\}\end{align*} $$
for every
$H\in \Omega $
. Then
$\vert \Omega \vert =\tbinom {\vert E\vert }{\lfloor (1-\epsilon ^2)\cdot \vert E\vert \rfloor }$
and
$\bigcup _{H\in \Omega }\mathcal {Q} (H)=\mathcal {E}$
. Thus, there exists
$H_0\in \Omega $
such that
$|\mathcal {Q} (H_0)|\geq {|\mathcal {E}|}/{|\Omega |}\geq {(1-\epsilon )2^{|E|}}/{\tbinom {\vert E\vert }{\lfloor (1-\epsilon ^2)\cdot \vert E\vert \rfloor }}.$
By (4.5), we can choose
$S\subseteq \mathcal {Q}(H_0)$
such that
$$ \begin{align} |S|\geq\frac{(1-\epsilon)2^{|E|}}{2^{|E|-\lfloor (1-\epsilon^2)\cdot\vert E\vert\rfloor} \cdot\tbinom{|E|}{\lfloor (1-\epsilon^2)\cdot\vert E\vert\rfloor} } \end{align} $$
and for any
$\sigma '\neq \sigma "\in S$
, there exists
$g\in H_0$
that satisfies
$\sigma '(g)\neq \sigma "(g)$
. That is,
$\vert S\vert _{H_0}\vert =\vert S\vert $
. Let
$t=1-\epsilon ^2$
and
$\unicode{x3bb} =\log _2(2^t\cdot t^t\cdot (1-t)^{(1-t)})>0$
. Then by Stirling’s formula, when
$\vert E\vert $
is large enough, one has
$$ \begin{align*} &\vert S\vert_{H_0}\vert \approx 2^{\lfloor t\vert E\vert\rfloor-1}\cdot \sqrt{2\pi t(1-t)\vert E\vert}\cdot t^{t\vert E\vert}\cdot(1-t)^{(1-t)\vert E\vert}\\ &\quad\geq 2^{t\vert E\vert-2}\cdot \sqrt{2\pi t(1-t)\vert E\vert}\cdot t^{t\vert E\vert}\cdot (1-t)^{(1-t)\vert E\vert}\\ &\quad\geq (2^t\cdot t^t\cdot(1-t)^{(1-t)})^{\vert E\vert}>2^{\unicode{x3bb}\vert E\vert}. \end{align*} $$
By Lemma 3.4, there exists a subset
$H_1\subseteq H_0$
with
$\vert H_1\vert>d\vert H_0\vert $
such that
$S\vert _{H_1}=\{1,2\}^{H_1}$
, where d is a positive constant independent with
$ E$
when
$\vert E\vert $
is large enough. By Remark 3.1,
$(\mathcal {V}_1,\mathcal {V}_2)$
is independent with respect to
$\pi $
. This ends our proof.
5
$\pi $
is open if and only if
$\widetilde {\pi }$
is open
In this section, we will prove Theorem 1.2. In fact, we have the following result.
Theorem 5.1. Let
$\pi : X\to Y$
be a surjective continuous map between two compact metrizable spaces,
$\widetilde {\pi }: \mathcal {M}(X)\to \mathcal {M}(Y)$
be the induced map of
$\pi $
and
$\widetilde {\pi }_n: \mathcal {M}_n(X)\to \mathcal {M}_n(Y)$
be the restriction of
$\widetilde {\pi }$
on
$\mathcal {M}_n(X)$
. Then the following are equivalent:
-
(1)
$\pi $
is open; -
(2)
$\widetilde {\pi }$
is open; -
(3)
$\widetilde {\pi }_n$
is open for each
$n\in \mathbb {N}$
; -
(4)
$\widetilde {\pi }_n$
is open for some
$n\in \mathbb {N}$
.
Proof.
$(3)\Rightarrow (4)$
is trivial. We will show
$(2)\Rightarrow (1)$
,
$(4)\Rightarrow (1)$
,
$(1)\Rightarrow (3)$
and
$(1)\Rightarrow (2)$
. Fix compatible metrics
$\rho _X$
for X and
$\rho _Y$
for Y.
(2)
$\Rightarrow $
(1). Suppose that
$\widetilde {\pi }$
is open. For every non-empty open subset U of X, we shall show that
$\pi (U)$
is an open subset of Y. That is, for every
$y\in \pi (U)$
, there exists
$r>0$
such that
$B_{\rho _Y}(y,r)\subseteq \pi (U)$
.
Now fix
$y_0\in \pi (U)$
. Since U is open, there exist
$x_0\in U$
and
$\delta>0$
with
$\pi (x_0)=y_0$
and
$\overline {B_{\rho _X}(x_0,\delta )}\subseteq U$
. Then by Urysohn’s lemma, there exists a countinuous map
$f: X\to [0,1]$
with
$f(z)=1$
when
$z\in B_{\rho _X}(x_0,{\delta}/{2})$
and
$f(z)=0$
when
$z\in X\backslash B_{\rho _X}(x_0,\delta )$
. We set
$$ \begin{align*}\widetilde{U}:=\bigg\{\mu\in\mathcal{M}(X): \int f\,d\mu>\frac{2}{3}\bigg\}.\end{align*} $$
Clearly,
$\widetilde {U}$
is an open subset of
$\mathcal {M}(X)$
and
$\delta _{x_0}\in \widetilde {U}$
.
Since
$\widetilde {\pi }$
is open,
$\widetilde {\pi }(\widetilde {U})$
is an open subset of
$\mathcal {M}(Y)$
. Note that
$\delta _{y_0}=\widetilde {\pi }(\delta _{x_0})\in \widetilde {\pi }(\widetilde {U})$
. Thus, there exists
$r>0$
such that
$$ \begin{align*}\{\delta_{y}: \ y\in B_{\rho_Y}(y_0,r)\}\subset \widetilde{\pi}(\widetilde{U}).\end{align*} $$
Then for every
$y'\in B_{\rho _{Y}}(y_0,r)$
, there exists
$\mu _{y'}\in \widetilde {U}$
such that
$\widetilde {\pi }(\mu _{y'})=\delta _{y'}$
. On the one hand, since
$\mu _{y'}(\{\pi ^{-1}(y')\})=\delta _{y'}(\{y'\})=1$
, we have
$$ \begin{align} \mathrm{supp}(\mu_{y'})\subseteq\pi^{-1}(\{y'\}). \end{align} $$
On the other hand, since
$\mu _{y'}\in \widetilde {U}$
, we have
$\int f\,d\mu _{y'}>\tfrac 23$
. Thus,
$$ \begin{align*}\emptyset\neq \mathrm{supp}(\mu_{y'})\cap B_{\rho_X}(x_0,\delta)\subseteq \mathrm{supp}(\mu_{y'})\cap U.\end{align*} $$
By (5.1), we have
$U\cap \pi ^{-1}(\{y'\})\neq \emptyset $
. That is,
$y'\in \pi (U)$
. Then by the arbitrariness of
$y'\in B_{\rho _{Y}}(y_0,r)$
, one has
$B_{\rho _{Y}}(y_0,r)\subseteq \pi (U)$
. Thus,
$\pi (U)$
is an open subset of Y and
$\pi $
is open.
