On Relative Metric Mean Dimension with Potential and Variational Principles

Abstract

In this article, we introduce a notion of relative mean metric dimension with potential for a factor map $\pi : (X,d, T)\rightarrow (Y, S)$ between two topological dynamical systems. To link it with ergodic theory, we establish four variational principles in terms of metric entropy of partitions, Shapira’s entropy, Katok’s entropy and Brin–Katok local entropy respectively. Some results on local entropy with respect to a fixed open cover are obtained in the relative case. We also answer an open question raised by Shi (On variational principles for metric mean dimension, 2021. arXiv:2101.02610) partially for a very well-partitionable compact metric space, and in general we obtain a variational inequality involving box dimension of the space. Corresponding inner variational principles given an invariant measure of (Y, S) are also investigated.

Appendix: Proof of Proposition 3.5

We prepare two lemmas before going to the proof of Proposition 3.5.

Lemma 6.1

Let $\pi : (X, T)\rightarrow (Y,S)$ be a factor map, $\mu \in {\mathcal {M}}_T(X)$ and $\mu =\int \mu _yd\nu (y)$ the disintegration of $\mu$ over $\nu =\pi \mu$. Then for any ${\mathcal {V}}\in {\mathcal {C}}_X$ and $0<\rho <1$, there exists $\beta \in {\mathcal {P}}_X$ such that $\beta \succeq {\mathcal {V}}$ and $N_{\mu _y}(\beta , \rho )\le N_{\mu _y}({\mathcal {V}}, \rho )$ for $\nu \text {-a.e.}\ y\in Y$.

Proof

Let ${\mathcal {V}}=\{V_1, \cdots , V_m\}$. For $\nu \text {-a.e.}\ y\in Y$, there exists $I_y \subset \{1,\cdots , m\}$ with cardinality $N_{\mu _y}({\mathcal {V}}, \rho )$ such that $\bigcup _{i\in I_y}V_i$ covers a subset of $\pi ^{-1}y$ up to a set of $\mu _y$-measure less than $\rho$. Hence we can find $y_1, \cdots , y_s \in Y$ such that for $\nu \text {-a.e.}\ y\in Y$, $I_y =I_{y_i}$ for some $i\in \{1,\cdots ,s\}$. For $i = 1,\cdots s$, define

$$\begin{aligned} D_i=\{y\in Y: \mu _y(\bigcup _{j\in I_{y_i}}V_j)>1-\rho \}. \end{aligned}$$

Let $C_1=D_1$, $C_i = D_i\setminus \cup _{j=1}^{i-1}D_j, i= 2,\cdots , s$.

Fix $i\in \{1,\cdots , s\}$. Assume $I_{y_i}=\{k_1< \cdots < k_{t_i}\}$ where $t_i=N_{\mu _{y_i}}({\mathcal {V}}, \rho )$. Take $\{W_1(y_i),\cdots ,W_{t_i}(y_i)\}$ where

$$\begin{aligned} W_1(y_i) =V_{k_1}, W_2(y_i) =V_{k_2}\setminus V_{k_1},\cdots , W_{t_i}(y_i)=V_{k_{t_i}}\setminus \cup _{j=1}^{t_i-1}V_{k_j}. \end{aligned}$$

Define $A:=X\setminus \left( \cup _{i=1}^s(\pi ^{-1}C_i\cap \cup _{j=1}^{t_i}W_j(y_j)\right)$ and $A_1=A\cap V_1, A_l:=A\cap (V_l\setminus \cup _{j=1}^{l-1}V_j), l=2, \cdots , m$. Finally, define

$$\begin{aligned} \begin{aligned}\beta =\{&\pi ^{-1}C_1\cap W_1(y_1), \cdots , \pi ^{-1}C_1\cap W_{t_1}(y_1), \cdots , \\&\pi ^{-1}C_s\cap W_1(y_s), \cdots , \pi ^{-1}C_s\cap W_{t_s}(y_s), A_1, \cdots , A_m\}. \end{aligned} \end{aligned}$$

Then $\beta \succeq {\mathcal {V}}$ and $N_{\mu _y}(\beta , \rho )\le N_{\mu _y}({\mathcal {V}}, \rho )$ for $\nu \text {-a.e.}\ y\in Y$. $\square$

Lemma 6.2

(Strong Rohlin Lemma, see Lemma 2.5 in [22]) Let $(X, {\mathcal {B}}, \mu , T)$ be an ergodic, aperiodic invertible system and let $\alpha \in {\mathcal {P}}_X$. Then for any $\delta >0$ and $n\in {\mathbb {N}}$, one can find a set $B\in {{\mathcal {B}}}$ such that $B, TB, \cdots , T^{n-1}B$ are mutually disjoint, $\mu (\cup _{i=0}^{n-1}T^iB)>1-\delta$ and the distribution of $\alpha$ is the same as the distribution of the partition $\alpha |_B$ that $\alpha$ induces on B.

