2.1 Hyperspaces

Let $(X,d)$ be a compact metric space. We write $\mathbf {B}_\epsilon ^d(x):=\, \{x'\in X\mid d(x,x')\leq \epsilon \}$ and $B_\epsilon ^d(x):=\{x'\in X\mid d(x,x')< \epsilon \}$ for $\epsilon>0$ and $x\in X$ . For $A\in \mathcal {H}(X)$ , we denote $\mathrm {diam}_d(A)=\max _{x,x'\in A}d(x,x')$ . If the metric is understood from the context, we simply write $\mathbf {B}_\epsilon (x)$ , $B_\epsilon (x)$ and $\mathrm {diam}(A)$ . For $A\in \mathcal {H}(X)$ and $\epsilon>0$ , we denote $\mathbf {B}_\epsilon (A):=\bigcup _{x\in A}\mathbf {B}_\epsilon (x)$ . The Hausdorff metric is given by

$$ \begin{align*} d_{\mathcal{H}}(A,A'):=\min\{\epsilon>0\mid A\subseteq \mathbf{B}_\epsilon(A') \,\text{ and }\,A'\subseteq \mathbf{B}_\epsilon(A)\} \end{align*} $$

for $A,A'\in \mathcal {H}(X)$ . Note that, as a straightforward consequence of the compactness of $X$ , the infimum in the above formula is attained. It is well known that $d_{\mathcal {H}}$ is a metric on $\mathcal {H}(X)$ and that $\mathcal {H}(X)$ equipped with the Hausdorff metric $d_{\mathcal {H}}$ is a compact metric space. We recommend [Reference NadlerNad92] for more details on hyperspaces and note that $\mathrm {diam}\colon \mathcal {H}(X)\to \mathbb {R}^+$ is continuous.

2.2 Amenable groups and Følner sequences

Let $G$ be a locally compact and $\sigma $ -compact group ( $\mathrm {lc}_\sigma $ group) and choose a (left) Haar measure $|\cdot |$ on $G$ . A sequence $(F_n)$ in $\mathcal {H}'(G)$ is called left Følner if $|F_n\Delta KF_n|/|F_n|\to 0$ holds for all $K\in \mathcal {H}(G)$ , where $\Delta $ denotes the symmetric difference. It is called right Følner whenever $|F_n\Delta F_nK|/|F_n|\to 0$ holds for all $K\in \mathcal {H}(G)$ . An $\mathrm {lc}_\sigma $ group that allows for a left Følner sequence is called amenable.

2.3 Actions

Let $G$ be a topological group and $X$ be a compact metric space. A (continuous) action of $G$ on $X$ is a group homomorphism $\alpha $ from $G$ into the group of homeomorphisms $X\to X$ such that $G\times X\to X\colon (g,x)\mapsto \alpha (g)(x)$ is continuous. For $g\in G$ and $x\in X$ , we suppress the symbol for the action by simply writing $g.x:=\alpha (g)(x)$ . This allows us to simply speak of an action X of G. Whenever the group $G$ is given by the context, we simply speak of an action $X$ . See [Reference AuslanderAus88] for more details on actions.

For $x\in X$ , we denote $O(x):=\{g.x\mid g\in G\}$ for the orbit of $x$ . A point $x\in X$ is called transitive if it has a dense orbit. An action is called transitive if it allows for a transitive point. It is called minimal if every point is transitive.

A subset $M\subseteq X$ is called invariant if $M=g.M:=\{g.x\mid x\in M\}$ holds for all $g\in G$ . For an invariant $A\in \mathcal {H}(X)$ , the action of $G$ on $X$ naturally induces an action of $G$ on $A$ . We speak of a subaction (of X) in this case. Whenever $(X_n)_{n\in \mathbb {N}}$ is a countable family of actions, then we consider the action of $G$ on $\prod _n X_n$ given by $g.(x_n)_n:=(g.x_n)_n$ for $g\in G$ and $(x_n)_n\in \prod _n X_n$ . Furthermore, whenever $X$ is an action, then $(g,A)\mapsto g.A$ induces an action of $G$ on the hyperspace $\mathcal {H}(X)$ . We simply write $\prod _n X_n$ and $\mathcal {H}(X)$ for the respective actions.

2.4 Factor maps

A continuous surjection $\pi \colon X\to Y$ between actions is called a factor map (also factor of X) if $\pi (g.x)=g.\pi (x)$ holds for all $g\in G$ and $x\in X$ . A factor map is called a conjugacy if it is a homeomorphism. Note that any factor map is closed, that is, that $\pi (A)$ is closed for all closed $A\subseteq X$ .

An action $X$ is called equicontinuous if there exists a continuous metric $d$ on $X$ such that $d(g.x,g.x')=d(x,x')$ for all $x,x'\in X$ . For any action $X$ , there exists a maximal equicontinuous factor of $X$ , that is, a factor map $\pi _{\mathrm {eq}}\colon X\to X_{\mathrm {eq}}$ onto an equicontinuous action $X_{\mathrm {eq}}$ such that for any factor map $\pi \colon X\to Y$ onto an equicontinuous action, there exists a factor map $\psi \colon X_{\mathrm {eq}}\to Y$ with $\pi =\psi \circ \pi _{\mathrm {eq}}$ . For details, see [Reference AuslanderAus88].

