# Scale pressure for amenable group actions

Dandan Cheng  ,  Qian Hao  and  Zhiming Li  School of Mathematics, Northwest
University, Xi’an, 710127, P.R.China<EMAIL_ADDRESS>
###### Abstract.

In this paper we introduce the notion of scale pressure and measure theoretic
scale pressure for amenable group actions. A variational principle for
amenable group actions is presented. We also describe these quantities by
pseudo-orbits. Moreover, we prove that if $G$ is a finitely generated
countable discrete amenable group, then the scale pressure of $G$ coincides
with the scale pressure of $G$ with respect to pseudo-orbits.

###### Key words and phrases:

Scale pressure;Variational principle;amenable group; Pseudo-orbits.

###### 2010 Mathematics Subject Classification:

37A15,37A35

##  1\. Introduction

The concept of entropy, introduced into the realm of dynamical systems more
than fifty years ago, has become an important ingredient in the
characterization of the complexity of dynamical systems. In the late nineties,
M. Gromov [3] proposed a new dynamical concept of dimension that was meant to
extend the usual topological dimension to broader contexts. This notion,
called mean dimension, is defined for continuous maps on compact metric spaces
in terms of the growth rate of refinements of coverings of the phase space and
was shown to be hardly computable in general. This invariant of topological
dynamical systems was used by E. Lindenstrauss and B. Weiss introduced in [8]
(it is a very interesting work to study which other properties of the entropy
are maintained by the mean dimension and the metric mean dimension) to answer
a long-standing question in topological dynamics: does every minimal
topological dynamical system embed into $([0,1]^{\mathbb{Z}},\sigma)$? The
answer is negative, since any system embeddable in
$([0,1]^{\mathbb{Z}},\sigma)$ has topological mean dimension at most one and
[8] constructed a minimal system with a mean dimension strictly greater than
one. This concept, inspired by the topological entropy, turns out to be a
metric version of Gromov’s notion. Moreover, it is an upper bound for the mean
dimension and, in general, easier to estimate. However, it depends on the
metric used, while the topological entropy is metric independent. The upper
and lower metric mean dimensions, unlike Gromov’s concept, depend on the
metric adopted on the space and are nonzero only if the topological entropy of
the dynamics is infinite. Since continuous systems on manifold with infinite
topological entropy are generic [17], the metric mean dimension exhibits
several intrinsic features which makes it a rather compelling notion to be
studied. There are many elegant approaches to mean dimensions [5, 7, 12] and
mean dimensions with potential [11].

This work was intended as an attempt to give a unified way to analyze systems
with infinite entropy by introducing a notion of pressure-like complexity
functions, which is called _scale pressure_ , with respect to a scale function
$s$ and a potential function $f$. A variational principle of scale pressure is
presented. We also describe these quantities by pseudo-orbits.

##  2\. Scale Pressures

More precisely, suppose that $X$ is a compact metric space with metric $d$,
$\mathcal{B}$ is the Borel $\sigma$-algebra, $G$ is a countable discrete
amenable group and $C(X,\mathbb{R})$ is the Banach algebra of real-valued
continuous functions on $X$ equipped with the supreme norm:
$\|\varphi\|:=\max\limits_{x\in X}|\varphi(x)|$, $\varphi\in C(X,\mathbb{R})$.
Let $(X,G)$ be a $G$-action topological dynamical system, $\mathcal{M}_{G}(X)$
be collection of the $G$-invariant Borel probability measures of $X$ and
$\mathcal{E}_{G}(X)$ be collection of the $G$-ergodic Borel probability
measures of $X$. Recall that a group $G$ is said to be _amenable_ if there
exists a sequence of finite subsets $\\{F_{n}\\}_{n\in\mathbb{N}}$ of $G$
which are asymptotically invariant, i.e.,

$\lim_{n\rightarrow+\infty}\frac{|F_{n}\vartriangle gF_{n}|}{|F_{n}|}=0,\text{
for all }g\in G.$

Such sequences $\\{F_{n}\\}_{n\in\mathbb{N}}$ are called Følner sequences. We
denote the collection of nonempty finite subsets of $G$ by $F(G)$. For each
$F\in F(G)$, set metrics $d_{F}(x,y)=\max\limits_{g\in F}d(gx,gy)$.

Recall that a Følner sequence $\\{F_{n}\\}_{n\in\mathbb{N}}$ in $G$ is
_tempered_ if there exists a constant $C$ which is independent of $n$ such
that

$|\bigcup_{k<n}F_{k}^{-1}F_{n}|\leqslant C|F_{n}|,\text{ for all
}n\in\mathbb{N}.$

######  Definition 2.1.

Let $\varepsilon>0$, $E,J,K\subseteq X$ and $F\in F(G)$. If for any $x\in K$
there exists $y\in J$ such that $d_{F}(x,y)<\varepsilon$, then we call $J$ is
an _ $(F,\varepsilon)$-spanning set of $K$_. If for any $x,y\in E$ we have
$d_{F}(x,y)>\varepsilon$, then we call $E$ is an _ $(F,\varepsilon)$-separated
set_.

Let $sep_{\varepsilon,F}(X,G)$ denote the maximal cardinality of an
$(F,\varepsilon)$-separated subset of $X$ and $spa_{\varepsilon,F}(X,G)$
denote the smallest cardinality of an $(F,\varepsilon)$-spanning subset of
$X$.

######  Definition 2.2.

A function $s:(0,1)\to(0,\infty)$ is called a _scale function_ , if for any
$\lambda\in(0,\infty)$, $\lim\limits_{x\rightarrow 0}\frac{s(\lambda
x)}{s(x)}=1.$ We denote by $\mathcal{S}$ (resp. $\mathcal{S}^{*}$) the set of
all (resp. non-increasing) scale functions.