Proof of Proposition 3.5

The proposition follows from the following two lemmas. $\square$

Lemma 6.3

Let $\mu \in {\mathcal {E}}_T(X)$ and ${\mathcal {U}}\in {\mathcal {C}}_X^o$. Then for any $0<\rho <1$,

$$\begin{aligned} {\overline{h}}^{S}_{\mu }(T,{\mathcal {U}},\rho |Y)\le h_\mu (T,{\mathcal {U}}|Y). \end{aligned}$$

Proof

Take any finite Borel partition $\alpha \succeq {\mathcal {U}}$. According to Theorem 3.1, as $\mu$ is ergodic, there exist $Y_1\in {\mathcal {B}}_Y$ with $\nu (Y_1)=1$ such that for each $y\in Y_1$ and $\mu _y\text {-a.e.}\ x,$

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{-\log \mu _y(\alpha _0^{n-1}(x))}{n}= h_\mu (T,\alpha |Y). \end{aligned}$$

Fix $y\in Y_1$. For $n\in {\mathbb {N}}$ and $\delta >0,$ set

$$\begin{aligned} I_n:=\{x\in \pi ^{-1}y:\mu _y(\alpha _0^{n-1}(x))>\exp (-(h_\mu (T,\alpha |Y)+\delta )n)\} =\pi ^{-1}y\cap \bigcup _{V\in {\mathcal {J}}_n}V, \end{aligned}$$

where ${\mathcal {J}}_n=\{V\in \alpha _0^{n-1}:\mu _y(V)>\exp (-(h_\mu (T,\alpha |Y)+\delta )n)$. Then for any $\delta >0$, $\lim _{n\rightarrow \infty }\mu _y(I_n)=1.$ Thus, for sufficiently large $n\in {\mathbb {N}}$, we have $\mu _y(I_n)>1-\rho .$ Since

$$\begin{aligned} \begin{aligned} \#{\mathcal {J}}_n&=\#\{V\in \alpha _0^{n-1}:\mu _y(V)>\exp (-(h_\mu (T,\alpha |Y)+\delta )n)\\&\le \exp ((h_\mu (T,\alpha |Y)+\delta )n), \end{aligned} \end{aligned}$$

the set $I_n$ can be covered by at most $\exp ((h_\mu (T,\alpha |Y)+\delta )n)$ elements of the partition $\alpha _0^{n-1}$. Then

$$\begin{aligned} N_{\mu _y}({\mathcal {U}}_0^{n-1},\rho )\le N_{\mu _y}(\alpha _0^{n-1},\rho )\le \exp ((h_\mu (T,\alpha |Y)+\delta )n). \end{aligned}$$

Thus for any $\delta >0$,

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\int \log N_{\mu _y}({\mathcal {U}}_0^{n-1},\rho )d\mu (y)\le h_\mu (T,\alpha |Y)+\delta . \end{aligned}$$

Letting $\delta \rightarrow 0$, we obtain ${\overline{h}}^{S}_{\mu }(T,{\mathcal {U}},\rho |Y)\le h_\mu (T,\alpha |Y).$ Taking infimum over $\alpha \succeq {\mathcal {U}}$, we have

$$\begin{aligned} {\overline{h}}^{S}_{\mu }(T,{\mathcal {U}},\rho |Y)\le h_\mu (T,{\mathcal {U}}|Y). \end{aligned}$$

$\square$

Lemma 6.4

Let $\mu \in {\mathcal {E}}_T(X)$ and ${\mathcal {U}}\in {\mathcal {C}}_X^o$. Then

$$\begin{aligned} h_\mu (T,{\mathcal {U}}|Y)\le {\underline{h}}^{S}_{\mu }(T,{\mathcal {U}}|Y). \end{aligned}$$

Proof

Let $\mu \in {\mathcal {E}}_T(X)$. If the system (X, T) is periodic, then $\mu$ is supported on a fixed point of T and

$$\begin{aligned} {\underline{h}}^{S}_{\mu }(T,{\mathcal {U}}|Y)=h_\mu (T,{\mathcal {U}}|Y)=0. \end{aligned}$$

Thus let us assume (X, T) is aperiodic.