2.5 Invariant measures

We denote by $\mathcal {M}(X)$ the set of all Borel probability measures on $X$ . Equipped with the weak*-topology, $\mathcal {M}(X)$ becomes a compact metric space [Reference Fuhrmann, Gröger and HauserFGH25, Reference GlasnerGla03, Reference VillaniVil03]. For $\mu \in \mathcal {M}(X)$ and a continuous surjection $\pi \colon X\to Y$ , we denote $\pi _*\mu $ for the pushforward measure, given by $\pi _*\mu (M):=\mu (\pi ^{-1}(M))$ for all measurable $M\subseteq Y$ . Clearly, we have $\pi _*\mu \in \mathcal {M}(Y)$ .

For $g\in G$ and $\mu \in \mathcal {M}(X)$ , we denote $g_*\mu $ for the pushforward under the homeomorphism induced by $g$ . A measure $\mu \in \mathcal {M}(X)$ is called invariant if $g_*\mu =\mu $ holds for all $g\in G$ . We denote by $\mathcal {M}_G(X)$ the set of all invariant $\mu \in \mathcal {M}(X)$ . An action is called uniquely ergodic if it allows for a unique invariant probability measure. An invariant measure $\mu \in \mathcal {M}(X)$ is called ergodic if for each invariant measurable $M\subseteq X$ , we have that $\mu (M)\in \{0,1\}$ . Let $\pi \colon X\to Y$ be a factor map. Note that for $\mu \in \mathcal {M}_G(X)$ , we have $\pi _*\mu \in \mathcal {M}_G(Y)$ and that ergodicity is preserved under pushing forward.

2.6 Actions of amenable $\mathrm {lc}_\sigma $ groups

Let $X$ be an action of an amenable $\mathrm {lc}_\sigma $ group $G$ . For $x\in X$ , we denote by $\delta _x$ the Dirac measure at $x$ . For $K\in \mathcal {H}'(G)$ and $\mu \in \mathcal {M}(X)$ , we denote $K_*\mu := ({1}/{|K|})\int _K (g^*\mu ) \, {d}g$ , that is,

$$ \begin{align*} (K_*\mu)(M) :=\frac{1}{|K|}\int_K \mu(g^{-1}.M) \, {d}g \end{align*} $$

for all $M\subseteq X$ measurable. Note that $K_*\mu \in \mathcal {M}(X)$ . For a left (or right) Følner sequence $\mathcal {F}=(F_n)$ in $G$ and $\mu \in \mathcal {M}(X)$ , we say that $x$ is $\mathcal {F}$ -generic for $\mu $ if $((F_n)_*\delta _x)_{n\in \mathbb {N}}$ converges to $\mu $ in $\mathcal {M}(X)$ . We say that $x$ is $\mathcal {F}$ -generic if it is $\mathcal {F}$ -generic for some $\mu \in \mathcal {M}(X)$ . Note that for $x\in X$ , any left Følner sequence $\mathcal {F}$ allows for a subsequence ${\mathcal {F}}'$ such that $x$ is ${\mathcal {F}}'$ -generic. Furthermore, from [Reference Fuhrmann, Gröger and LenzFGL22, Theorem 2.4], we know that for an ergodic measure $\mu \in \mathcal {M}_G(X)$ and any left Følner sequence $\mathcal {F}$ , there exists a subsequence $\mathcal {F}'$ such that $\mu $ -almost every point is $\mathcal {F}$ -generic for $\mu $ .

It follows from a standard Krylov–Bogolyubov argument that for a left Følner sequence $\mathcal {F}=(F_n)$ and $x\in X$ , any cluster point of $((F_n)_*\delta _x)$ is invariant. Thus, any point in a uniquely ergodic action is $\mathcal {F}$ -generic with respect to all left Følner sequences $\mathcal {F}$ . Furthermore, from the sequential compactness of $\mathcal {M}(X)$ , it is straightforward to observe that $\mathcal {M}_G(X)\neq \emptyset $ . Another straightforward argument yields that for a factor map $\pi \colon X\to Y$ and an ergodic $\nu \in \mathcal {M}_G(Y)$ , there exists $\mu \in \mathcal {M}_G(X)$ with $\pi _*\mu =\nu $ . These observations rely on the amenability of $G$ .

2.7 Subadditivity techniques

A function $H\colon \mathcal {H}'(G)\to \mathbb {R}^+$ is called right dominated if for all $K\in \mathcal {H}'(G)$ , we have $\sup _{g\in G}H(Kg)/|Kg|<\infty $ . It is called right measurable if the function $G\to \mathbb {R}^+$ defined by $g\mapsto H(Kg)$ is measurable for all $K\in \mathcal {H}'(G)$ . A right measurable function $H\colon \mathcal {H}'(G)\to \mathbb {R}^+$ is said to satisfy (𝔖), if for all $K,F\in \mathcal {H}'(G)$ , we have

(𝔖) $$ \begin{align} H(F)\leq \frac{1}{|K|}\int_{K^{-1}F} H(Kg) \, {d}g. \end{align} $$

Note that the integral in the above formula is well defined, since $g\mapsto H(Kg)\in \mathbb {R}^+$ is measurable.