Let
$\mathcal{S}^{**}=\\{s\in\mathcal{S}^{*}:\limsup\limits_{\varepsilon\rightarrow
0}\frac{\varepsilon\log\varepsilon}{s(\varepsilon)}=0\\}.$

For any Følner sequence $\\{F_{n}\\}_{n\in\mathbb{N}}$, let
$S_{F_{n},\varphi}(x)=\sum\limits_{g\in F_{n}}\varphi(gx).$ For any
$\varepsilon>0$, $s\in\mathcal{S}$, we put

$\displaystyle Q_{\varepsilon,F_{n}}(\varphi,s):=$
$\displaystyle\inf\\{\sum_{x\in J}\exp(s(\varepsilon)S_{F_{n},\varphi}(x)):\
J\ is\ an\ (F_{n},\varepsilon)\ spanning\ set\ of\ X\\};$ $\displaystyle
P_{\varepsilon,F_{n}}(\varphi,s):=$ $\displaystyle\sup\\{\sum_{x\in
E}\exp(s(\varepsilon)S_{F_{n},\varphi}(x)):\ E\ is\ an\ (F_{n},\varepsilon)\
separated\ set\ of\ X\\}.$

Similarly, let

$\displaystyle q_{\varepsilon,F_{n}}(\varphi,s):=$
$\displaystyle\inf\\{\sum_{A\in\alpha}\inf_{x\in
A}\exp(s(\varepsilon)S_{F_{n},\varphi}(x)):\ \alpha\ \mbox{is\ a\ finite\
cover\ with\ mesh}(\alpha,d_{F_{n}})<\varepsilon\\};$ $\displaystyle
p_{\varepsilon,F_{n}}(\varphi,s):=$
$\displaystyle\inf\\{\sum_{A\in\alpha}\sup_{x\in
A}\exp(s(\varepsilon)S_{F_{n},\varphi}(x)):\ \alpha\ \mbox{is\ a\ finite\
cover\ with\ mesh}(\alpha,d_{F_{n}})<\varepsilon\\},$

where
mesh$(\alpha,d_{F_{n}}):=\max\limits_{A\in\alpha}\mbox{diam}_{d_{F_{n}}}(A)$.

######  Remark 2.1.

(1) $0\leqslant
Q_{\varepsilon,F_{n}}(\varphi,s)\leqslant\|\exp(s(\varepsilon)S_{F_{n},\varphi})\|spa_{\varepsilon,F_{n}}(X,G)$,
$0\leqslant
P_{\varepsilon,F_{n}}(\varphi,s)\leqslant\|\exp(s(\varepsilon)S_{F_{n},\varphi})\|sep_{\varepsilon,F_{n}}(X,G)$.

(2) $P_{\varepsilon,F_{n}}(0,s)=sep_{\varepsilon,F_{n}}(X,G)$,
$Q_{\varepsilon,F_{n}}(0,s)=spa_{\varepsilon,F_{n}}(X,G)$.

(3) $Q_{\varepsilon,F_{n}}(\varphi,s)\leqslant
P_{\varepsilon,F_{n}}(\varphi,s)\leqslant p_{\varepsilon,F_{n}}(\varphi,s)$.

(4) If $d(x,y)<\varepsilon$ implies $|\varphi(x)-\varphi(y)|\leqslant\delta$,
then

$p_{\varepsilon,F_{n}}(\varphi,s)\leqslant e^{|F_{n}|\delta
s(\varepsilon)}q_{\varepsilon,F_{n}}(\varphi,s).$

For $\varphi\in C(X,\mathbb{R})$, $s\in\mathcal{S}$ and $\varepsilon>0$, put

$Q_{\varepsilon}(\varphi,s):=\limsup_{n\rightarrow\infty}\frac{1}{|F_{n}|}\log
Q_{\varepsilon,F_{n}}(\varphi,s);$
$P_{\varepsilon}(\varphi,s):=\limsup_{n\rightarrow\infty}\frac{1}{|F_{n}|}\log
P_{\varepsilon,F_{n}}(\varphi,s);$

$p_{\varepsilon}(\varphi,s):=\limsup_{n\rightarrow\infty}\frac{1}{|F_{n}|}\log
p_{\varepsilon,F_{n}}(\varphi,s);$

$q_{\varepsilon}(\varphi,s):=\limsup_{n\rightarrow\infty}\frac{1}{|F_{n}|}\log
q_{\varepsilon,F_{n}}(\varphi,s).$

Let

$Q(\varphi,s):=\limsup_{\varepsilon\rightarrow
0}\frac{Q_{\varepsilon}(\varphi,s)}{s(\varepsilon)};$
$P(\varphi,s):=\limsup_{\varepsilon\rightarrow
0}\frac{P_{\varepsilon}(\varphi,s)}{s(\varepsilon)};$
$p(\varphi,s):=\limsup_{\varepsilon\rightarrow
0}\frac{p_{\varepsilon}(\varphi,s)}{s(\varepsilon)};$
$q(\varphi,s):=\limsup_{\varepsilon\rightarrow
0}\frac{q_{\varepsilon}(\varphi,s)}{s(\varepsilon)}.$

The following Proposition is a direct consequence of Remark 2.1 (3) and (4).

######  Proposition 2.1.

For $s\in\mathcal{S}$, and $\varphi\in C(X,\mathbb{R})$,

$Q(\varphi,s)\leqslant P(\varphi,s)\leqslant p(\varphi,s)\leqslant
q(\varphi,s).$

Denote by $C(X,\mathbb{R}^{+})$ the set of nonnegative continuous functions.

######  Proposition 2.2.

For $\varphi\in C(X,\mathbb{R}^{+})$, $s\in\mathcal{S}^{*}$,

$Q(\varphi,s)=P(\varphi,s)=p(\varphi,s)=q(\varphi,s).$

###### Proof.

By Proposition 2.1 and Remark 1.1 (4), we are left to show that
$Q(\varphi,s)\geqslant q(\varphi,s)$. For any $\varepsilon>0$ and
$(F_{n},\frac{\varepsilon}{2})$ spanning set $J$, then the $d_{F_{n}}$ balls
centered at $J$ of radius $\frac{\varepsilon}{2}$ cover $X$ and these balls
form an open cover $\alpha_{J}$ with mesh$(\alpha,d_{F_{n}})<\varepsilon$.

$\displaystyle\limsup_{\varepsilon\to
0}\frac{\limsup\limits_{n\to\infty}\frac{1}{|F_{n}|}\log\sum\limits_{y\in
J}\exp(s(\frac{\varepsilon}{2})S_{F_{n},\varphi}(y))}{s(\frac{\varepsilon}{2})}$
$\displaystyle\geqslant$ $\displaystyle\limsup_{\varepsilon\to
0}\frac{\limsup\limits_{n\to\infty}\frac{1}{|F_{n}|}\log\sum\limits_{A\in\alpha}\inf\limits_{x\in
A}\exp(s(\varepsilon)S_{F_{n},\varphi}(x))}{s(\frac{\varepsilon}{2})}.$

In the above inequality, we use the fact that $\varphi\in C(X,\mathbb{R}^{+})$
and $s\in\mathcal{S}^{*}$. Therefore, $Q(\varphi,s)\geqslant q(\varphi,s)$.