(4)
$\Rightarrow $
(1). We assume that there exists
$n\in \mathbb {N}$
such that
$\widetilde {\pi }_n$
is open. Let U be an open subset of X. We shall show that for every
$y\in \pi (U)$
, there exists
$r>0$
such that
$y\in B_{\rho _Y}(y,r)\subseteq \pi (U)$
.
Let
$y\in \pi (U)$
, there exists
$x\in U$
with
$\pi (x)=y$
. We set
$$ \begin{align*}\widetilde{U}=\mathcal{M}_n(X)\cap\{\mu\in\mathcal{M}(X): \mu(U)>0\}.\end{align*} $$
Here,
$\widetilde {U}$
is an open subset of
$\mathcal {M}_n(X)$
which contains
$\delta _x$
. Since
$\widetilde {\pi }_n$
is open,
$\widetilde {\pi }_n(\widetilde {U})$
is open which contains
$\delta _y$
. Then there exists
$r>0$
such that
$\{\delta _{z}: \rho _{Y}(z,y)<r\}\subseteq \widetilde {\pi }_n(\widetilde {U})$
. Hence, for every
$z\in B_{\rho _Y}(y,r)$
, there exist
$x_1,x_2,\ldots ,x_n\in X$
such that
$$ \begin{align*}\mu=\frac{1}{n}\sum_{i=1}^n\delta_{x_i}\in\widetilde{U}\quad \text{and}\quad \widetilde{\pi}_n(\mu)=\delta_z.\end{align*} $$
Then one has
$\pi (x_i)=z$
for every
$i\in [n]$
. Since
$\mu \in \widetilde {U}$
, there exists
$i_0\in [n]$
with
$x_{i_0}\in U$
. That is,
$z=\pi (x_{i_0})\in \pi (U)$
. Hence,
$B_{\rho _Y}(y,r)\subseteq \pi (U)$
. This implies
$\pi $
is open.
(1)
$\Rightarrow $
(3). Now we assume that
$\pi $
is open. Let
$n\in \mathbb {N}$
and
$\widetilde {U}$
be an open subset of
$\mathcal {M}_n(X)$
. We shall show that for every
$\nu \in \widetilde {\pi }_n(\widetilde {U})\subseteq \mathcal {M}_n(Y)$
, there exists an open neighbourhood of
$\nu $
in
$\mathcal {M}_n(Y)$
contained in
$\widetilde {\pi }_n(\widetilde {U})$
.
For any
$\nu \in \widetilde {\pi }_n(\widetilde {U})\subseteq \mathcal {M}_n(Y)$
, there exist positive integers h,
$k_1,k_2,\ldots ,k_h$
with
$\sum _{i\in [h]}k_i=n$
and pairwise distinct
$y_1,y_2,\ldots ,y_h\in Y$
such that
$$ \begin{align*}\nu=\frac{1}{n}(k_1\delta_{y_1}+k_2\delta_{y_2}+\cdots+k_h\delta_{y_h}).\end{align*} $$
Since
$\nu \in \widetilde {\pi }_n(\widetilde {U})$
, there exists
$\mu \in \widetilde {U}\subseteq \mathcal {M}_n(X)$
such that
$\widetilde {\pi }_n(\mu )=({1}/{n})\sum _{i=1}^{h}k_i\delta _{y_i}$
. Then for every
$i\in [h]$
, there exist integers
$\ell _i$
,
$m_{i,j}$
, and points
$x_{i,j}\in X$
for
$j\in [\ell _i]$
satisfying:
-
(a)
$m_{i,1}+m_{i,2}+\cdots +m_{i,\ell _i}=k_i$
for every
$i\in [h]$
; -
(b)
$x_{i,1},x_{i,2},\ldots ,x_{i,\ell _i}$
are pairwise distinct and
$\pi (x_{i,j})=y_i$
for every
$i\in [h]$
and
$j\in [\ell _i]$
; -
(c)
$\mu =({1}/{n})\sum \nolimits _{i\in [h]}\sum \nolimits _{j\in [\ell _i]}m_{i,j}\delta _{x_{i,j}}$
.
Since
$\widetilde {U}$
is an open neighbourhood of
$\mu $
, there exists
$r_0>0$
such that if
$z_{i,j}^1,z_{i,j}^2,\ldots , z_{i,j}^{m_{i,j}}\in B_{\rho _X}(x_{i,j},r_0)$
for every
$i\in [h]$
,
$j\in [\ell _i]$
, then
$$ \begin{align} \frac{1}{n}\sum\limits_{i\in[h]}\sum\limits_{j\in[\ell_i]}\bigg(\sum_{t\in[m_{i,j}]}\delta_{z_{i,j}^t}\bigg)\in\widetilde{U}. \end{align} $$
Note that
$y_1,y_2,\ldots ,y_{h}$
are pairwise distinct, then there exists
$\delta>0$
such that
$\{B_{\rho _Y}(y_i,\delta )\}_{i\in [h]}$
are pairwise disjoint. By item (b) and the continuity of
$\pi $
, there exists
$r\in (0,r_0)$
such that
$$ \begin{align} \pi(B_{\rho_X}(x_{i,j},r))\subseteq B_{\rho_X}(y_i,\delta)\ \ \ \text{and}\ \ \ B_{\rho_X}(x_{i,t},r)\cap B_{\rho_X}(x_{i,j},r)=\emptyset \end{align} $$
for every
$i\in [h]$
and different
$j,t\in [\ell _i]$
.
