Zero entropy actions of amenable groups are not dominant

It is immediate from the definition of isomorphism that for $\mu$-a.e. $x,x^{\prime}\in X,$

$$d_{F}\left(Q_{T,F}(x),Q_{T,F}(x^{\prime})\right)\ =\ d_{F}\left((\varphi Q)_{S,F}(\varphi x),(\varphi Q)_{S,F}(\varphi x^{\prime})\right).$$

Therefore it follows that $\varphi\left(B_{\operatorname{Ham}}(Q,T,F,x,\epsilon)\right)=B_{\operatorname{Ham}}(\varphi Q,S,F,\varphi x,\epsilon)$ for $\mu$-a.e. $x$. So any collection of $(Q,T,F)$-Hamming balls in $X$ covering a set of $\mu$-measure $1-\epsilon$ is directly mapped by $\varphi$ to a collection of $(\varphi Q,S,F)$-Hamming balls in $Y$ covering a set of $\nu$-measure $1-\epsilon$. Therefore $\operatorname{cov}(Q,T,F,\mu,\epsilon)\geq\operatorname{cov}(\varphi Q,S,F,\nu,\epsilon)$. The reverse inequality holds by doing the same argument with $\varphi^{-1}$ in place of $\varphi$. ∎

Let $h=h(\mu,T,P)$. Let $\gamma>0$. By Theorem 2.4, for $n$ sufficiently large depending on $\gamma$, we have

$$\mu\left\{x\in X:\mu(P_{T,F_{n}}(x))\geq\exp((-h-\gamma)|F_{n}|)\right\}\ >\ 1-\epsilon.$$

Let $X^{\prime}$ denote the set in the above equation. Let $\mathcal{G}$ be the family of cells of the partition $P_{T,F_{n}}$ that meet $X^{\prime}$. Then clearly $\mu\left(\bigcup\mathcal{G}\right)>1-\epsilon$ and $|\mathcal{G}|<\exp((h+\gamma)|F_{n}|)$. Therefore

$$\limsup_{n\to\infty}\frac{1}{|F_{n}|}\log\ell(n)\ \leq\ h+\gamma,$$

and this holds for arbitrary $\gamma$, so we are done. ∎

First, because $A$ is finite and $(F_{n})$ is Følner we have

$$\lim_{n\to\infty}\frac{|AF_{n}|}{|F_{n}|}\ =\ 1.$$

Now fix any $g\in G$ and observe that

$$\frac{|gAF_{n}\,\triangle\,AF_{n}|}{|AF_{n}|}\ \leq\ \frac{|gAF_{n}\,\triangle\,F_{n}|+|F_{n}\,\triangle\,AF_{n}|}{|F_{n}|}\cdot\frac{|F_{n}|}{|AF_{n}|}\ \to\ 0\qquad\text{as $n\to\infty$},$$

which shows that $(AF_{n})$ is a Følner sequence. ∎

For each $m$, let $N_{m}$ be such that $b(m,n)<1/m$ for all $n>N_{m}$. Without loss of generality, we may assume that $N_{m}<N_{m+1}$. Then we define the sequence $(a_{n})$ by $a_{n}=b(1,n)$ for $n\leq N_{2}$ and $a_{n}=b(m,n)$ for $N_{m}<n\leq N_{m+1}$. We have $a_{n}\to 0$ because $a_{n}<1/m$ for all $n>N_{m}$. Finally, the fact that $b(m+1,n)\geq b(m,n)$ implies that for every fixed $m$, $a_{n}\geq b(m,n)$ as soon as $n>N_{m}$. ∎