∎

######  Definition 2.3.

The map

$SP(\cdot,\cdot):C(X,\mathbb{R})\times\mathcal{S}\rightarrow\mathbb{R}\cup\\{\infty\\},\varphi\mapsto
SP(\varphi,s)=Q(\varphi,s)$

is called _scale pressure of_ $G$ with respect to $s$ and $\varphi$.

For any invariant measure $\mu\in\mathcal{M}_{G}(X)$, $\varepsilon>0$,
$0<\delta<1$, $s\in\mathcal{S}$ we put

$\displaystyle P_{\mu,\delta,\varepsilon,F_{n}}(\varphi,s):=$
$\displaystyle\inf\\{\sum_{x\in E}\exp(s(\varepsilon)S_{F_{n},\varphi}(x)):\
E\mbox{ is\ an}\ (F_{n},\varepsilon)\mbox{\ spanning\ set\ of}\ D\subset X\
\mbox{with}\ \mu(D)\geqslant 1-\delta\\};$

Let

$P_{\mu,\delta,\varepsilon}(\varphi,s):=\limsup_{n\rightarrow\infty}\frac{1}{|F_{n}|}\log
P_{\mu,\delta,\varepsilon,F_{n}}(\varphi,s),$

and

$P_{\mu,\delta}(\varphi,s):=\limsup_{\varepsilon\to
0}\frac{P_{\mu,\delta,\varepsilon}(\varphi,s)}{s(\varepsilon)}.$

For any finite measurable partition $\xi$ of $(X,\mathscr{B})$ and any
$\mu\in\mathcal{M}_{G}(X)$, write
$h_{\mu}(G,\xi):=\lim\limits_{n\rightarrow\infty}-\frac{1}{|F_{n}|}\sum\limits_{A\in\xi_{F_{n}}}\mu(A)\log\mu(A)$,
where $\xi_{F_{n}}=\bigvee\limits_{g\in F_{n}}g^{-1}\xi$.

######  Definition 2.4.

Let $(X,G)$ be a $G$-action topological dynamical system,
$\mu\in\mathcal{M}_{G}(X)$, we call

$h_{\mu}(G):=\sup\\{h_{\mu}(G,\xi):\ \xi\ \mbox{is\ a\ finite\ measurable\
partition\ of\ }(X,\mathscr{B})\\}$

the _measure-theoretic entropy_ of $G$ with respect to $\mu$.

For $\mu\in\mathcal{M}_{G}(X)$ define $N_{\mu}(F_{n},\varepsilon,\delta)$ as
the minimal number of dynamical balls $B_{F_{n}}(x,\varepsilon)$ needed to
cover a set of measure strictly bigger than $1-\delta$. Then define
$h_{\mu,\delta,\varepsilon}(X,G)=\limsup\limits_{n\to\infty}\frac{1}{|F_{n}|}\log
N_{\mu}(F_{n},\varepsilon,\delta)$. It was proven by Y. Zhao in [14] that
$h_{\mu}(G)=\lim\limits_{\varepsilon\to 0}h_{\mu,\delta,\varepsilon}(X,G)$ for
any $\mu\in\mathcal{E}_{T}(G)$, $\delta\in(0,1)$ and tempered Følner sequence
$\\{F_{n}\\}_{n\in\mathbb{N}}$ in $G$ with
$\lim\limits_{n\rightarrow\infty}\frac{|F_{n}|}{\log n}=\infty$.

######  Proposition 2.3.

Let $\\{F_{n}\\}_{n\in\mathbb{N}}$ be a tempered Følner sequence in $G$ with
$\lim\limits_{n\rightarrow\infty}\frac{|F_{n}|}{\log n}=\infty$,
$s\in\mathcal{S}$ and $\varepsilon\in(0,1)$. For $\varphi\in C(X,\mathbb{R})$,
$\mu\in\mathcal{E}_{G}(X)$,

$\lim_{\delta\rightarrow 0}P_{\mu,\delta}(\varphi,s)=\lim_{\delta\rightarrow
0}\limsup_{\varepsilon\to
0}\frac{h_{\mu,\delta,\varepsilon}(X,G)}{s(\varepsilon)}+\int\varphi d\mu.$

###### Proof.

The proof is further divided into two steps.

_Step 1_ First, we show that

(2.1) $\lim_{\delta\rightarrow
0}P_{\mu,\delta}(\varphi,s)\leqslant\lim_{\delta\rightarrow
0}\limsup_{\varepsilon\to
0}\frac{h_{\mu,\delta,\varepsilon}(X,G)}{s(\varepsilon)}+\int\varphi d\mu.$

For any $\tau>0$, there is $0<\varepsilon<\tau$ such that if $x,y\in X$,
$d(x,y)<\varepsilon$, then $|\varphi(x)-\varphi(y)|<\tau.$ Since $\mu$ is
ergodic, by Birkhoff ergodic theorem and Egorov theorem, there is a measurable
set $B\in\mathscr{B}$ with $\mu(B)>1-\frac{\delta}{2}$ satisfies the
following: For every $i\in\mathbb{N}$ there exists $N_{i}\in\mathbb{N}$ such
that $N_{i+1}>N_{i}$ and for any $n\geqslant N_{i}$, $x\in B$,
$|\frac{1}{|F_{n}|}s(\varepsilon)S_{F_{n},\varphi}(x)-\int
s(\varepsilon)\varphi d\mu|<\frac{1}{i}.$

We choose a sequence of positive integers $\\{n_{i}\\}_{i\in\mathbb{N}}$
satisfying:

(1) $n_{i}\geqslant N_{i}$, for any $i\in\mathbb{N}$;

(2)
$P_{\mu,\delta,\varepsilon}(\varphi,s)=\lim\limits_{i\rightarrow\infty}\frac{1}{|F_{n_{i}}|}\log
P_{\mu,\delta,\varepsilon,F_{n_{i}}}(\varphi,s)$.

Let $C_{n_{i}}$ be a set with $\mu(C_{n_{i}})>1-\frac{\delta}{2}$ and
$D_{n_{i}}$ be an $(F_{n_{i}},\varepsilon)$ spanning set of $C_{n_{i}}$ with
card$(D_{n_{i}})=N_{\mu}(F_{n_{i}},\varepsilon,\frac{\delta}{2})$.