Since
$\pi $
is open,
$\bigcap _{j=1}^{\ell _i}\pi (B_{\rho _X}(x_{i,j},r))$
for every
$i\in [h]$
is open. We set
$$ \begin{align*}\widetilde{V}:=\bigg\{\tau\in\mathcal{M}_n(Y):\tau\bigg(\bigcap_{j=1}^{\ell_i} \pi(B_{\rho_X}(x_{i,j},r))\bigg)>\frac{k_i}{n}-\frac{1}{2n},\ i\in[h]\bigg\}.\end{align*} $$
It is an open subset of
$\mathcal {M}_n(Y)$
. Moreover, for every
$i_0\in [h]$
,
$$ \begin{align*} \nu\bigg(\bigcap_{j=1}^{\ell_{i_0}}\pi(B_{\rho_X}(x_{i_0,j},r))\bigg) &=\frac{1}{n}\sum_{i\in[h]}k_i\delta_{y_i}\bigg(\bigcap_{j=1}^{\ell_{i_0}}\pi(B_{\rho_X}(x_{i_0,j},r))\bigg)\nonumber\\ &=\frac{1}{n}k_{i_0}\delta_{y_{i_0}}\bigg(\bigcap_{j=1}^{\ell_{i_0}}\pi(B_{\rho_X}(x_{i_0,j},r))\bigg)\nonumber\\ &=\frac{1}{n}k_{i_0}>\frac{k_{i_0}}{n}-\frac{1}{2n}. \end{align*} $$
Thus,
$\nu \in \widetilde {V}$
. Next, we shall show that
$\widetilde {V}\subseteq \widetilde {\pi }_n(\widetilde {U})$
.
Now fix any
$\tau \in \widetilde {V}\subseteq \mathcal {M}_n(Y)$
. We have
$\tau =({1}/{n})\sum \nolimits _{s=1}^{n}\delta _{u_{s}}$
for some
$u_{s}\in Y$
. For every
$i\in [h]$
, we set
$$ \begin{align} L(i):=\bigg\{s\in[n]: u_{s}\in \bigcap_{j=1}^{\ell_i} \pi(B_{\rho_X}(x_{i,j},r))\bigg\}. \end{align} $$
By
$\tau \in \widetilde {V}$
, one has
$$ \begin{align} \vert L(i)\vert=n\cdot\tau\bigg(\bigcap_{j=1}^{\ell_i} \pi(B_{\rho_X}(x_{i,j},r))\bigg)>k_i-\frac{1}{2} \end{align} $$
for every
$i\in [h]$
. Since
$\vert L(i)\vert \in \mathbb {N}$
, by (5.5),
$\vert L(i)\vert \geq k_i$
. We note that
$L(i)$
,
$i\in [h]$
are pairwise disjoint since
$\bigcap _{j=1}^{\ell _i}\pi (B_{\rho _X}(x_{i,j},r))$
,
$i\in [h]$
are pairwise disjoint. Moreover, by
$\sum _{i\in [h]}k_i=n$
, one has
$\vert L(i)\vert =k_i$
for every
$i\in [h]$
. Hence,
$$ \begin{align} \bigcup_{i\in[h]}L(i)=\bigsqcup_{i\in[h]} L(i)=[n]\quad\text{and}\quad \tau=\frac{1}{n}\sum_{i\in[h]}\bigg(\sum_{s\in L(i)}\delta_{u_s}\bigg). \end{align} $$
For every
$i\in [h]$
, since
$\vert L(i)\vert =k_i\overset {(a)}{=}\sum _{j\in [\ell _i]}m_{i,j}$
, we can rewrite
$L(i)=\{s_1,s_2,\ldots , s_{k_i}\}$
. For every
$i\in [h]$
and
$j\in [\ell _i]$
, we denote
$R_i(j)=\sum _{t=1}^{j}m_{i,t}$
and
$R_i(0)=0$
. Then
$R_i(\ell _i)=k_i$
. By (5.4), for every
$j\in [\ell _i]$
and integer q with
$R_i(j-1)+1\leq q\leq R_i(j)$
, there exists
$x_{i,q}'\in B(x_{i,j},r)$
such that
$\pi (x_{i,q}')=u_{s_q}$
. Then by (5.2), one has
$$ \begin{align*}\mu':=\frac{1}{n}\sum_{i\in[h]}\sum_{j\in[\ell_i]}\sum_{q=R_i(j-1)+1}^{R(j)} \delta_{x_{i,q}'}\in\widetilde{U}\end{align*} $$
and
$$ \begin{align*} \widetilde{\pi}_n(\mu')&=\frac{1}{n}\sum_{i\in[h]}\sum_{j\in[\ell_i]}\sum_{q=R_i(j-1)+1}^{R_i(j)} \delta_{u_{s_q}}=\frac{1}{n}\sum_{i\in[h]}\sum_{q\in[k_i]}\delta_{u_{s_q}}\\ &=\frac{1}{n}\sum_{i\in[h]}\sum_{s\in L(i)}\delta_{u_s}\overset{(5.6)}{=}\tau. \end{align*} $$
This implies
$\widetilde {V}\subset \widetilde {\pi }_n(\widetilde {U})$
. Hence,
$\widetilde {\pi }_n(\widetilde {U})$
is an open subset of
$\mathcal {M}_n(Y)$
and
$\widetilde {\pi }_n$
is open.
$(1)\,\Rightarrow (2)$
. Now we assume that
$\pi $
is open. Let
$\widetilde {U}$
be an open subset of
$\mathcal {M}(X)$
. We shall show
$\widetilde {\pi }(\widetilde {U})$
is open in
$\mathcal {M}(Y)$
.
For every
$\nu \in \widetilde {\pi }(\widetilde {U})$
, there exists
$\mu \in \widetilde {U}$
such that
$\nu =\widetilde {\pi }(\mu )$
. Next we shall show that there exists
$\delta>0$
small enough such that if we set
$$ \begin{align*}\widetilde{V}:=\{\tau\in\mathcal{M}(Y): d_{P}(\nu,\tau)<\delta\},\end{align*} $$
where
$$ \begin{align*} {d_P(\tau,\nu):=\mathrm{inf}\{\delta>0:\tau(A)\leq\nu(A^{\delta})+\delta \text{ and } \nu(A)\leq\tau(A^{\delta})+\delta \text{ for all } A\in\mathcal{B}_Y\}},\end{align*} $$
then
$\widetilde {V}$
is an open neighbourhood of
$\nu $
contained in
$\widetilde {\pi }(\widetilde {U})$
.