Fix $n\in {\mathbb {N}}$. Let $\beta$ be constructed as in the proof of Lemma 6.1 for ${\mathcal {V}}= {\mathcal {U}}_0^{n-1}$. We also use the notation from that proof, for example, A is the subset of X such that $\mu (A)<\rho$ and for any $x\notin A$, $N_{\mu _x}(\beta , \rho )\le N_{\mu _x}({\mathcal {U}}_0^{n-1}, \rho )$. Choose $\delta >0$ such that $0<\rho +\delta <1/4$. By Lemma 6.2, we can construct a strong Rohlin tower with respect to $\beta$, with height n and error $<\delta$. Let ${\tilde{B}}$ denote the base of the tower and $B={\tilde{B}} \setminus A$. Clearly, $\mu (B)>(1-\rho )\mu ({\tilde{B}})$ and $\mu (E)\ge 1-(\rho +\delta )$ where $E=\cup _{i=0}^{n-1}T^iB$. Consider $\beta |_{{\tilde{B}}}$ and index its elements by sequences $i_0, \cdots , i_{n-1}$ such that if $B_{i_0, \cdots , i_{n-1}}\in \beta |_{{\tilde{B}}}$, then $T^jB_{i_0, \cdots , i_{n-1}} \subset U_{i_j}$ for every $0\le j\le n-1$. Let ${\hat{\alpha }}:=\{{\hat{A}}_1, \cdots , {\hat{A}}_M\}$ be a partition of E defined by

$$\begin{aligned} {\hat{A}}_m:=\cup \{T^jB_{i_0, \cdots , i_{n-1}}: 0\le j\le n-1, i_j=m\}. \end{aligned}$$

Note that ${\hat{A}}_m \subset U_m$ for every $1\le m\le M$. Extend ${\hat{\alpha }}$ to a partition $\alpha$ of X in some way such that $\alpha \succeq {\mathcal {U}}$ and $\#\alpha =2M$.

Set $\eta ^4=\rho +\delta$ and define for every $k>n$ large enough, $f_k(x)=\frac{1}{k}\sum _{i=0}^{k-1}\chi _E(T^ix)$ and $L_k:=\{x\in X: f_k(x)>1-\eta ^2\}$. Then by Birkhoff ergodic theorem $\int f_k >1-\eta ^4$, and

$$\begin{aligned} \eta ^2\cdot \mu (L_k^c)\le \int _{L_k^c}1-f_k\le \int _{X}1-f_k\le \eta ^4 \end{aligned}$$

which gives $\mu (L_k)\ge 1-\eta ^2$. Put $J_k$ to be the set of $x\in X$ such that for any $j\ge k$,

$$\begin{aligned} \mu _x(\alpha _0^{j-1}(x))<\exp (-(h_\mu (T,\alpha |Y)-\eta )j), \end{aligned}$$

(4)

and

$$\begin{aligned} |\frac{1}{j}\sum _{i=0}^{j-1}\log N_{\mu _{T^{i}x}} ({\mathcal {U}}_0^{n-1},\rho ) \chi _B(T^ix)-\int _B \log N_{\mu _{z}} ({\mathcal {U}}_0^{n-1},\rho ) d\mu (z)|\le \eta . \end{aligned}$$

(5)

By Theorem 3.1 and the Birkhoff ergodic theorem, $\mu (J_k)>1-\eta ^2$ for k large enough. Set $G_k=L_k\cap J_k$ and then $\mu (G_k)>1-2\eta ^2$. Define ${\tilde{G}}_k=\{x\in G_k:\mu _x(G_k)\ge 1-4\eta \},$ then

$$\begin{aligned} {\tilde{G}}_k^c=\{x\in G_k:\mu _x(G_k)<1-4\eta \}\cup G_k^c=\{x\in G_k:\mu _x(G^c_k)>4\eta \}\cup G_k^c. \end{aligned}$$

Therefore,

$$\begin{aligned} \mu ({\tilde{G}}_k^c)\cdot 4\eta \le \int \mu _x(G_k^c)d\mu (x)+\mu (G_k^c)=2\mu (G_k^c)\le 4\eta ^2, \end{aligned}$$

i.e., $\mu ({\tilde{G}}_k^c)\le \eta$.