Put $E_{n_{i}}=B\cap C_{n_{i}}$, then $\mu(E_{n_{i}})>1-\delta.$ Let
$J_{n_{i}}\subset D_{n_{i}}$ be an $(F_{n_{i}},\varepsilon)$ spanning set of
$E_{n_{i}}$ with smallest cardinality. Therefore,

$\displaystyle\sum_{x\in
J_{n_{i}}}\exp(s(\varepsilon)S_{F_{n_{i}},\varphi}(x))$
$\displaystyle\leqslant$ $\displaystyle\sum_{x\in
J_{n_{i}}}\exp[s(\varepsilon)|F_{n_{i}}|(\int\varphi d\mu+\frac{1}{i})]$
$\displaystyle\leqslant$ $\displaystyle
card(D_{n_{i}})\exp[s(\varepsilon)|F_{n_{i}}|(\int\varphi d\mu+\frac{1}{i})].$

Therefore

$P_{\mu,\delta,\varepsilon,F_{n_{i}}}(\varphi,s)\leqslant
card(D_{n_{i}})\exp[|F_{n_{i}}|(s(\varepsilon)\int\varphi
d\mu+s(\varepsilon)\frac{1}{i})].$

Hence,

$P_{\mu,\delta,\varepsilon}(\varphi,s)\leqslant
h_{\mu,\frac{\delta}{2},\varepsilon}(X,G)+s(\varepsilon)\int\varphi d\mu.$

Dividing by $s(\varepsilon)$ and letting $\varepsilon\rightarrow 0$, we have
(2.1).

_Step 2_ Next, we show that

(2.2) $\lim_{\delta\rightarrow
0}P_{\mu,\delta}(\varphi,s)\geqslant\lim_{\delta\rightarrow
0}\limsup_{\varepsilon\to
0}\frac{h_{\mu,\delta,\varepsilon}(X,G)}{s(\varepsilon)}+\int\varphi d\mu.$

By Birkhoff ergodic theorem and Egorov theorem, similarly to _Step 1_ we can
find a measurable set $B\in\mathscr{B}$ with $\mu(B)>1-\delta$ and a sequence
of positive integers $\\{n_{i}\\}_{i\in\mathbb{N}}$ satisfying the following:

For every $i\in\mathbb{N}$ there exists $N_{i}\in\mathbb{N}$ such that for any
$i\in\mathbb{N}$, $x\in B$,
$|\frac{1}{|F_{n_{i}}|}s(\varepsilon)S_{F_{n_{i}},\varphi}(x)-\int
s(\varepsilon)\varphi d\mu|<\frac{1}{i},$ and

$h_{\mu,2\delta,\varepsilon}(X,G)=\lim\limits_{i\rightarrow\infty}\frac{1}{|F_{n_{i}}|}\log
N_{\mu,2\delta,\varepsilon,F_{n_{i}}}(X,G).$

Let $C_{n_{i}}$ be a set with $\mu(C_{n_{i}})>1-\delta$ and $D_{n_{i}}$ be an
$(F_{n_{i}},\varepsilon)$ spanning set of $C_{n_{i}}$. Put $E_{n_{i}}=B\cap
C_{n_{i}}$, then $\mu(E_{n_{i}})>1-2\delta.$ Let $J_{n_{i}}\subset D_{n_{i}}$
be an $(F_{n_{i}},\varepsilon)$ spanning set of $E_{n_{i}}$ with smallest
cardinality. Therefore,

$\displaystyle\sum_{x\in
D_{n_{i}}}\exp(s(\varepsilon)S_{F_{n_{i}},\varphi}(x))$
$\displaystyle\geqslant$ $\displaystyle\sum_{x\in
J_{n_{i}}}\exp(s(\varepsilon)S_{F_{n_{i}},\varphi}(x))$
$\displaystyle\geqslant$ $\displaystyle\sum_{x\in
J_{n_{i}}}\exp[s(\varepsilon)|F_{n_{i}}|(\int\varphi d\mu-\frac{1}{i})]$
$\displaystyle=$ $\displaystyle
card(J_{n_{i}})\exp[s(\varepsilon)|F_{n_{i}}|(\int\varphi d\mu-\frac{1}{i})]$
$\displaystyle\geqslant$ $\displaystyle
N_{\mu}(F_{n_{i}},\varepsilon,2\delta)\exp[s(\varepsilon)|F_{n_{i}}|(\int\varphi
d\mu-\frac{1}{i})].$

Hence,

$\displaystyle P_{\mu,\delta,\varepsilon}(\varphi,s)$ $\displaystyle\geqslant$
$\displaystyle h_{\mu,2\delta,\varepsilon}(X,G)+s(\varepsilon)\int\varphi
d\mu.$

Dividing by $s(\varepsilon)$ and letting $\varepsilon\rightarrow 0$, we have
(2.2).

∎

######  Definition 2.5.

For any $0<\delta<1$, $\mu\in\mathcal{M}_{G}(X)$, $s\in\mathcal{S}$, the map

$SP_{\mu}(\cdot,\cdot):C(X,\mathbb{R})\times\mathcal{S}\rightarrow\mathbb{R}\cup\\{\infty\\},\varphi\mapsto
SP_{\mu,\delta}(\varphi,s):=P_{\mu,\delta}(\varphi,s)$

is called _measure-theoretic scale pressure of_ $G$ with respect to $s$ and
$\varphi$.

One of our main results is the following variational principle, we need a
condition. For any finite measurable partition $\xi$ and $r>0$, let
$U_{r}(A):=\\{x\in A:\exists y\in A^{c},\ \mbox{with}\ d(x,y)<r\\}$ and
$U_{r}(\xi):=\bigcup\limits_{A\in\xi}U_{r}(A)$. Since
$\bigcap\limits_{r>0}U_{r}(\xi)=\partial\xi$, $\lim\limits_{r\rightarrow
0}\mu(U_{r}(\xi))=\mu(\partial\xi)$, for any $\mu\in\mathcal{M}_{G}(X)$, where
$\partial\xi:=\bigcup\limits_{A\in\xi}\partial A$ and $\partial A$ is the
boundary of $A$.

If a finite measurable partition $\xi$ satisfies $\mu(\partial\xi)=0$ for some
$\mu\in\mathcal{E}_{G}(X)$, then for any $\gamma>0$, we can find $0<r<\gamma$
such that $\mu(U_{r}(\xi))<\gamma.$ Let
$r_{\mu,\gamma}:=\sup\\{r\in\mathbb{R}^{+}:\exists\,\mbox{finite\,measurable\,partition}\,\xi\,\mbox{with}\,\mu(\partial\xi)=0,diam(\xi)<\gamma\,\mbox{and}\,\mu(U_{r}(\xi))<\gamma\\}$
and $r_{\gamma}:=\inf\limits_{\mu\in\mathcal{E}_{G}(X)}r_{\mu,\gamma}$.