Example 2.2. For bounded and measurable $f\colon G\to \mathbb {R}^+$ , the function

$$ \begin{align*}H\colon \mathcal{H}'(G)\to \mathbb{R}^+\colon K\mapsto\int_K f(g) \, {dg} \end{align*} $$

is right dominated, right measurable and satisfies (𝔖).

Proof. Since compact subsets of $G$ have finite Haar measure, and $f$ is bounded and measurable, the integral defining $H$ is well defined and finite. Furthermore, the boundedness of $f$ yields

$$ \begin{align*}\sup_{g\in G}\frac{H(Kg)}{|Kg|}=\sup_{g\in G} \frac{1}{|Kg|}\int_{Kg} f(h) \, {dh} \leq \|f\|_\infty\end{align*} $$

for all $K\in \mathcal {H}'(G)$ , that is, that $H$ is right dominated. From the continuity of the multiplication $G\times G\to G$ , we know that $(h,g)\mapsto f(hg)$ is measurable. Using the Fubini–Tonelli theorem, we observe that $H$ is right measurable and compute the following for $K,F\in \mathcal {H}'(G)$ :

$$ \begin{align*} \int_{K^{-1}F} H(Kg) \, {dg} &=\int_{K^{-1}F} \int_{K} f(hg) \, {dh} \, {dg} =\int_{K}\int_{K^{-1}F} f(hg) \, {dg} \, {dh}\\ &\geq \int_{K}\int_{h^{-1}F} f(hg) \, {dg} \, {dh} = |K|\int_{F} f(g) \, {dg} = |K|H(F). \end{align*} $$

This verifies that $H$ satisfies (𝔖).

Proposition 2.3. Let $H\colon \mathcal {H}'(G)\to \mathbb {R}^+$ be a right dominated and right measurable function that satisfies (𝔖), and $(F_n')$ a Følner sequence in $G$ . We have

$$ \begin{align*} \lim_{n\to \infty} \sup_{g\in G} \frac{H(F_n'g)}{|F_n'g|} = \inf_{K\in \mathcal{H}'(G)}\sup_{g\in G} \frac{H(Kg)}{|Kg|} &= \sup_{(F_n)} \limsup_{n\to \infty} \frac{H(F_n)}{|F_n|} <\infty, \end{align*} $$

where the supremum, taken over all Følner sequences $(F_n)$ , is attained. Moreover, the same statement holds with $\limsup $ replaced by $\liminf $ .

Remark 2.5. A function $H\colon \mathcal {H}'(G)\to \mathbb {R}^+$ is called right invariant if $H(Kg)=H(K)$ holds for all $g\in G$ . If $G$ is unimodular, and $H$ is right invariant and satisfies (𝔖), then $H$ is right dominated and right measurable, and Proposition 2.3 yields that for any Følner sequence $(F_n)$ in $G$ , we have

$$ \begin{align*}\frac{H(F_n)}{|F_n|} \to \inf_{K\in \mathcal{H}'(G)}\frac{H(K)}{|K|}. \end{align*} $$

Recall from Remark 2.1 that, for countable discrete groups, Shearer’s inequality implies (𝔖). Thus, Proposition 2.3 extends [Reference Downarowicz, Frej and RomagnoliDFR16, Proposition 3.3], from which the present result was inspired.

Proof of Proposition 2.3

Since $H$ is right dominated, for each $n\in \mathbb {N}$ , we have $ \sup _{g\in G} ({H(F_n'g)}/{|F_n'g|}) < \infty $ and find $g_n\in G$ with

$$ \begin{align*} \sup_{g\in G} \frac{H(F_n'g)}{|F_n'g|}\leq \frac{H(F_n'g_n)}{|F_n'g_n|}+\frac{1}{n}. \end{align*} $$

Since $(F_n'g_n)$ is a left Følner sequence and $F_n'\in \mathcal {H}'(G)$ for every $n$ , we have

$$ \begin{align*} \inf_{K\in \mathcal{H}'(G)}\sup_{g\in G} \frac{H(Kg)}{|Kg|} &\leq \liminf_{n\to \infty} \sup_{g\in G} \frac{H(F_n'g)}{|F_n'g|} \leq \limsup_{n\to \infty} \sup_{g\in G} \frac{H(F_n'g)}{|F_n'g|}\\ &\leq \limsup_{n\to \infty} \frac{H(F_n'g_n)}{|F_n'g_n|} \leq \sup_{(F_n)} \limsup_{n\to \infty} \frac{H(F_n)}{|F_n|}, \end{align*} $$

where the supremum is taken over all Følner sequences $(F_n)$ . Similarly, we observe

$$ \begin{align*} \inf_{K\in \mathcal{H}'(G)}\sup_{g\in G} \frac{H(Kg)}{|Kg|} \leq \sup_{(F_n)} \liminf_{n\to \infty} \frac{H(F_n)}{|F_n|} \leq \sup_{(F_n)} \limsup_{n\to \infty} \frac{H(F_n)}{|F_n|}. \end{align*} $$