Since
$\widetilde {U}$
is open, by Proposition 2.2, there exist
$k\in \mathbb {N}$
and an open set of the form
$\mathbb {W}(U_1,U_2,\ldots ,U_k; \ \eta _1,\eta _2,\ldots ,\eta _k)$
of
$\mathcal {M}(X)$
, where
$U_1,U_2,\ldots ,U_k$
are disjoint non-empty open subsets of X and
$\eta _1,\eta _2, \ldots ,\eta _k$
are positive real numbers with
$\eta _1+\eta _2+\cdots +\eta _k<1$
, such that
$$ \begin{align*} \mu\in\mathbb{W}(U_1,U_2,\ldots,U_k;\eta_1, \eta_2,\ldots,\eta_k)\subset\overline{\mathbb{W}(U_1,U_2, \ldots,U_k;\eta_1,\eta_2,\ldots,\eta_k)}\subset\widetilde{U}. \end{align*} $$
For any
$t_1,t_2\in \{0,1\}^{[k]}$
, we denote
$t_1>t_2$
if
$t_1\neq t_2$
and
$t_1(i)\geq t_2(i)$
for every
$i\in [k]$
. For every
$\sigma \in \{0,1\}^{[k]}$
, we set
$$ \begin{align*}V_{\sigma}:=\bigcap_{\substack{i\in[k]\\ \sigma(i)=1}}\pi(U_i), \quad V^{\prime}_{\sigma}:=V_{\sigma}\backslash\bigcup\limits_{\substack{\alpha\in\{0,1\}^{[k]}\\\alpha>\sigma}}V_{\alpha}\end{align*} $$
and
$$ \begin{align} \mathcal{E}:=\{t\in\{0,1\}^{[k]}: \nu(V_t')>0\}. \end{align} $$
Recall that for any subset A of Y and
$a>0$
, we denote
$A^{a}=\{y\in Y :\rho _Y(y,A)<a\}$
, where
$\rho _Y$
is the compatible metric on Y. For every
$i\in [k]$
, since
$\pi $
is open and
$U_i$
is open in X, then
$\pi (U_i)$
is an open subset of Y. Then by inner regularity, there exist
$\varepsilon>0$
small enough,
$\delta \in (0,\varepsilon )$
and compact subsets
$C_i$
of Y for
$i\in [k]$
such that:
-
(c1)
$\mu (U_i)>\eta _i+6^{k}\varepsilon $
for every
$i\in [k]$
; -
(c2)
$\nu (V_{\sigma }')>5k\varepsilon $
, for every
$\sigma \in \mathcal {E}$
; -
(c3)
$C_i\subseteq C_i^{\delta }\subseteq C_i^{2\delta }\subseteq \pi (U_i)$
for every
$i\in [k]$
; -
(c4)
$\nu (C_i)>\nu (\pi (U_i))-\varepsilon $
for
$i\in [k]$
.
Now we set
$$ \begin{align*}\widetilde{V}:=\{\tau\in\mathcal{M}(Y): d_{P}(\nu,\tau)<\delta\}.\end{align*} $$
Clearly,
$\widetilde {V}$
is an open subset of
$\mathcal {M}(Y)$
containing
$\nu $
. Now it is sufficient to prove that
$\widetilde {V}\subset \widetilde {\pi }(\widetilde {U})$
.
For every
$\sigma \in \{0,1\}^{[k]}$
, we set
$$ \begin{align*}C_{\sigma}(\delta):=\bigcap\limits_{\substack{i\in[k]\\\sigma(i)=1}}C_i^{\delta}\quad \text{and} \quad C_{\sigma}':= C_{\sigma}(\delta)\backslash\bigcup_{\substack{\alpha\in\{0,1\}^{[k]}\\\alpha>\sigma}}C_{\alpha}(\delta).\end{align*} $$
Then for every
$\sigma \in \{0,1\}^{[k]}$
, by items (c3) and (c4), we have
$$ \begin{align} \nu(V_{\sigma})&=\nu\bigg(\bigcap_{\substack{i\in[k]\\\sigma(i)=1}}\pi(U_i)\bigg)\nonumber\\ &\overset{(c3)}{=}\nu\bigg(\bigcap_{\substack{i\in[k]\\\sigma(i)=1}}((\pi(U_i)\backslash C_i)\cup C_i)\bigg) \overset{(c4)}{\leq}\nu\bigg(\bigcap_{\substack{i\in[k]\\\sigma(i)=1}}C_i\bigg)+k\varepsilon. \end{align} $$
We note that for any
$t_1\neq t_2\in \{0,1\}^{[k]}$
, one has
$C_{t_1}'\cap C_{t_2}'=\emptyset $
. In fact, if we define
$t_1\vee t_2\in \{0,1\}^{[k]}$
by
$$ \begin{align*}(t_1\vee t_2)(i)=\max\{t_1(i),t_2(i)\}\end{align*} $$
for every
$i\in [k]$
, then it is clear that
$t_1\vee t_2>t_1$
or
$t_1\vee t_2>t_2$
. Without loss of generality, we can assume
$t_1\vee t_2>t_1$
, then
$C_{t_1}'\subseteq C_{t_1}(\delta )\backslash {C_{t_1\vee t_2}(\delta )}$
. However,
$$ \begin{align*} C_{t_1}'\cap C_{t_2}'\subseteq C_{t_1}(\delta)\cap C_{t_2}(\delta)= C_{t_1\vee t_2}(\delta). \end{align*} $$
Hence,
$C_{t_1}'\cap C_{t_2}'=\emptyset $
.