Because $T$ has zero entropy, there exists a finite generating partition for $T$ (see for example [Sew19, Corollary 1.2] or [Ros88, Theorem 2^′]). Fix such a partition $P$ and let $Q=\{Q_{1},\dots,Q_{r}\}$ be any given partition. Because $P$ is generating, there is an integer $m$ and another partition $Q^{\prime}=\{Q^{\prime}_{1},\dots,Q^{\prime}_{r}\}$ such that $Q^{\prime}$ is refined by $P_{T,F_{m}}=\bigvee_{g\in F_{m}}(T^{g})^{-1}P$ and

$$\mu\{x:Q(x)\neq Q^{\prime}(x)\}\ <\ \frac{\epsilon}{4}.$$

By the mean ergodic theorem, we can write

$$d_{F_{n}}(Q_{T,F_{n}}(x),Q^{\prime}_{T,F_{n}}(x))\ =\ \frac{1}{|F_{n}|}\sum_{f\in F_{n}}1_{\{y:Q(y)\neq Q^{\prime}(y)\}}(T^{f}x)\ \xrightarrow{L^{1}(\mu)}\ \mu\{y:Q(y)\neq Q^{\prime}(y)\}\ <\ \frac{\epsilon}{4},$$

so in particular, for $n$ sufficiently large, we have

$$\mu\{x:d_{F_{n}}(Q_{T,F_{n}}(x),Q^{\prime}_{T,F_{n}}(x))<\epsilon/2\}\ >\ 1-\frac{\epsilon}{4}.$$

Let $Y$ denote the set $\{x:d_{F_{n}}(Q_{T,F_{n}}(x),Q^{\prime}_{T,F_{n}}(x))<\epsilon/2\}$.

Recall that $Q^{\prime}$ is refined by $P_{T,F_{m}}$, so $Q^{\prime}_{T,F_{n}}$ is refined by $(P_{T,F_{m}})_{T,F_{n}}=P_{T,F_{m}F_{n}}$. Let $\ell=\ell(m,n)$ be the minimum number of $P_{T,F_{m}F_{n}}$ cells required to cover a set of $\mu$-measure at least $1-\epsilon/4$, and let $C_{1},\dots,C_{\ell}$ be such a collection of cells satisfying $\mu\left(\bigcup_{i}C_{i}\right)\geq 1-\epsilon/4$. If any of the $C_{i}$ do not meet the set $Y$, then drop them from the list. Because $\mu(Y)>1-\epsilon/4$ we can still assume after dropping that $\mu\left(\bigcup_{i}C_{i}\right)>1-\epsilon/2$. Choose a set of representatives $y_{1},\dots y_{\ell}$ with each $y_{i}\in C_{i}\cap Y$.

Now we claim that $Y\cap\bigcup_{i}C_{i}\subseteq\bigcup_{i=1}^{\ell}B_{\operatorname{Ham}}(Q,T,F_{n},y_{i},\epsilon)$. To see this, let $x\in Y\cap\bigcup_{i}C_{i}$. Then there is one index $j$ such that $x$ and $y_{j}$ are in the same cell of $P_{T,F_{m}F_{n}}$. We can then estimate

	$\displaystyle d_{F_{n}}\left(Q_{T,F_{n}}(x),Q_{T,F_{n}}(y_{j})\right)\$	$\displaystyle\leq\ d_{F_{n}}\left(Q_{T,F_{n}}(x),Q^{\prime}_{T,F_{n}}(x)\right)+d_{F_{n}}\left(Q^{\prime}_{T,F_{n}}(x),Q^{\prime}_{T,F_{n}}(y_{j})\right)$
		$\displaystyle\quad+d_{F_{n}}\left(Q^{\prime}_{T,F_{n}}(y_{j}),Q_{T,F_{n}}(y_{j})\right)$
		$\displaystyle<\ \frac{\epsilon}{2}+0+\frac{\epsilon}{2}\ =\ \epsilon.$

The bounds for the first and third terms come from the fact that $x,y_{j}\in Y$. The second term is $0$ because $Q^{\prime}_{F_{n}}$ is refined by $P_{T,F_{m}F_{n}}$ and $y_{j}$ was chosen so that $x$ and $y_{j}$ are in the same $P_{T,F_{m}F_{n}}$-cell. Therefore, $\operatorname{cov}(Q,T,F_{n},\mu)\leq\ell(m,n)$. So, the proof is complete once we find a fixed sequence $(a_{n})$ that is subexponential in $|F_{n}|$ and eventually dominates $\ell(m,n)$ for each fixed $m$.