Since $H$ is right dominated, we have $ \inf _{K\in \mathcal {H}'(G)}\sup _{g\in G} ({H(Kg)}/{|Kg|})<\infty. $ Thus, it remains to show that for every Følner sequence $(F_n)$ , we have

(♠) $$ \begin{align} \limsup_{n\to \infty} \frac{H(F_n)}{|F_n|} &\leq \inf_{K\in \mathcal{H}'(G)}\sup_{g\in G} \frac{H(Kg)}{|Kg|}. \end{align} $$

For this, consider $K\in \mathcal {H}'(G)$ and $\epsilon>0$ . Recall from Remark 2.4 that $\chi _G$ denotes the modular function. Since $K^{-1}F_n$ is compact and $\chi _G$ is continuous, we find $h_n\in K^{-1}F_n$ with $\chi _G(h_n)=\max _{g\in K^{-1}F_n}\chi _G(g)$ . For $g\in (Kh_n)^{-1}F_n=h_n^{-1}K^{-1}F_n$ , we have $h_ng\in K^{-1}F_n$ and, hence,

$$ \begin{align*} \chi_G(g)=\chi_G(h_n^{-1})\chi_G(h_n g)\leq \chi_G(h_n^{-1})\chi_G(h_n)= 1. \end{align*} $$

Thus, from (𝔖), we observe

$$ \begin{align*} H(F_n) &\leq \frac{1}{|Kh_n|}\int_{(Kh_n)^{-1}F_n} H(Kh_ng) \, {dg} = \int_{(Kh_n)^{-1}F_n} \frac{|Kh_ng|}{|Kh_n|} \frac{H(Kh_ng)}{|Kh_ng|} \, {dg}\\ &\leq \bigg(\int_{(Kh_n)^{-1}F_n}\chi_G(g) \, {dg}\bigg) \bigg(\sup_{g\in G} \frac{H(Kg)}{|Kg|}\bigg)\\ &\leq |(Kh_n)^{-1}F_n| \bigg(\sup_{g\in G} \frac{H(Kg)}{|Kg|}\bigg). \end{align*} $$

For large $n$ , we have $|K^{-1}F_n\Delta F_n|\leq \epsilon |F_n|$ and, hence,

$$ \begin{align*} |(Kh_n)^{-1}F_n| = |h_n^{-1}K^{-1}F_n| = |K^{-1}F_n|\leq |K^{-1}F_n\Delta F_n|+|F_n|\leq (1+\epsilon)|F_n|. \end{align*} $$

This shows

$$ \begin{align*} H(F_n) \leq |(Kh_n)^{-1}F_n| \bigg(\sup_{g\in G} \frac{H(Kg)}{|Kg|}\bigg) \leq (1+\epsilon)|F_n| \bigg(\sup_{g\in G} \frac{H(Kg)}{|Kg|}\bigg) \end{align*} $$

for large $n$ and, hence,

$$ \begin{align*} \limsup_{n\to \infty} \frac{H(F_n)}{|F_n|} \leq (1+\epsilon) \sup_{g\in G} \frac{H(Kg)}{|Kg|}. \end{align*} $$

The claim (♠) follows from the arbitrariness of $\epsilon>0$ .

2.8 Upper densities

Let $G$ be an amenable $\mathrm {lc}_\sigma $ group. Consider a measurable subset $G'\subseteq G$ . For a left (or right) Følner sequence $\mathcal {F}=(F_n)$ , we define the upper asymptotic density as $\mathrm {ua}{-}\mathrm{dens}_{\mathcal {F}}(G'):=\limsup _{n\to \infty }|G'\cap F_n|/|F_n|$ and the lower asymptotic density as $\mathrm {la}\text{-}\mathrm{dens}_{\mathcal {F}}(G'):=\liminf _{n\to \infty }|G'\cap F_n|/|F_n|$ . Furthermore, we define the upper Banach density as $\mathrm {ub}\text{-}\mathrm{dens}(G'):=\sup _{\mathcal {F}}\mathrm {ua}\text{-}\mathrm{dens}_{\mathcal {F}}(G')$ , where the supremum is taken over all left Følner sequences $\mathcal {F}$ in $G$ . The following lemma shows that our definition agrees with the definition in the literature. See [Reference Downarowicz, Huczek and ZhangDHZ19, §2.2] and the references therein for more details in the context of actions of countable amenable groups.

Lemma 2.6. For any measurable $G'\subseteq G$ , we have

$$ \begin{align*} \mathrm{ub}\text{-}\mathrm{dens}(G') =\sup_{\mathcal{F}}\mathrm{ua}\text{-}\mathrm{dens}_{\mathcal{F}}(G') =\sup_{\mathcal{F}}\mathrm{la}\text{-}\mathrm{dens}_{\mathcal{F}}(G') =\inf_{K\in \mathcal{H}'(G)}\sup_{g\in G}\frac{|G'\cap Kg|}{|Kg|}, \end{align*} $$

where the suprema, taken over all left Følner sequences $\mathcal {F}$ in $G$ , are attained. Furthermore, for every left Følner sequence $(F_n)$ , we have

$$ \begin{align*} \sup_{g\in G}\frac{|G'\cap F_n g|}{|F_ng|} \to \mathrm{ub}\text{-}\mathrm{dens}(G'). \end{align*} $$