######  Condition A.

For any $\gamma>0$, $r_{\gamma}>0$ and $\limsup\limits_{\gamma\rightarrow
0}\frac{s(r_{\gamma})}{s(\gamma)}=1.$

######  Remark 2.2.

If we choose $s\equiv 1$, then Condition A holds for the system such that for
any $\gamma>0$, $r_{\gamma}>0$. If we take $s(x)=-\log x$, for example, a
trivial example satisfying Condition A is one dimensional uniquely ergodic
systems whose ergodic measure is Lebesgue measure.

For any $\varepsilon\in(0,1)$, $s\in\mathcal{S}^{**}$, $\varphi\in
C(X,\mathbb{R})$ and $0<\delta<1$, let

$sp(\varphi,s,\delta):=\limsup\limits_{\varepsilon\to
0}\frac{\sup\limits_{\mu\in\mathcal{M}_{G}(X)}P_{\mu,\delta,\varepsilon}(\varphi,s)}{s(\varepsilon)}.$

######  Theorem A.

Let $(X,G)$ be a $G$-action topological dynamical system satisfying Condition
A. $\\{F_{n}\\}_{n\in\mathbb{N}}$ is a tempered Følner sequence in $G$ with
$\lim\limits_{n\rightarrow\infty}\frac{|F_{n}|}{\log n}=\infty$. For any
$S\in\mathcal{S}^{**}$, $\varphi\in C(X,\mathbb{R})$ and $0<\delta<1$,

$SP(\varphi,s)=sp(\varphi,s,\delta).$

######  Remark 2.3.

The condition $\lim\limits_{n\rightarrow\infty}\frac{|F_{n}|}{\log n}=\infty$
on the tempered Følner sequence $\\{F_{n}\\}_{n\in\mathbb{N}}$ is only to
guarantee the application of Shannon-McMillan-Breiman theorem (see for example
[6]). Therefore, neither the quantity $P_{\mu,\delta,\varepsilon}(\varphi)$
nor the result depends on the choice of Følner sequence.

To prove theorem A, we need the following lemma.

######  Lemma 2.1.

For any $\gamma>0,$ there exists $\mu_{\gamma}\in\mathcal{M}_{G}(X)$ such that
for all finite measurable partition $\xi$ with diam$(\xi)<\gamma$ and
$\mu_{\gamma}(\partial\xi)=0$, we have $P_{\gamma}(\varphi,s)\leqslant
h_{\mu_{\gamma}}(G,\xi)+S(\gamma)\int\varphi d\mu_{\gamma}$. Moreover,
$\mu_{\gamma}$ can be chosen to be ergodic.

###### Proof.

The proof follows the same line of [15, Theorem 1.2], with slight
modification. For any $\gamma>0$, any tempered Følner sequence
$\\{F_{n}\\}_{n\in\mathbb{N}}$ with
$\lim\limits_{n\to\infty}\dfrac{|F_{n}|}{\log n}=\infty$, let
$E_{n}=\\{x_{1},...,x_{sep_{\gamma,F_{n}}(X,G)}\\}$ be a maximal collection of
$(F_{n},\gamma)$-separated points in $X$ with

$\log\sum\limits_{x\in E_{n}}\exp(s(\gamma)S_{F_{n},\varphi}(x))\geqslant\log
P_{\gamma,F_{n}}(\varphi,s)-1.$

Define

$\sigma_{n}=\dfrac{\sum\limits_{y\in
E_{n}}\exp(s(\gamma)S_{F_{n},\varphi}(y))\delta_{y}}{\sum\limits_{x\in
E_{n}}\exp(s(\gamma)S_{F_{n},\varphi}(x))},$

where $\delta_{x}$ is the probability measure supported at $x$. Then define

$\mu_{n}=\dfrac{1}{|F_{n}|}\sum_{g\in{F_{n}}}\sigma_{n}\circ g^{-1}.$

Consider a subsequence $\\{n_{k}\\}_{k\in\mathbb{N}}$ such that

$P_{\gamma}(\varphi,s)=\lim_{k\to\infty}\dfrac{1}{|F_{n_{k}}|}\log
P_{\gamma,n_{k}}(\varphi,s)$

and $\\{\mu_{n_{k}}\\}_{k\in\mathbb{N}}$ converges to some
$\mu_{\gamma}\in\mathcal{M}_{G}(X)$. Let $\xi$ be a finite measurable
partition with diam$(\xi)<\gamma$ and $\mu(\partial\xi)=0$. Recall that
$\xi_{F_{n}}=\bigvee\limits_{g\in{F_{n}}}g^{-1}\xi$. Since each element of
$\xi_{F_{n}}$ contains at most one element of $E_{n}$, by the definition of
$\sigma_{n}$, we have

$\displaystyle H_{\sigma_{n}}(\xi_{F_{n}})+s(\gamma)\int
S_{F_{n},\varphi}(x)d\sigma_{n}$ $\displaystyle=$ $\displaystyle\sum_{y\in
E_{n}}\sigma_{n}(\\{y\\})[s(\gamma)S_{F_{n},\varphi}(y)-\log\sigma_{n}(\\{y\\})]$
$\displaystyle=$ $\displaystyle\log\sum_{y\in
E_{n}}\exp(s(\gamma)S_{F_{n},\varphi}(y)),$

where
$H_{\sigma_{n}}(\xi_{F_{n}}):=\sum\limits_{A\in\xi_{F_{n}}}-\sigma_{n}(A)\log\sigma_{n}(A)$.
By a similar argument of [4, Lemma 3.1(3)], for any finite subset $F$ of $G$,
we have

$H_{\sigma_{n}}(\xi_{F_{n}})\leqslant\frac{1}{|F|}\sum_{g\in
F_{n}}H_{\sigma_{n}\circ
g^{-1}}(\xi_{F})+\frac{1}{|F_{n}|}|F^{-1}F_{n}\backslash
F_{n}|\log\mbox{card}(\xi),$

where $\mbox{card}(\xi)$ denotes the number of cells of partition $\xi$.
Hence,

$\displaystyle\frac{1}{|F_{n}|}H_{\sigma_{n}}(\xi_{F_{n}})$
$\displaystyle\leqslant$
$\displaystyle\frac{1}{|F|}\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}H_{\sigma_{n}\circ g^{-1}}(\xi_{F})+\frac{|F^{-1}F_{n}\backslash
F_{n}|}{|F_{n}|}\log\mbox{card}(\xi)$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{|F|}H_{\mu_{n}}(\xi_{F})+\frac{|F^{-1}F_{n}\backslash
F_{n}|}{|F_{n}|}\log\mbox{card}(\xi).$