Because $T$ has zero entropy, Lemmas 2.5 and 2.6 imply that

$$\limsup_{n\to\infty}\frac{1}{|F_{n}|}\log\ell(m,n)\ =\ \limsup_{n\to\infty}\frac{|F_{m}F_{n}|}{|F_{n}|}\cdot\frac{1}{|F_{m}F_{n}|}\log\ell(m,n)\ =\ 0\qquad\text{for each fixed $m$}.$$

Note also that because $P_{T,F_{m+1}F_{n}}$ refines $P_{T,F_{m}F_{n}}$, we have $\ell(m+1,n)\geq\ell(m,n)$ for all $m,n$. Therefore, we can apply Lemma 2.7 to the numbers $b(m,n)=|F_{n}|^{-1}\log\ell(m,n)$ to produce a sequence $(a_{n}^{\prime})$ satisfying $a_{n}^{\prime}\to 0$ and $a_{n}^{\prime}\geq b(m,n)$ eventually for each fixed $m$. Then $a_{n}:=\exp(|F_{n}|a_{n}^{\prime})$ is the desired sequence. ∎

Choose a sequence $(a_{n})$ as in Proposition 2.8 such that for any partition $Q$, $\operatorname{cov}(Q,T,F_{n},\mu)\leq a_{n}$ for sufficiently large $n$. Let $\mathcal{U}$ be the dense $G_{\delta}$ set of cocycles associated to $(a_{n})$ as guaranteed by Theorem 3.1, and let $\alpha\in\mathcal{U}$, so we know that $\operatorname{cov}(\overline{Q},T_{\alpha},F_{n},\mu\times m)\geq 2a_{n}$ for infinitely many $n$. Now if $\varphi:(X\times I,T_{\alpha},\mu\times m)\to(X,T,\mu)$ were an isomorphism, then by Lemma 2.3, $\varphi\overline{Q}$ would be a partition of $X$ satisfying $\operatorname{cov}(\varphi\overline{Q},T,F_{n},\mu)=\operatorname{cov}(\overline{Q},T_{\alpha},F_{n},\mu\times m)>2a_{n}$ for infinitely many $n$, contradicting the conclusion of Proposition 2.8. Therefore, we have produced a dense $G_{\delta}$ set of cocycles $\alpha$ such that $T_{\alpha}\not\simeq T$, which implies Theorem 1.1. ∎

For the names $\overline{Q}_{T_{\beta^{(n)}},F}(x,t)$ and $\overline{Q}_{T_{\alpha},F}(x,t)$ to be the same means that for every $g\in F$,

$$\overline{Q}\left(T^{g}x,\beta^{(n)}_{g}(x)t\right)\ =\ \overline{Q}\left(T^{g}x,\alpha_{g}(x)t\right),$$

which is equivalent to

(1)

$$\pi\left(\beta^{(n)}_{g}(x)t\right)\ =\ \pi\left(\alpha_{g}(x)t\right).$$

The idea is the following. For fixed $g$ and $x$, if $\alpha_{g}(x)$ and $\beta^{(n)}_{g}(x)$ are close in $d_{A}$, then (1) fails for only a small measure set of $t$. And if $\beta^{(n)}$ is very close to $\alpha$ in the cocycle topology, then $\beta^{(n)}_{g}(x)$ and $\alpha_{g}(x)$ are close for all $g\in F$ and most $x\in X$. Then, by Fubini’s theorem, we will get that the measure of the set of $(x,t)$ failing (1) is small.