7.1 Mean equicontinuity

Let $X$ be an action and let $d$ be a continuous metric on $X$ . We consider the induced action of $G$ on $X^2$ . The Weyl pseudo-metric $D$ is defined by $D:=\mathbf {A}(d)$ . An action is called mean equicontinuous whenever for all $x\in X$ and $\epsilon>0$ , there exists $\delta>0$ such that for all $x'\in \mathbf {B}_\delta (x)$ , we have $D(x,x')\leq \epsilon $ . Similarly, we define the $\mathcal {F}$ -Besicovitch pseudo-metric $D_{\mathcal {F}}:=\mathbf {A}_{\mathcal {F}}(d)$ and the concept of $\mathcal {F}$ -mean equicontinuity for any left Følner sequence. Clearly, any diam-mean equicontinuous action is mean equicontinuous. Furthermore, any $\mathcal {F}$ -diam-mean equicontinuous action is $\mathcal {F}$ -mean equicontinuous. With an argument similar to that in the proof of Proposition 4.2, we observe the following. See [Reference Łacka and StraszakŁS18, §3.3] for the statement in the context of countable amenable groups.

Remark 7.1. For $(x,x')\in X^2$ , we have

$$ \begin{align*} D(x,x') &= \sup_{\mathcal{F}}D_{\mathcal{F}}(x,x') = \sup_{\mathcal{F}}\liminf_{n\to \infty}\frac{1}{|F_n|}\int_{F_n}d(g.x,g.x') \, {dg} \\ &= \inf_{K\in \mathcal{H}'(G)}\sup_{h\in G}\frac{1}{|Kh|}\int_{Kh}d(g.x,g.x') \, {dg}, \end{align*} $$

where the suprema, taken over all left Følner sequences $\mathcal {F}=(F_n)$ , are attained. Furthermore, for every left Følner sequence $(F_n)$ , we have

$$ \begin{align*} \sup_{h\in G}\frac{1}{|F_nh|}\int_{F_nh}d(g.x,g.x') \, {dg} \to D(x,x'). \end{align*} $$

A factor map $\pi \colon X\to Y$ is called Banach proximal if for all $y\in Y$ and $x,x'\in \pi ^{-1}(y)$ , we have $D(x,x')=0$ [Reference HauserHau26, Reference Li, Siming and YeLTY15]. It follows from Theorem 5.2 that any regular factor map is Banach proximal.

We will need the following results of [Reference Fuhrmann, Gröger and LenzFGL22].

If $X$ is an action, then there exists a smallest closed subset $A\subseteq X$ such that $\mu (A)=1$ for all $\mu \in \mathcal {M}_G(X)$ . We call $A$ the maximal support of X. For reference regarding the following, see [Reference Fuhrmann, Gröger and LenzFGL22, Proposition 5.7] and [Reference Fuhrmann, Gröger and LenzFGL22, Theorem 5.8].

7.2 The maximal equicontinuous factor of a diam-mean equicontinuous action

To relate diam-mean equicontinuity to regularity, we need to relate the mean diameter of fibres to that of balls. For Banach proximal factor maps, this will be achieved in the following lemmas.

Lemma 7.5. Let $X$ be an action. For $A\in \mathcal {H}(X)$ , $K\in \mathcal {H}'(G)$ and $\epsilon>0$ , there exists a finite non-empty subset $E\subseteq A$ such that

$$ \begin{align*}K^*\mathrm{diam}(A)\leq K^*\mathrm{diam}(E) +\epsilon.\end{align*} $$

Proof. For any $g\in G$ , we choose $x_g,x_g'\in A$ with $d(g.x_g,g.x_g')=\mathrm {diam}(g.A)$ . Recall that $\mathrm {diam}\colon \mathcal {H}(X)\to \mathbf {R}^+$ is continuous. Furthermore, note that $K\times X\to X$ and $K\times \mathcal {H}(X)\to ~\mathcal {H}(X)$ are uniformly continuous. Thus, there exists a neighbourhood $V$ of the identity element of $G$ such that for all $x\in X$ , $A'\in \mathcal {H}(X)$ and $g,g'\in K$ with $g'g^{-1}\in V$ , we have $d(g.x,g'.x)\leq \epsilon /3$ and $|\mathrm {diam}(g.A')-\mathrm {diam}(g'.A')|\leq \epsilon /3$ . Choose a finite subset $F$ of $K$ with $K\subseteq VF$ and denote $E:=\{x_h,x_h'\mid h\in F\}$ . For $g\in K$ , there exists $h\in F$ with $gh^{-1}\in V$ . Such $h$ satisfies

$$ \begin{align*}\mathrm{diam}(h.A)=d(h.x_h,h.x_h')=\mathrm{diam}(h.E)\leq \mathrm{diam}(g.E)+\tfrac{1}{3}\epsilon\end{align*} $$

and allows to compute

$$ \begin{align*} \mathrm{diam}(g.A) &=d(g.x_g,g.x_g')\\ &\leq d(g.x_g,h.x_g) + d(h.x_g,h.x_g') + d(h.x_g',g.x_g')\\ &\leq \mathrm{diam}(h.A) + \tfrac{2}{3}\epsilon \leq \mathrm{diam}(g.E) + \epsilon. \end{align*} $$

This shows that for all $g\in K$ , we have $\mathrm {diam}(g.A)\leq \mathrm {diam}(g.E)+\epsilon $ . We thus conclude that $K^*\mathrm {diam}(A)\leq K^*\mathrm {diam}(E)+\epsilon $ .