Since $\mu(\partial\xi_{F})=0$, we have

$\displaystyle\limsup_{n\rightarrow\infty}\frac{1}{|F_{n}|}\log\sum_{y\in
E_{n}}\exp(s(\gamma)S_{F_{n},\varphi}(y))$ $\displaystyle=$
$\displaystyle\lim_{j\rightarrow\infty}\frac{1}{|F_{n_{j}}|}\log\sum_{y\in
E_{n_{j}}}\exp(s(\gamma)S_{F_{n_{j}},\varphi}(y))$ $\displaystyle=$
$\displaystyle\lim_{j\rightarrow\infty}\frac{1}{|F_{n_{j}}|}(H_{\sigma_{n_{j}}}(\xi_{F_{n_{j}}})+s(\gamma)\int
S_{F_{n_{j}},\varphi}(x)d\sigma_{n_{j}})$ $\displaystyle\leqslant$
$\displaystyle\lim_{j\rightarrow\infty}(\frac{1}{|F|}H_{\mu_{n_{j}}}(\xi_{F})+\frac{|F^{-1}F_{n_{j}}\backslash
F_{n_{j}}|}{|F_{n_{j}}|}\log\mbox{card}(\xi)+\lim_{j\rightarrow\infty}s(\gamma)\int\varphi
d\mu_{n_{j}}$ $\displaystyle=$
$\displaystyle\frac{1}{|F|}H_{\mu_{\gamma}}(\xi_{F})+s(\gamma)\int\varphi
d\mu_{\gamma}.$

Taking infimum over all finite subsets of $G$, by [2, Theorem 4.2],

$\limsup\limits_{n\rightarrow\infty}\frac{1}{|F_{n}|}\log\sum\limits_{y\in
E_{n}}\exp(s(\gamma)S_{F_{n},\varphi}(y))\leqslant
h_{\mu_{\gamma}}(G,\xi)+s(\gamma)\int\varphi d\mu_{\gamma}.$

We can get that $P_{\gamma}(\varphi,s)\leqslant
h_{\mu_{\gamma}}(G,\xi)+s(\gamma)\int\varphi d\mu_{\gamma}.$

By ergodic decomposition theorem [4], we can choose
$\mu_{\gamma}\in\mathcal{E}_{G}(X)$ satisfying the desired properties. ∎

###### proof of theorem A.

The inequality $SP(\varphi,s)\geqslant sp(\varphi,s,\delta)$ follows directly
from definitions.

For any $\tau>0$, there is $0<\varepsilon<\min\\{\tau,\frac{1-\delta}{2}\\}$
such that if $x,y\in X$, $d(x,y)<\varepsilon$, then
$|\varphi(x)-\varphi(y)|<\tau.$

For any finite measurable partition $\xi$ with $\mu(\partial\xi)=0$ for some
$\mu\in\mathcal{E}_{G}(X)$, we can find $0<r_{\mu}<\varepsilon$ such that
$\mu(U_{r}(\xi))<\varepsilon.$ Suppose that $N_{\mu}(F_{n},r,\delta)$ is the
minimal number of $B_{F_{n}}(x,r)$ balls that cover a set of measure more than
$1-\delta$. According to the proof of [15, Theorem 3.1], by a combinational
arguments on the numbers of the cells $\xi_{F_{n}}$ and the bowen balls
$B_{F_{n}}(x,r)$, there is $N_{1}\in\mathbb{N}$ such that for any $n>N_{1}$,

$N_{\mu}(F_{n},r,\delta)\geqslant\frac{\exp[|F_{n}|(h_{\mu}(G,\xi)-\varepsilon)]}{D(\frac{\varepsilon}{2},N_{0},F_{n})}\frac{1-\delta}{4},$

where
$D(\frac{\varepsilon}{2},N_{0},F_{n}):=\sum\limits_{m=0}^{[|F_{n}|\frac{\varepsilon}{2}]}(N_{0}-1)^{m}C_{|F_{n}|}^{m},$
and $N_{0}=card(\xi)$. By Stirling formula,

$\lim\limits_{n\rightarrow\infty}\frac{1}{|F_{n}|}\log
D(\frac{\varepsilon}{2},N_{0},F_{n})=\frac{\varepsilon}{2}\log(N_{0}-1)-\frac{\varepsilon}{2}\log\frac{\varepsilon}{2}-(1-\frac{\varepsilon}{2})\log(1-\frac{\varepsilon}{2}).$

Again by Birkhoff ergodic theorem and Egorov theorem, similarly to the proof
in _Step 1_ of Proposition 1.2, we can find a measurable set $B\in\mathscr{B}$
with $\mu(B)>1-\frac{\delta}{2}$ and a sequence of positive integers
$\\{n_{i}\\}_{i\in\mathbb{N}}$ satisfies the following:

For every $i\in\mathbb{N}$ there exists $N_{i}\in\mathbb{N}$ such that for any
$i\in\mathbb{N}$, $x\in B$,
$|\frac{1}{|F_{n_{i}}|}s(r)S_{F_{n_{i}},\varphi}(x)-\int s(r)\varphi
d\mu|<\frac{1}{i},$ and

$P_{\mu,\delta,r}(\varphi,s)=\lim\limits_{i\rightarrow\infty}\frac{1}{|F_{n_{i}}|}\log
P_{\mu,\delta,r,F_{n_{i}}}(\varphi,s).$

Let $C_{n_{i}}$ be a set with $\mu(C_{n_{i}})>1-\frac{\delta}{2}$ and
$D_{n_{i}}$ be an $(F_{n_{i}},r)$ spanning set of $C_{n_{i}}$. Put
$E_{n_{i}}=B\cap C_{n_{i}}$, then $\mu(E_{n_{i}})>1-\delta.$ Let
$J_{n_{i}}\subset D_{n_{i}}$ be an $(F_{n_{i}},r)$ spanning set of $E_{n_{i}}$
with smallest cardinality. Thus for any $x\in J_{n_{i}}$, there is $\pi(x)\in
E_{n_{i}}$ such that $d_{F_{n_{i}}}(x,\pi(x))<r$. Therefore,