Here are the details. Fix $\rho>0$; we will show that the measure of the desired set is at least $1-\rho$ for $n$ sufficiently large. First, let $\sigma$ be so small that for any $\phi,\psi\in\operatorname{Aut}(I,m)$,

$$d_{A}(\phi,\psi)<\sigma\quad\text{implies}\quad m\left\{t:\pi(\phi t)=\pi(\psi t)\right\}>1-\rho/2.$$

This is possible because

$$\{t:\pi(\phi t)\neq\pi(\psi t)\}\ \subseteq\ (\phi^{-1}[0,1/2)\,\triangle\,\psi^{-1}[0,1/2))\cup(\phi^{-1}[1/2,1]\,\triangle\,\psi^{-1}[1/2,1]).$$

Then, from the definition of the cocycle topology, we have

$$\mu\left\{x\in X:d_{A}\left(\beta^{(n)}_{g}(x),\alpha_{g}(x)\right)<\sigma\quad\text{for all $g\in F$}\right\}\ \to\ 1\qquad\text{as $n\to\infty$}.$$

Let $n$ be large enough so that the above is larger than $1-\rho/2$. Then, by Fubini’s theorem, we have

		$\displaystyle(\mu\times m)\left\{(x,t):\overline{Q}_{T_{\beta^{(n)}},F}(x,t)\ =\ \overline{Q}_{T_{\alpha},F}(x,t)\right\}$
	$\displaystyle=\$	$\displaystyle\int m\left\{t:\pi\left(\beta^{(n)}_{g}(x)t\right)=\pi\left(\alpha_{g}(x)t\right)\quad\text{for all $g\in F$}\right\}\,d\mu(x).$

We have arranged things so that the integrand above is $>1-\rho/2$ on a set of $x$ of $\mu$-measure $>1-\rho/2$, so the integral is at least $(1-\rho/2)(1-\rho/2)>1-\rho$ as desired. ∎

Suppose $\beta^{(n)}$ is a sequence of cocycles converging to $\alpha$ and satisfying $\operatorname{cov}(\overline{Q},T_{\beta^{(n)}},F,\mu\times m)\leq L$ for all $n$. We will show that $\operatorname{cov}(\overline{Q},T_{\alpha},F,\mu\times m)\leq L$ as well. The covering number $\operatorname{cov}(\overline{Q},T_{\beta^{(n)}},F,\mu\times m)$ is a quantity which really depends only on the measure $\left(\overline{Q}_{T_{\beta^{(n)}},F}\right)_{*}(\mu\times m)\in\operatorname{Prob}\left(\{0,1\}^{F}\right)$, which we now call $\nu_{n}$ for short. The assumption that $\operatorname{cov}(\overline{Q},T_{\beta^{(n)}},F,\mu\times m)\leq L$ for all $n$ says that for each $n$, there is a collection of $L$ words $w_{1}^{(n)},\dots,w_{L}^{(n)}\in\{0,1\}^{F}$ such that the Hamming balls of radius $\epsilon$ centered at these words cover a set of $\nu_{n}$-measure at least $1-\epsilon$. Since $\{0,1\}^{F}$ is a finite set, there are only finitely many possibilities for the collection $(w_{1}^{(n)},\dots,w_{L}^{(n)})$. Therefore by passing to a subsequence and relabeling we may assume that there is a fixed collection of words $w_{1},\dots,w_{L}$ with the property that if we let $B_{i}$ be the Hamming ball of radius $\epsilon$ centered at $w_{i}$, then $\nu_{n}\left(\bigcup_{i=1}^{L}B_{i}\right)\geq 1-\epsilon$ for every $n$.

Now, by Lemma 3.2, the map $\overline{Q}_{T_{\beta^{(n)}},F}$ agrees with $\overline{Q}_{T_{\alpha},F}$ on a set of measure converging to $1$ as $n\to\infty$. This implies that the measures $\nu_{n}$ converge in the total variation norm on $\operatorname{Prob}\left(\{0,1\}^{F}\right)$ to $\nu:=\left(\overline{Q}_{T_{\alpha},F}\right)_{*}(\mu\times m)$. Since $\nu_{n}\left(\bigcup_{i=1}^{L}B_{i}\right)\geq 1-\epsilon$ for every $n$, we conclude that $\nu\left(\bigcup_{i=1}^{L}B_{i}\right)\geq 1-\epsilon$ also, which implies that $\operatorname{cov}(\overline{Q},T_{\alpha},F,\mu\times m)\leq L$ as desired. ∎