We fix an element $C_y$ of this partition of $G_k\cap \pi ^{-1}\pi y$ and want to estimate the number of $\alpha _0^{n-1}$-elements needed to cover it. If $0 \le i_1< \cdots < i_m\le k-n$ are the times elements of $C_y$ visit B, then we need at most $N_{\mu _{T^{i_j}y}} ({\mathcal {U}}_0^{n-1},\rho )$ $\alpha _{i_j}^{i_j+n-1}$-elements to cover $C_y$. Because the size of $[0, k-1]\setminus \cup _j[i_j, i_j+n-1]$ is at most $\eta ^2 k+2n$, we need at most $\prod _{j=1}^mN_{\mu _{T^{i_j}y}} ({\mathcal {U}}_0^{n-1},\rho )\cdot (2M)^{\eta ^2 k+2n}$ $\alpha _0^{k-1}$-elements to cover $C_y$. Finally, in view of (5), we know that $G_k\cap \pi ^{-1}\pi y$ can be covered by no more than

$$\begin{aligned} e^{kH(\eta ^2+2n/k)}\cdot (2M)^{\eta ^2 k+2n}\cdot e^{k(\int _B \log N_{\mu _{z}} ({\mathcal {U}}_0^{n-1},\rho )d\mu (z)+\eta )} \end{aligned}$$

(6)

$\alpha _0^{k-1}$-elements, where $H(t):=-t\log t-(1-t)\log (1-t)$. Since $y\in G_k\subset J_k$, any $V\in \alpha _0^{k-1}$ intersecting nontrivially with $G_k\cap \pi ^{-1}\pi y$ has $\mu _y$-measure less than $\exp (-(h_\mu (T,\alpha |Y)-\eta )k)$ by (4). Thus we have

$$\begin{aligned} \begin{aligned}&1-4\eta \le \mu _y(G_k\cap \pi ^{-1}\pi y)\\&\quad \le e^{-(h_\mu (T,\alpha |Y)-\eta )k}e^{kH(\eta ^2+2n/k)}\cdot (2M)^{\eta ^2 k+2n} \cdot e^{k(\int _B \log N_{\mu _{z}} ({\mathcal {U}}_0^{n-1},\rho )d\mu (z)+\eta )}. \end{aligned} \end{aligned}$$

(7)

Recall that the distribution of $\beta$ is the same as the distribution of the partition $\beta |_{{\tilde{B}}}$ and $z\mapsto N_{\mu _z} ({\mathcal {U}}_0^{n-1},\rho )$ is constant on each atom of $\beta |_{X\setminus A}$ by Lemma 6.1. Then by (7) and setting $k\rightarrow \infty$, we get

$$\begin{aligned} \begin{aligned}&h_\mu (T,\alpha |Y)\\&\quad \le \eta +H(\eta ^2)+\eta ^2 \log (2M)+\int _B \log N_{\mu _{z}} ({\mathcal {U}}_0^{n-1},\rho )d\mu (z)+\eta \\&\quad \le 2\eta +H(\eta ^2)+\eta ^2 \log (2M)+\frac{1}{n}\int \log N_{\mu _{z}} ({\mathcal {U}}_0^{n-1},\rho ) d\mu (z). \end{aligned} \end{aligned}$$

By letting $\delta \rightarrow 0$, we obtain

$$\begin{aligned} \begin{aligned} h_\mu (T,\alpha |Y) \le 2\rho ^{\frac{1}{4}}+H(\rho ^{\frac{1}{2}})+\rho ^{\frac{1}{2}} \log (2M)+\frac{1}{n}\int \log N_{\mu _{z}} ({\mathcal {U}}_0^{n-1},\rho ) d\mu (z). \end{aligned} \end{aligned}$$

Taking $\liminf _{n\rightarrow \infty }$ and then $\lim _{\rho \rightarrow 0}$, we have

$$\begin{aligned} \begin{aligned} h_\mu (T,{\mathcal {U}}|Y)\le h_\mu (T,\alpha |Y)\le {\underline{h}}^{S}_{\mu }(T,{\mathcal {U}}|Y). \end{aligned} \end{aligned}$$

$\square$