Now for any fixed
$\tau \in \widetilde {V}$
, we shall show
$\tau \in \widetilde {\pi }(\widetilde {U})$
. By
$d_P(\nu ,\tau )<\delta $
, one has
$\tau (A^\delta )\geq \nu (A)-\delta $
for every
$A\in \mathcal {B}_Y$
. Then for every
$\sigma \in \mathcal {E}$
,
$$ \begin{align} \tau(C_{\sigma}(\delta))&=\tau\bigg(\bigcap_{\substack{i\in[k]\\\sigma(i)=1}}C_{i}^{\delta} \bigg)\geq \tau\bigg(\bigg(\bigcap_{\substack{i\in[k]\\\sigma(i)=1}}C_i\bigg)^{\delta}\bigg)\nonumber\\ &\geq\nu\bigg(\bigcap_{\substack{i\in[k]\\\sigma(i)=1}}C_i\bigg)-\delta\overset{(5.8)}{\geq} \nu(V_{\sigma})-k\varepsilon-\delta. \end{align} $$
Moreover, for every
$\sigma \in \{0,1\}^{[k]}$
,
$$ \begin{align} &\bigg(\bigcup_{\substack{\alpha\in\{0,1\}^{[k]}\\ \alpha>\sigma}} C_{\alpha}(\delta)\bigg)^{\delta} =\bigcup_{\substack{\alpha\in\{0,1\}^{[k]}\\ \alpha>\sigma}}(C_{\alpha}(\delta))^{\delta} =\bigcup_{\substack{\alpha\in\{0,1\}^{[k]}\\ \alpha>\sigma}} \bigg(\bigcap_{\substack{i\in[k]\\\alpha(i)=1}}C_{i}^{\delta}\bigg)^{\delta}\nonumber\\&\quad\subseteq \bigg(\bigcup_{\substack{\alpha\in\{0,1\}^{[k]}\\ \alpha>\sigma}}\bigg(\bigcap_{\substack{i\in[k]\\\alpha(i)=1}}C_i^{2\delta}\bigg)\bigg) \overset{(c3)}{\subseteq}\bigg(\bigcup_{\substack{\alpha\in\{0,1\}^{[k]}\\ \alpha>\sigma}}\bigg(\bigcap_{\substack{i\in[k]\\\alpha(i)=1}}\pi(U_i)\bigg)\bigg) \nonumber\\&\quad=\bigcup_{\substack{\alpha\in\{0,1\}^{[k]}\\\alpha>\sigma}}V_{\alpha}. \end{align} $$
Since
$d_P(\nu ,\tau )<\delta $
, one has
$$ \begin{align} \tau(A)\leq \nu(A^\delta)+\delta\quad \text{for every } A\in\mathcal{B}_Y. \end{align} $$
Note that for every
$\alpha ,\sigma \in \{0,1\}^{[k]}$
with
$\alpha>\sigma $
, one has
$C_{\alpha }(\delta )\subseteq C_{\sigma }(\delta )$
and
$V_{\alpha }\subseteq V_{\sigma }$
. Then for every
$\sigma \in \mathcal {E}$
,
By
$\overline {\bigcup _{n\in N}\mathcal {M}_n(Y)}=\mathcal {M}(Y)$
, there exist
$\tau _n=({1}/{n})\sum \nolimits _{j=1}^{n}\delta _{y_{n,j}}\in \mathcal {M}_n(Y)$
for
$n\in \mathbb {N}$
and some
$y_{n,j}\in Y$
,
$j\in [n]$
, such that
$\tau _n\to \tau $
as
$n\to \infty $
. Moreover, since
$\widetilde {V}$
is open in
$\mathcal {M}(Y)$
, we can find
$N_0\in \mathbb {N}$
such that
$\tau _n\in \widetilde {V}$
for
$n\geq N_0$
.
Let
$n\geq N_0$
. For every
$\sigma \in \{0,1\}^{[k]}$
, we set
$$ \begin{align*}S_\sigma^{n}:=\{h\in[n]: y_{n,h}\in C_{\sigma}'\}.\end{align*} $$
Since
$\tau _n\in \widetilde {V}$
, by (5.12) and recall that for any
$t_1\neq t_2\in \{0,1\}^{[k]}$
, one has
$C_{t_1}'\cap C_{t_2}'=\emptyset $
, then:
-
(i)
$S_{t_1}^n\cap S_{t_2}^n=\emptyset $
for any
$t_1\neq t_2\in \{0,1\}^{[k]}$
; -
(ii)
$\vert S_{\sigma }^n\vert \geq n(\nu (V_{\sigma }')-3k\varepsilon )$
for every
$\sigma \in \mathcal {E}$
, where
$\mathcal {E}$
is defined as (5.7).
Now, for every
$\sigma \in \{0,1\}^{[k]}$
and
$i\in [k]$
, we set
$$ \begin{align} U_{i,\sigma}:=U_i\cap\pi^{-1}(V_{\sigma}')\quad \text{and} \quad a_{i,\sigma}:=\mu(U_{i,\sigma}). \end{align} $$
Fix any
$\sigma \in \{0,1\}^{[k]}$
. We can rewrite
$\{i\in [k]: \sigma (i)=1\}$
as
$\{i_1<i_2<\cdots <i_{q}\}$
for some
$q\in \mathbb {N}$
. For
$i_1$
, we choose arbitrarily a subset
$S_{\sigma ,i_1}^{n}$
of
$S_{\sigma }^n$
with
$\vert S_{\sigma ,i_1}^n\vert =\lfloor {a_{i_1,\sigma }}/{\sum \nolimits _{\ell \in [k],\sigma (\ell )=1}a_{\ell ,\sigma }}\vert S_{\sigma }^n\vert \rfloor $
, where we note:
$\tfrac 00=0$
. For
$i_2$
, we choose arbitrarily a subset
$S_{\sigma ,i_2}^{n}$
of
$S_{\sigma }^n\setminus S_{\sigma ,i_1}^n$
with
$\vert S_{\sigma ,i_2}^n\vert =\lfloor {a_{i_2,\sigma }}/{\sum \nolimits _{\ell \in [k],\sigma (\ell )=1}a_{\ell ,\sigma }}\vert S_{\sigma }^n\vert \rfloor $
. We continue inductively obtaining
$$ \begin{align*}S_{\sigma,i_j}^n\subseteq S_{\sigma}^n\setminus(S_{\sigma,i_1}^n\cup S_{\sigma,i_2}^n \cup\cdots\cup S_{\sigma,i_{j-1}}^n)\end{align*} $$
for
$j=3,4,\ldots ,q-1$
, with
$\vert S_{\sigma ,i_j}^n\vert =\lfloor {a_{i_j,\sigma }}/{\sum \nolimits _{\ell \in [k],\sigma (\ell )=1}a_{\ell ,\sigma }}\vert S_{\sigma }^n\vert \rfloor $
. We set
$S_{\sigma ,i_q}^n=S_{\sigma }^n\setminus (\bigcup _{j=1}^{q-1}S_{\sigma ,i_j}^n)$
. Additionally, we note that
$$ \begin{align*}y_{n,h}\in C_{\sigma}'\subseteq\bigcap\limits_{\substack{i\in[k]\\\sigma(i)=1}}\pi(U_i)=\bigcap_{\ell=1}^{q} \pi(U_{i_\ell})\end{align*} $$
for every
$h\in S_{\sigma }^n$
. Then we have the following properties for
$S_{\sigma ,i}^n$
,
$i\in [k]$
.