Proof. For $g\in G$ , we have

$$ \begin{align*}\mathrm{diam}(g.E)=\max_{x,x'\in E}d(g.x,g.x')\leq \sum_{x,x'\in E}d(g.x,g.x').\end{align*} $$

We thus observe $\mathrm {Diam}(E)\leq \sum _{x,x'\in E}D(x,x')=0$ from a straightforward subadditivity argument.

Lemma 7.7. Let $\pi \colon X\to Y$ be a Banach proximal factor map, $y\in Y$ and $\delta>0$ . For $K\in ~\mathcal {H}'(G)$ and $\epsilon>0$ , there exist $g\in G$ and $x\in X$ with

$$ \begin{align*}K^*\mathrm{diam}(\pi^{-1}(y))\leq (Kg)^*\mathrm{diam}(\mathbf{B}_\delta(x))+\epsilon.\end{align*} $$

Proof. Let $\hat {\mathcal {F}}=(\hat {F}_n)$ be a left Følner sequence. Consider a subsequence $\mathcal {F}=(F_n)$ such that $\mathrm {Diam}_{\hat {\mathcal {F}}}(\pi ^{-1}(y))=\lim _n F_n^*\mathrm { diam}(\pi ^{-1}(y))$ and a sequence $(\epsilon _n)$ in $(0,1)$ satisfying $\epsilon _n\to 0$ . From Lemma 7.7, we observe the existence of sequences $(g_n)$ in $G$ and $(x_n)$ in $X$ such that

$$ \begin{align*}F_n^*\mathrm{diam}(\pi^{-1}(y))\leq (F_ng_n)^*\mathrm{diam}(\mathbf{B}_{\delta/2}(x_n))+\epsilon_n.\end{align*} $$

Since $X$ is compact (and by possibly considering a further subsequence of $\mathcal {F}$ ), we assume without loss of generality that $(x_n)$ converges to some $x\in X$ . For large $n$ , we have $\mathbf {B}_{\delta /2}(x_n)\subseteq \mathbf {B}_\delta (x)$ and, hence,

$$ \begin{align*} F_n^*\mathrm{diam}(\pi^{-1}(y))-\epsilon_n \leq (F_ng_n)^*\mathrm{diam}(\mathbf{B}_{\delta/2}(x_n)) \leq (F_ng_n)^*\mathrm{diam}(\mathbf{B}_{\delta}(x)). \end{align*} $$

Since $\mathcal {F}':=(F_n g_n)$ is a left Følner sequence, it follows that

$$ \begin{align*} \mathrm{Diam}_{\hat{\mathcal{F}}}(\pi^{-1}(y)) =\mathrm{Diam}_{\mathcal{F}}(\pi^{-1}(y)) \leq \mathrm{Diam}_{\mathcal{F}'}(\mathbf{B}_\delta(x)) \leq \sup_{x\in X}\mathrm{Diam}(\mathbf{B}_\delta(x)). \end{align*} $$

Taking the supremum over all left Følner sequences $\hat {\mathcal {F}}$ yields the statement.

7.3 Lifting of diam-mean equicontinuity along regular factor maps

Proof. Consider $x\in \pi ^{-1}(y)$ . Furthermore, denote by $\nu $ the measure for which $y$ is $\mathcal {F}$ -generic. Recall that the notion of a diam-mean equicontinuous point does not depend on the choice of a continuous metric of the respective action. We thus assume without loss of generality that the metrics on $X$ and $Y$ satisfy $d(x_1,x_2)\geq d(\pi (x_1),\pi (x_2))$ for all $x_1,x_2\in X$ . Denote $X_0$ for the set of injectivity points of $\pi $ and $Y_0:=\pi (X_0)$ . Since $\pi $ is regular, we observe that $\nu (Y_0)=1$ .

Let $\epsilon>0$ . For $\delta>0$ , we denote

$$ \begin{align*}G_x^\delta:=\{g\in G\mid \mathrm{diam}(g.\mathbf{B}_{\delta}(x))>\epsilon\}.\end{align*} $$

Considering the characterization presented in Proposition 6.8, it suffices to show that there exists $\delta>0$ with $\mathrm {ua}\text{-}\mathrm{dens}_{\mathcal {F}}(G_x^\delta )\leq \epsilon $ .