$\displaystyle\sum_{x\in D_{n_{i}}}\exp(s(\gamma)S_{F_{n_{i}},\varphi}(x))$
$\displaystyle\geqslant$ $\displaystyle\sum_{x\in
J_{n_{i}}}\exp(s(\gamma)S_{F_{n_{i}},\varphi}(x))$ $\displaystyle\geqslant$
$\displaystyle\sum_{x\in J_{n_{i}}}\exp[s(\gamma)|F_{n_{i}}|(\int\varphi
d\mu-\frac{1}{i})]$ $\displaystyle=$ $\displaystyle
card(J_{n_{i}})\exp[s(\gamma)|F_{n_{i}}|(\int\varphi d\mu-\frac{1}{i})]$
$\displaystyle\geqslant$ $\displaystyle
N_{\mu}(F_{n_{i}},\gamma,\delta)\exp[s(\gamma)|F_{n_{i}}|(\int\varphi
d\mu-\frac{1}{i})]$ $\displaystyle\geqslant$
$\displaystyle\frac{1-\delta}{4D(\frac{\varepsilon}{2},N_{0},F_{n_{i}})}\exp[|F_{n_{i}}|(h_{\mu}(G,\xi)-\varepsilon)]\exp[s(\gamma)|F_{n_{i}}|(\int\varphi
d\mu-\frac{1}{i})]$ $\displaystyle=$
$\displaystyle\frac{1-\delta}{4D(\frac{\varepsilon}{2},N_{0},F_{n_{i}})}\exp[|F_{n_{i}}|(h_{\mu}(G,\xi)-\varepsilon+s(\gamma)\int\varphi
d\mu-s(\gamma)\frac{1}{i})].$

Hence, for any $\delta>0$, $\mu\in\mathcal{E}_{G}(X)$ and any finite partition
$\xi$ with $\mu(\partial\xi)=0$, we can find $0<r<\varepsilon$ such that

$\displaystyle P_{\mu,\delta,r}(\varphi,s)$ $\displaystyle\geqslant$
$\displaystyle h_{\mu}(G,\xi)+s(r)\int\varphi
d\mu+s(r)\tau-\varepsilon-[\frac{\varepsilon}{2}\log(N_{0}-1)-\frac{\varepsilon}{2}\log\frac{\varepsilon}{2}-(1-\frac{\varepsilon}{2})\log(1-\frac{\varepsilon}{2})].$

For any $0<\gamma<\varepsilon$, by Lemma 2.1, we have

$\displaystyle P_{\gamma}(\varphi,s)$ $\displaystyle\leqslant$ $\displaystyle
h_{\mu_{\gamma}}(G,\xi)+s(\gamma)\int\varphi d\mu_{\gamma}$
$\displaystyle\leqslant$ $\displaystyle
P_{\mu_{\gamma},\delta,r_{\mu_{\gamma}}(\varphi,s)}+s(r_{\mu_{\gamma}})\tau+\varepsilon+[\frac{\varepsilon}{2}\log(N_{0}-1)+\frac{\varepsilon}{2}\log\frac{\varepsilon}{2}+(1-\frac{\varepsilon}{2})\log(1-\frac{\varepsilon}{2})]$
$\displaystyle\leqslant$
$\displaystyle\sup_{\mu\in\mathcal{M}_{G}(X)}P_{\mu,\delta,r_{\mu_{\gamma}}(\varphi,s)}+s(r_{\mu_{\gamma}})\tau+\varepsilon+[\frac{\varepsilon}{2}\log(N_{0}-1)+\frac{\varepsilon}{2}\log\frac{\varepsilon}{2}+(1-\frac{\varepsilon}{2})\log(1-\frac{\varepsilon}{2})]$

Dividing by $s(\gamma)$ and letting $\tau\rightarrow 0$. Since
$\tau>\varepsilon>\gamma>0$, by Condition A, we have

$SP(\varphi,s)\leqslant sp(\varphi,s).$

∎

At last, we pose an open problem, namely, whether we can exchange the order of
the limit and supremum in Theorem A.

######  Open problem.

Let $(X,G)$ be a $G$-action topological dynamical system, $\varphi\in
C(X,\mathbb{R})$, and $0<\delta<1$. Whether

$SP(\varphi,s)=\sup_{\mu\in\mathcal{M}_{G}(X)}SP_{\mu,\delta}(\varphi,s).$

##  3\. Pseudo Orbits

Pseudo orbits have proved to be a powerful conceptual tool in dynamical
systems. For example, if group action is expansive and has the pseudo orbit
tracing property then it is topologically stable [1]; every expansive action
of an amenable group with positive entropy that has the pseudo orbit tracing
property must admit off-diagonal asymptotic pairs [9]. In this section, we
investigate scale pressure with respect to pseudo orbits.

In this section, we assume that $G$ is a finitely generated countable discrete
amenable group with a _finite_ generating set
$\mathfrak{G}=\\{\mathbf{g}_{1},\ldots,\mathbf{g}_{m}\\}$, $m\in\mathbb{N}$.

######  Definition 3.1.

For $\epsilon>0$ we say that a sequence $\\{y_{g}\\}_{g\in G}$ is an
_$\epsilon$ -pseudo orbit_ of an action $G$ (with respect to the generating
set $\mathfrak{G}$) if $d(y_{tg},t(y_{g}))<\epsilon,\ \forall
t\in\mathfrak{G},g\in G$.

Let $PO_{\epsilon}(\mathfrak{G})$ denote all $\epsilon$ pseudo orbits for $G$,
let $X^{G}=\prod\limits_{g\in G}X_{g}$ with $X_{g}=X$, and denote the point
$\overline{x}$ in $X^{G}$ by $(x_{g})_{g\in G}.$

######  Definition 3.2.

Let $\varepsilon>0$, $E,J,K\subseteq X^{G}$ and $F\in F(G)$. If for any
$\overline{x}\in K$ there exists $\overline{y}\in J$ such that
$d(x_{g},y_{g})\leqslant\varepsilon$, for all $g\in F$, then we call $J$ is an
_ $(F,\varepsilon)$ spanning set of $K$_. If for any
$\overline{x},\overline{y}\in E$ there is a $g\in F$ such that
$d(x_{g},y_{g})>\varepsilon$, then we call $E$ is an _ $(F,\varepsilon)$
separated set_.