-
(i*)
$\vert S_{\sigma ,i}^n\vert \geq \lfloor {a_{i,\sigma }}/{\sum \nolimits _{\ell \in [k],\sigma (\ell )=1}a_{\ell ,\sigma }}\vert S_{\sigma }^n\vert \rfloor $
for every
$i\in [k]$
with
$\sigma (i)=1$
. (ii*) For every
$i\in [k]$
with
$\sigma (i)=1$
, if
$h\in S_{\sigma ,i}^n$
, then there exists
$x_{n,h}^{\sigma }\in U_i$
satisfying
$\pi (x_{n,h}^{\sigma })=y_{n,h}$
.(iii*)
$S_{\sigma ,i'}^n\cap S_{\sigma ,i"}^n=\emptyset $
for every
$i'\neq i"\in \{i\in [k]:\sigma (i)=1\}$
and
$\bigcup \nolimits _{i\in [k],\sigma (i)=1} S_{\sigma ,i}^n=S_{\sigma }^n$
.
Since
$\pi $
is surjective, for every
$h'\in S_0^n:=[n]\backslash (\bigcup \nolimits _{\sigma \in \{0,1\}^{[k]}}S_{\sigma }^n)$
, there exists
$x_{n,h'}\in X$
such that
$\pi (x_{n,h'})=y_{n,h'}$
. Now we set
$$ \begin{align} \mu_n:&=\frac{1}{n}\bigg(\sum_{\sigma\in \{0,1\}^{[k]}}\sum_{h\in S_{\sigma}^n}\delta_{x_{n,h}^{\sigma}}+\sum_{h'\in S_0^n}\delta_{x_{n,h'}}\bigg)\nonumber\\ &\overset{\text{(iii*)}}{=}\frac{1}{n}\bigg(\sum_{\sigma\in \{0,1\}^{[k]}}\sum_{\substack{i\in[k]\\ \sigma(i)=1}}\sum_{h\in S_{\sigma,i}^n} \delta_{x_{n,h}^{\sigma}}+\sum_{h'\in S_0^n}\delta_{x_{n,h'}}\bigg). \end{align} $$
Clearly,
$\widetilde {\pi }(\mu _n)=\tau _n$
. We claim that
$\mu _{n}(U_{i_0})>\eta _{i_0}$
for every
$i_0\in [k]$
when n is sufficiently large. Once it is true, we have
$$ \begin{align*} \mu_n\in\mathbb{W}(U_1,U_2,\ldots,U_k;\eta_1,\eta_2,\ldots,\eta_k). \end{align*} $$
Then we can find a sequence
$n_1<n_2<\cdots $
such that
$\lim \nolimits _{i\to \infty }\mu _{n_i}=\mu '$
for some
$\mu '\in \mathcal {M}(X)$
. Thus,
$$ \begin{align*} \mu'\in\overline{\mathbb{W}(U_1,U_2,\ldots,U_k;\eta_1,\eta_2,\ldots,\eta_k)}\subset\widetilde{U}\end{align*} $$
and
$\widetilde {\pi }(\mu ')=\lim \nolimits _{i\to \infty }\widetilde {\pi }(\mu _{n_i})=\lim \nolimits _{i\to \infty }\tau _{n_i}=\tau $
. By the arbitrariness of
$\tau $
, one has
$\widetilde {V}\subseteq \widetilde {\pi }(\widetilde {U})$
. This will end our proof.
Now, we shall show the claim:
$\mu _{n}(U_{i_0})>\eta _{i_0}$
for every
$i_0\in [k]$
when n is sufficiently large. To show that, for any fixed
$i_0\in \{1,2,\ldots ,k\}$
, we need the following facts.
Fact 1:
$\sum \nolimits _{\sigma \in \mathcal {E},\sigma (i_0)=1}\mu (U_{i_0}\cap \pi ^{-1}(V_{\sigma }'))=\mu (U_{i_0}\cap \bigcup \nolimits _{\sigma \in \mathcal {E},\sigma (i_0)=1}\pi ^{-1}(V_{\sigma }'))$
. In fact, for any
$t_1\neq t_2\in \{0,1\}^{[k]}$
, if
$y\in V_{t_1}'\cap V_{t_2}'\subseteq V_{t_1}\cap V_{t_2}$
, then
$y\in V_{t_1\vee t_2}$
. Since
$t_1\vee t_2>t_1\ \text {or}\ t_2$
, one has
$y\notin V_{t_1}'$
or
$y\notin V_{t_2}'$
, which is a contradiction of
$y\in V_{t_1}'\cap V_{t_2}'$
. Hence,
$V_{t_1}'\cap V_{t_2}'=\emptyset $
. Then Fact 1 follows.
Fact 2:
$\nu ((\bigcup \nolimits _{\substack {\sigma \in \mathcal {E}\\\sigma (i_0)=1}}V_{\sigma }')\Delta (\bigcup \nolimits _{\substack {\sigma \in \{0,1\}^{[k]}\\\sigma (i_0)=1}}V_{\sigma }))=0$
, where
$A\Delta B$
denotes
$(A\setminus B)\cup (B\setminus A)$
for every
$A,B\in \mathcal {B}_Y$
. In fact, it is clear that
$$ \begin{align} \bigcup\limits_{\substack{\sigma\in\mathcal{E}\\\sigma(i_0)=1}}V_{\sigma}'\subseteq \bigcup\limits_{\substack{\sigma\in\{0,1\}^{[k]}\\\sigma(i_0)=1}}V_{\sigma}. \end{align} $$
Since
$\sigma \in \{0,1\}^{[k]}\backslash \mathcal {E}$
implies
$\nu (V_{\sigma }')=0$
, one has
$$ \begin{align} \nu\bigg(\bigcup\limits_{\substack{\sigma\in\mathcal{E}\\\sigma(i_0)=1}}V_{\sigma}'\bigg) =\nu\bigg(\bigcup\limits_{\substack{\sigma\in\{0,1\}^{[k]}\\\sigma(i_0)=1}}V_{\sigma}'\bigg). \end{align} $$
Clearly,
$\bigcup \nolimits _{\substack {\sigma \in \{0,1\}^{[k]}\\\sigma (i_0)=1}}V_{\sigma }\supseteq \bigcup \nolimits _{\substack {\sigma \in \{0,1\}^{[k]}\\\sigma (i_0)=1}} V_{\sigma }'$
. Moreover, for any given
$x\in \bigcup \nolimits _{\substack {\sigma \in \{0,1\}^{[k]}\\\sigma (i_0)=1}}V_{\sigma }$
, if we define
$\sigma '$
as
$$ \begin{align*}\sigma'(i)=\max\{t(i): t\in\{0,1\}^{[k]} \text{ with } x\in V_{t}\}\end{align*} $$
for every
$i\in [k]$
, then
$\sigma '(i_0)=1$
and
$x\in V_{\sigma '}^{'}\subseteq \bigcup \nolimits _{\substack {\sigma \in \{0,1\}^{[k]}\\\sigma (i_0)=1}} V_{\sigma }'$
. Hence,
$$ \begin{align} \bigcup\limits_{\substack{\sigma\in\{0,1\}^{[k]}\\\sigma(i_0)=1}}V_{\sigma}=\bigcup\limits_{\substack{\sigma\in\{0,1\}^{[k]}\\\sigma(i_0)=1}} V_{\sigma}'. \end{align} $$
By (5.15), (5.16) and (5.17), Fact 2 holds.