Since any factor map is closed, we know that for $y'\in Y$ , there exists $\delta _{y'}\in (0,\epsilon /2)$ such that $\pi ^{-1}(\mathbf {B}_{2\delta _{y'}}(y'))\subseteq \mathbf {B}_{\epsilon /2}(\pi ^{-1}(y'))$ . For details, see [Reference Aliprantis and BorderAB06, Theorem 17.7(1)]. Exploring the regularity of the Borel measure $\nu $ , we choose a compact subset $A\subseteq ~Y_0$ with ${\nu (A)\geq 1-\epsilon /2}$ . Furthermore, we choose a finite subset $E\subseteq A$ satisfying ${A\subseteq \bigcup _{y'\in E}B_{\delta _{y'}}(y')=:O}$ and denote $\delta ':=\min _{y'\in E}\delta _{y'}$ . Since $y$ is assumed to be $\mathcal {F}$ -diam-mean equicontinuous, the characterization presented in Proposition 6.8 allows to observe that there exists $\delta>0$ such that

$$ \begin{align*}G_y^\delta:=\{g\in G\mid \mathrm{diam}(g.\mathbf{B}_{\delta}(y))>\delta'\}\end{align*} $$

satisfies $\mathrm {ua}\text{-}\mathrm{dens}_{\mathcal {F}}(G_y^\delta )\leq \delta '$ .

We will next show that, with this choice of $\delta $ , we also have $\mathrm {ua}\text{-}\mathrm{dens}_{\mathcal {F}}(G_x^\delta )\leq \epsilon $ . This will be achieved by considering

$$ \begin{align*}G':=\{g\in G\mid g.y\notin O\}\end{align*} $$

and showing: (i) $\mathrm {ua}\text{-}\mathrm{dens}_{\mathcal {F}}(G'\cup G_y^\delta )\leq \epsilon $ and (ii) $G_x^\delta \subseteq G'\cup G_y^\delta $ .

(i) Clearly, $O=\bigcup _{y'\in E}B_{\delta _{y'}}(y')$ is open and satisfies $\nu (O)\geq \nu (A)\geq 1-\epsilon /2$ . Thus, $A':=Y\setminus O$ is closed and satisfies $\nu (A')\leq \epsilon /2$ . We thus observe from $(F_n)_*\delta _y\to \nu $ that

$$ \begin{align*} \mathrm{ua}\text{-}\mathrm{dens}_{\mathcal{F}}(G')=\limsup_{n\to \infty}\frac{|F_n\cap G'|}{|F_n|} =\limsup_{n\to \infty} (F_n)_*\delta_y(A') \leq \nu(A')\leq \frac{\epsilon}{2}. \end{align*} $$

Since $\delta '=\min _{y'\in E}\delta _{y'}\leq \epsilon /2$ , we conclude

$$ \begin{align*} \mathrm{ua}\text{-}\mathrm{dens}_{\mathcal{F}}(G'\cup G_y^\delta) \leq \mathrm{ua}\text{-}\mathrm{dens}_{\mathcal{F}}(G') + \mathrm{ua}\text{-}\mathrm{dens}_{\mathcal{F}}(G_y^\delta) \leq \frac{\epsilon}{2} + \delta' \leq \epsilon. \end{align*} $$

(ii) Consider $g\in G_x^\delta $ and assume that $g\notin G'\cup G_y^\delta $ . From $g\notin G_y^\delta $ , we observe $\mathrm {diam}(g.\mathbf {B}_\delta (y))\leq \delta '$ and, hence, $g.\mathbf {B}_\delta (\pi (x))=g.\mathbf {B}_\delta (y)\subseteq \mathbf {B}_{\delta '}(g.y)$ . Furthermore, since $g\notin G'$ , we have $g.y\in O=\bigcup _{y'\in E}B_{\delta _{y'}}(y')$ . Thus, there exists $y'\in E$ with $d(g.y,y')<~\delta _{y'}$ . For such $y'$ , we observe $\mathbf {B}_{\delta '}(g.y) \subseteq \mathbf {B}_{\delta _{y'}}(g.y) \subseteq \mathbf {B}_{2\delta _{y'}}(y')$ . Using our particular choice of the metrics on $X$ and $Y$ , we compute

$$ \begin{align*}\pi(g.\mathbf{B}_\delta(x)) = g.\pi(\mathbf{B}_\delta(x)) \subseteq g.\mathbf{B}_\delta(\pi(x)) \subseteq \mathbf{B}_{\delta'}(g.y) \subseteq \mathbf{B}_{2\delta_{y'}}(y') \end{align*} $$

and, hence, the choice of $\delta _{y'}$ yields

$$ \begin{align*} g.\mathbf{B}_\delta(x) \subseteq \pi^{-1}(\pi(g.\mathbf{B}_\delta(x))) \subseteq \pi^{-1}(\mathbf{B}_{2\delta_{y'}}(y')) \subseteq \mathbf{B}_{\epsilon/2}(\pi^{-1}(y')). \end{align*} $$

Since $y'\in E\subseteq A\subseteq Y_0$ , we observe that $\pi ^{-1}(y')$ is a singleton and, hence, that $\mathrm { diam}(g.\mathbf {B}_\delta (x))\leq \epsilon $ . This shows $g\notin G_x^\delta $ , which is a contradiction.

We observe Theorem 1.1 from Proposition 7.9 and the following theorem.

An action $X$ is called fully supported if $X$ equals its maximal support. The maximal support is always a closed invariant and non-empty subset. Thus, any minimal action is fully supported. Whenever $X$ is a diam-mean equicontinuous action that is not equicontinuous, then $\mathfrak {F}_1(X)$ is a proper subset of the diam-mean equicontinuous action $\mathcal {H}_\pi (X)$ , where $\pi \colon X\to X_{\mathrm {eq}}$ denotes the factor map onto the maximal equicontinuous factor of $X$ . With the arguments above, it follows that $\pi _{\mathcal {H}}\colon \mathcal {H}_\pi (X)\to X_{\mathrm {eq}}$ is regular and Theorem 5.2 yields that the maximal support of $\mathcal {H}_\pi (X)$ is contained in $\mathfrak {F}_1(X)$ . We thus observe the following.