Fact 3: For every
$\sigma \in \mathcal {E}$
,
$\sum \nolimits _{\ell \in [k],\sigma (\ell )=1}a_{\ell ,\sigma }\leq \nu (V_{\sigma }')$
. Note that
$U_1,U_2,\ldots ,U_k$
are disjoint. Then by (5.13), we have
Thus, Fact 3 holds.
Now by Facts 1–3, we have
We define
$t\in \{0,1\}^{[k]}$
as
$t(i_0)=1$
and
$t(i)=0$
for each
$i\in [k]\backslash \{i_0\}$
. Then
$\pi (U_{i_0})=V_{t}\subseteq \bigcup \nolimits _{\substack {\sigma \in \{0,1\}^{[k]}\\\sigma (i_0)=1}}V_{\sigma }$
. Thus,
$U_{i_0}\subseteq \pi ^{-1}(\bigcup \nolimits _{\substack {\sigma \in \{0,1\}^{[k]}\\\sigma (i_0)=1}}V_{\sigma })$
and by (5.18), we have
$$ \begin{align} \sum_{\sigma\in\mathcal{E},\sigma(i_0)=1}a_{i_0,\sigma}=\mu(U_{i_0}). \end{align} $$
Then for any
$n>N_0$
, we have
Then by letting
$n\to \infty $
, for every
$i_0\in [k]$
since
$\mu (U_{i_0})>\eta _{i_0}+6^{k}\varepsilon $
by (c1), we have
$\mu _n(U_{i_0})>\eta _{i_0}$
. This ends the proof of the claim.
Acknowledgements
The authors would like to thank Prof. Wen Huang, Prof. Hanfeng Li, Dr Yixiao Qiao and Dr Lei Jin for their useful comments and suggestions. K.L. is supported by the China Postdoctoral Science Foundation (No. 2022M710527).
A Appendix. Proof of Theorem 2.3
Let
$\pi : (X,G)\to (Y,G)$
be a factor map between two G-systems. For any
$n\in \mathbb {N}$
and a tuple
$\mathcal {V}=(V_1,V_2,\ldots ,V_n)$
of subsets of X, recall that we denote by
$\mathcal {P}_{\mathcal {V}}^{\pi }$
the set of all independence sets of
$\mathcal {V}$
with respect to
$\pi $
.
Identifying subsets of G with elements of
$\{0,1\}^G$
by taking indicator functions, we may think of
$\mathcal {P}_{\mathcal {V}}^{\pi }$
as a subset of
$\{0,1\}^G$
. Endow
$\{0,1\}^G$
with the shift given by
$$ \begin{align*}(s\sigma)(t)=\sigma(ts)\end{align*} $$
for all
$\sigma \in \{0,1\}^G$
and
$s,t\in G$
. It is clear that
$\mathcal {P}_{\mathcal {V}}^{\pi }$
is shift-invariant. Moreover, when
$V_1,V_2,\ldots ,V_n$
are closed subsets of X,
$\mathcal {P}_{\mathcal {V}}^{\pi }$
is also closed in
$\{0,1\}^G$
.
We say a closed and shift-invariant subset
$\mathcal {P}\subseteq \{0,1\}^G$
has positive density if there exists constant
$c>0$
such that for every non-empty subset F of G, there exists
$I\in \mathcal {P}$
with
$I\subseteq F$
such that
$\vert I\vert>c\vert F\vert $
. Then by Corollary 3.3, we immediately have the following property.
Proposition A.1. Let
$\pi : (X,G)\to (Y,G)$
be a factor map between two G-systems,
$(x_1,x_2)\in X\times X\backslash \Delta (X)$
. Then
$(x_1,x_2)\in E(X,G|\pi )$
if and only if for any disjoint open subsets
$V_1, V_2$
of X with
$x_i\in V_i$
for
$i=1,2$
,
$\mathcal {P}_{\{V_1,V_2\}}^{\pi }$
has positive density.
The following lemma is useful.
Lemma A.2. [Reference Kerr and Li25, Lemma 12.6]
Let A be a closed subset of X. Then
$\mathcal {P}_A:=\{I\subseteq G: \bigcap _{g\in I}g^{-1}A\neq \emptyset \}$
has positive density if and only if there exists
$\mu \in \mathcal {M}(X,G)$
with
$\mu (A)>0$
.
The following lemma is proved when
$G=\mathbb {Z}$
in [Reference Huang, Ye and Zhang19, Proposition 3.9]. We omit the proof.
Lemma A.3. Let
$\pi : (X,G)\to (Z,G)$
,
$\pi _1: (X,G)\to (Y,G)$
and
$\pi _2: (Y,G)\to (Z,G)$
be three factor maps such that
$\pi =\pi _2\cdot \pi _1$
. Then
$\pi $
has rel-u.p.e. implies
$\pi _2$
has rel-u.p.e.
For a factor map
$\pi : (X,\mathbb {Z})\to (Y,\mathbb {Z})$
between two
$\mathbb {Z}$
-systems, the authors in [Reference Huang, Ye and Zhang19] proved that if
$\pi $
has rel-u.p.e., then
$\mathrm {supp}(Y)=Y$
implies
$\mathrm {supp}(X)=X$
(see [Reference Huang, Ye and Zhang19, Theorem 5.4]). For discrete countable amenable group G, we have the same result.
Proposition A.4. Let
$\pi :(X,G)\to (Y,G)$
be a factor map between two G-systems. If
$\pi $
has rel-u.p.e. and
$\mathrm {supp}(Y)=Y$
, then
$\mathrm {supp}(X)=X$
.