Proposition 7.10 also allows to say something about regular extensions of $\mathcal {F}$ -diam-mean equicontinuous actions. An action $X$ is called pointwise uniquely ergodic if $\overline {O(x)}$ is uniquely ergodic for any $x\in X$ . In particular, any uniquely ergodic action is pointwise uniquely ergodic.

8.1 Product stability

For a sequence $(\pi _n\colon X_n\to Y_n)_{n\in \mathbb {N}}$ of factor maps, we consider the factor map $\prod _{n=1}^\infty \pi _n\colon \prod _{n=1}^\infty X_n \to \prod _{n=1}^\infty Y_n$ given by $(x_n)_n\mapsto (\pi _n(x_n))_n$ . See [Reference Cortez and HauserCH26] for further reading.

Proof. We denote $X:= \prod _{n=1}^\infty X_n$ and $Y:= \prod _{n=1}^\infty Y_n$ , and write $\pi ^{(n)}$ for the projection $X\to ~X_n$ .

‘ $\Rightarrow $ ’ Let $m\in \mathbb {N}$ . We assume without loss of generality that the metrics on $X_m$ and $X$ satisfy $d(x_1,x_2)\geq d(\pi ^{(m)}(x_1),\pi ^{(m)}(x_2))$ for all $x_1,x_2\in X$ . This implies $\mathrm {Diam}(\pi ^{(m)}(A))\leq \mathrm {Diam}(A)$ for $A\in \mathcal {H}(X)$ .

For $y\in Y_m$ , we choose a sequence $(y_n)_n\in Y$ with $y_m=y$ . Observe that $\pi ^{-1}((y_n)_n)=\prod _{n=1}^\infty \pi ^{-1}_n(y_n)$ and, hence, $\pi ^{(m)}(\pi ^{-1}((y_n)_n))=\pi _m^{-1}(y)$ . Since $\pi $ is assumed to be regular, we know from Theorem 5.2 that

$$ \begin{align*} \mathrm{Diam}(\pi_m^{-1}(y)) =\mathrm{Diam}(\pi^{(m)}(\pi^{-1}((y_n)_n))) \leq \mathrm{Diam}(\pi^{-1}((y_n)_n)) =0. \end{align*} $$

This shows $\mathrm {Diam}(\pi _m^{-1}(y))=0$ for all $y\in Y_m$ and we observe $\pi _m$ to be regular from another application of Theorem 5.2.

‘ $\Leftarrow $ ’ For $n\in \mathbb {N}$ , we denote $X_{n,0}$ for the set of injectivity points of $\pi _n$ . We observe that $X_0:=\prod _{n=1}^\infty X_{n,0}=\bigcap _{n=1}^\infty (\pi ^{(n)})^{-1}(X_{n,0})$ is the set of injectivity points of $\pi $ .

Let $\mu \in \mathcal {M}_G(X)$ and note that $\pi ^{(n)}_*\mu \in \mathcal {M}_G(X_n)$ for $n\in \mathbb {N}$ . From the regularity of $\pi _n$ , we obtain

$$ \begin{align*} \mu((\pi^{(n)})^{-1}(X_{n,0}))=(\pi^{(n)}_*\mu)(X_{n,0})=1. \end{align*} $$

Thus, $ \mu (X_0) =\mu (\bigcap _{n=1}^\infty (\pi ^{(n)})^{-1}(X_{n,0})) =1.$

8.2 (De-)Composition stability

The following proposition shows that regularity is stable under composition and decomposition.

8.3 $\mathcal {H}$ -(in)stability

Recall that a factor map $\pi \colon X\to Y$ is called almost one-to-one if the set of injectivity points is dense in $X$ . In [Reference Cortez and HauserCH26], it was presented that the properties discussed in the Propositions 8.1 and 8.2 also hold for almost one-to-one factor maps between actions (on compact metric spaces).

For a factor map $\pi \colon X\to Y$ , the image mapping $\mathcal {H}(X)\ni A\mapsto \pi (A)$ establishes a factor map $\mathcal {H}(\pi )\colon \mathcal {H}(X)\to \mathcal {H}(Y)$ [Reference Cortez and HauserCH26]. It was also shown in [Reference Cortez and HauserCH26, Reference Dai and XieDX24] that a factor map $\pi $ between actions (on compact metric spaces) is almost one-to-one if and only if $\mathcal {H}(\pi )$ is almost one-to-one. We next present that a similar statement does not hold in the context of regularity.

Let $\pi \colon X\to Y$ be a factor map and denote $X_0$ and $\mathcal {H}(X)_0$ for the sets of injectivity points of $\pi $ and $\mathcal {H}(\pi )$ , respectively. One readily verifies that $ \mathfrak {F}_1(X_0)=\mathfrak {F}_1(X)\cap \mathcal {H}(X)_0. $ From this, the following statement follows directly.