Proof. Assume that
$\mathrm {supp}(X)\not =X$
, then there exist
$x_1\in X$
and an open neighbourhood V of
$x_1$
such that
$V\cap \mathrm {supp}(X)=\emptyset $
. Let
$U=\bigcup _{g\in G}g^{-1}V$
, then U is open and
$\mu (U)=0$
for every
$\mu \in \mathcal {M}(X,G)$
. Thus,
$\mathrm {supp}(X)\subseteq U^c$
, where
$U^c=X\setminus U$
.
Let
$y=\pi (x_1)$
. We note that
$\pi ^{-1}\{y\}\cap U^c\neq \emptyset $
. In fact, since
$\mathrm {supp}(Y)=Y$
, there exits
$\nu \in \mathcal {M}(Y,G)$
such that
$y\in \mathrm {supp}(\nu )$
. Then there exists
$\tilde {\mu }\in \mathcal {M}(X,G)$
such that
$\widetilde {\pi }(\tilde {\mu })=\nu $
. If
$\pi ^{-1}\{y\}\subseteq U$
, there exists
$\delta>0$
such that
$\pi ^{-1}B(y,\delta )\subseteq U$
. Then
$\nu (B(y,\delta ))=\tilde {\mu }(\pi ^{-1}B(y,\delta ))=0$
. This contradicts
$y\in \mathrm {supp}(\nu )$
. Thus, there exists
$x_2\in U^c$
such that
$\pi (x_2)=y$
.
By Urysohn’s lemma, there exists continuous function
$f: X\to [0,1]$
such that
$f(x_1)=0$
and
$f(x)=1$
for any
$x\in U^{c}$
. We set
$$ \begin{align*}F: X\to [0,1]^G \ \ \text{by}\ \ (F(x))(g)=f(gx).\end{align*} $$
Consider the G-action on
$[0,1]^G$
defined by
$(g\omega )(h)=\omega (hg)$
for every
$\omega \in [0,1]^G$
and
$g,h\in G$
. We define a factor map
$$ \begin{align*}\phi: (X,G)\to ([0,1]^G\times Y,G) \ \ \text{by}\ \ \phi(x)=(F(x),\pi(x)).\end{align*} $$
Let
$W=\phi (X)$
and
$\pi _2: (W,G)\to (Y,G)$
be the projection map to the second coordinate. Then
$\pi =\pi _2\circ \phi $
. By Proposition A.3,
$\pi _2$
has rel-u.p.e. Note that
$\pi _2(\phi (x_1))=\pi (x_1)=\pi (x_2)=\pi _2(\phi (x_2))$
and
$\phi (x_1)\neq \phi (x_2)$
. Thus,
$$ \begin{align*}(\phi(x_1),\phi(x_2))\in R_{\pi_2}\setminus \Delta(W)=E(W,G\vert \pi_2).\end{align*} $$
Then, by Lemma A.2, one has
$\phi (x_1)\in \mathrm {supp}(W)$
. However,
$\phi (x_1)\notin \{1^G\}\times Y$
and for every
$\mu \in \mathcal {M}(X,G)$
, one has
$\mathrm {supp}(\mu )\subseteq U^c$
, which implies
$\mathrm {supp}(W)\subseteq \phi (U^c)\subseteq \{1^G\}\times Y$
. Thus,
$\phi (x_1)\notin \mathrm {supp}(W)$
. This is a contradiction.
Now we are ready to give the proof of Theorem 2.3.
Theorem A.5. Let
$\pi _i: (X_i,G)\to (Y_i,G)$
be two factor maps between G-systems and
$\mathrm {supp}(Y_i)=Y_i$
for
$i=1,2$
. Then
$\pi _1$
and
$\pi _2$
has rel-u.p.e. if and only if
$\pi _1\times \pi _2: (X_1\times X_2,G)\to (Y_1\times Y_2,G)$
has rel-u.p.e.
Proof. For the non-trivial direction, if
$\pi _1$
and
$\pi _2$
have rel-u.p.e., for any
$u_1=(x_1,z_1)$
and
$u_2=(x_2,z_2)$
in
$ X_1\times X_2$
with
$(u_1,u_2)\in R_{\pi _1\times \pi _2}\setminus \Delta (X_1\times X_2)$
, we shall prove
$(u_1,u_2)\in E(X_1\times X_2,G\vert \pi _1\times \pi _2)$
. Without loss of generality, we assume
$x_1\neq x_2$
.
Let
$\widetilde {U}_1=U_1\times V_1,\widetilde {U}_2=U_2\times V_2$
be neighbourhoods of
$u_1$
and
$u_2$
, respectively. Note that
$(x_1,x_2)\in R_{\pi _1}\setminus \Delta (X_1)=E(X_1,G\vert \pi _1)$
since
$\pi _1$
has rel-u.p.e. Then by Corollary 3.3, there exists
$c_1>0$
such that for every
$F\in \mathcal {F}(G)$
, there exists
$E\subseteq F$
with
$\vert E\vert>c_1\vert F\vert $
, which is an independence set of
$\{U_1,U_2\}$
with respect to
$\pi _1$
. For
$z_1$
and
$z_2$
, there are two cases.
Case 1:
$z_1\neq z_2$
. In this case,
$(z_1,z_2)\in R_{\pi _2}\setminus \Delta (X_2)=E(X_2,G\vert \pi _2)$
since
$\pi _2$
has rel-u.p.e. Then there exists
$c_2>0$
such that for every
$F\in \mathcal {F}(G)$
, there exists
$F_0\subseteq F$
with
$\vert F_0\vert>c_1\cdot c_2\vert F\vert $
, which is an independence set of
$\{\widetilde {U}_1,\widetilde {U}_2\}$
with respect to
$\pi _1\times \pi _2$
. This implies
$(u_1,u_2)\in E(X_1\times X_2,G\vert \pi _1\times \pi _2)$
.
Case 2:
$z_1=z_2=z$
for some
$z\in X_2$
. We set
$V=V_1\cap V_2$
. Then V is an open neighbourhood of z. Since
$\mathrm {supp}(Y_2)=Y_2$
and
$\pi _2$
has rel-u.p.e., by Proposition A.4, we have
$\mathrm {supp}(X_2)=X_2$
. Thus, there exists
$\nu \in \mathcal {M}(X_2,G)$
such that
$\nu (V)>0$
. By Lemma A.2,
$\mathcal {P}_V^{\pi _2}$
has positive density. Then by similar analysis in Case 1, we can also obtain that
$(u_1,u_2)\in E(X_1\times X_2,G\vert \pi _1\times \pi _2)$
. This ends our proof.