Denote by $sep_{\varepsilon,F}(X^{G},K)$ the maximal cardinality of an
$(F,\varepsilon)$-separated subset of $K\subseteq X^{G}$, by
$spa_{\varepsilon,F}(X^{G},K)$ the smallest cardinality of an
$(F,\varepsilon)$-spanning subset of $K\subseteq X^{G}$. Suppose that
$Y\subset PO_{\epsilon}(\mathfrak{G})$ is a nonempty subset, $\varphi\in
C(X,\mathbb{R})$. For $\varepsilon\in(0,1)$, $s\in\mathcal{S}$ and the Følner
sequence $\\{F_{n}\\}_{n\in\mathbb{N}}$, we put

$\displaystyle POQ_{\varepsilon,F_{n}}(Y,G,\varphi,s):=$
$\displaystyle\inf\\{\sum_{(x_{g})_{g\in G}\in J}\exp(s(\varepsilon)\sum_{g\in
F_{n}}\varphi(x_{g})):\ J\ is\ an\ (F_{n},\varepsilon)\ spanning\ set\ of\
Y\\};$ $\displaystyle POP_{\varepsilon,F_{n}}(Y,G,\varphi,s):=$
$\displaystyle\sup\\{\sum_{(x_{g})_{g\in G}\in E}\exp(s(\varepsilon)\sum_{g\in
F_{n}}\varphi(x_{g})):\ E\ is\ an\ (F_{n},\varepsilon)\ separated\ set\ of\
Y\\}.$

For $\varphi\in C(X,\mathbb{R})$ $\varepsilon\in(0,1)$, $s\in\mathcal{S}$ and
the Følner sequence $\\{F_{n}\\}_{n\in\mathbb{N}}$ in $G$, let

$POP(\varphi,s):=\limsup_{\varepsilon\rightarrow
0}\frac{\limsup\limits_{n\rightarrow\infty}\frac{1}{|F_{n}|}\log(\lim\limits_{\epsilon\rightarrow
0}POP_{\varepsilon,F_{n}}(PO_{\epsilon}(\mathfrak{G}),G,\varphi,s))}{s(\varepsilon)};$
$POQ(\varphi,s):=\limsup_{\varepsilon\rightarrow
0}\frac{\limsup\limits_{n\rightarrow\infty}\frac{1}{|F_{n}|}\log(\lim\limits_{\epsilon\rightarrow
0}POQ_{\varepsilon,F_{n}}(PO_{\epsilon}(\mathfrak{G}),G,\varphi,s))}{s(\varepsilon)}.$

######  Proposition 3.1.

For $\varphi\in C(X,\mathbb{R})$ with $\varphi\geqslant 0$, and for
$\varepsilon\in(0,1)$, $s\in\mathcal{S}^{*}$, $POP(\varphi,s)=POQ(\varphi,s)$.

###### Proof.

By a similar argument of Proposition 2.2, for any $\delta>0$,

$POQ(\varphi,s)\leqslant POP(\varphi,s)\leqslant POQ(\varphi,s)+\delta.$

∎

######  Definition 3.3.

The map

$PSP(\cdot,\cdot):C(X,\mathbb{R})\times\mathcal{S}\rightarrow\mathbb{R}\cup\\{\infty\\},\varphi\mapsto
PSP(\varphi,s):=POQ(\varphi,s)$

is called _scale pressure of_ $G$ _with respect to pseudo-orbits_.

######  Definition 3.4.

We call $(X,G)$ a _Lipschitz $G$-system_ if any $t\in\mathfrak{G}$ is a bi-
Lipschitz homeomorphism.

For $0\leqslant\epsilon\leqslant 1$, let $X_{\epsilon}$ be the closure of
$PO_{\epsilon}$. Since $G$ is countable, we can arrange $G$ as
$G:=\\{g_{i}\\}_{i\in\mathbb{N}}$ with $g_{0}=e$, then $(X_{1},\overline{d})$
is a compact metric space where
$\overline{d}(\overline{x},\overline{y}):=\sum\limits_{i\in\mathbb{N}}\frac{d(x_{g_{i}},y_{g_{i}})}{2^{i}}$,
$\overline{x},\overline{y}\in X_{1}$. For any $0\leqslant\epsilon\leqslant 1$,
$X_{\epsilon}$ is a compact subset of $X_{1}$. Then $G$ act on $X_{\epsilon}$
naturally by $t(x_{g})_{g\in G}=(x_{tg})_{g\in G}$. Set
$\pi_{\epsilon}:X_{\epsilon}\rightarrow X$, $(x_{g})_{g\in G}\mapsto x_{e}$.
It is clear that $\pi_{\epsilon}$ is continuous and
$\pi_{\epsilon}=\pi_{1}|_{X_{\epsilon}}$.

######  Lemma 3.1.

For any $\eta>0$, there exists $\epsilon_{0}>0$ such that for any
$0\leqslant\epsilon\leqslant\epsilon_{0}$ and $\overline{x}\in X_{\epsilon}$,
there exists $\overline{y}\in X_{0}$ with
$\overline{d}(\overline{x},\overline{y})<\eta,$ where
$X_{0}=\\{\overline{x}=(x_{g})_{g\in G}\in X^{G}:\ x_{tg}=t(x_{g}),\ \forall
t\in\mathfrak{G},g\in G\\}$.

###### Proof.

For any $\eta>0$, there exists $N\in\mathbb{N}$ such that
$\sum\limits_{i=N}^{\infty}\frac{D}{2^{i}}<\frac{\eta}{3}$, where
$D:=\max\\{d(x,y):x,y\in X\\}$. Choose $\epsilon_{0}>0$ so that
$d(x,y)<\epsilon_{0}$ implies $d_{N}(x,y)<\frac{\eta}{3}$, where
$d_{N}(x,y):=\max\limits_{0\leqslant i\leqslant N}d(g_{i}(x),g_{i}(y))$. For
any $\overline{y}\in X_{\rho}$ where $0\leqslant\rho\leqslant\epsilon_{0}$,
$\overline{x}:=(x_{g})_{g\in G}$ with $d(x_{e},y_{e})<\epsilon$ and
$x_{g}=g(x_{e})$, $\overline{x}$ will satisfy
$\overline{d}(\overline{x},\overline{y})<\eta.$ ∎

In the following Lemmas, we will add $X$ and $G$ to the notations $p$ and $Q$
in order to distinguish different spaces.

######  Lemma 3.2.

There exists $\eta_{0}>0$ such that for any $0<\eta<\eta_{0}$,
$s\in\mathcal{S}$, $F\in F(G)$ and $\varphi\in C(X,\mathbb{R})$,

$\lim_{\epsilon\rightarrow
0}p_{\eta,F}(X_{\epsilon},G,\varphi\circ\pi_{\epsilon},s)=p_{\eta,F}(X_{0},G,\varphi\circ\pi_{0},s).$

###### Proof.

For $\eta>0$ and $\epsilon>0$. Let $\alpha_{1,\eta}$ be a finite open cover of
$X_{1}$ with mesh $(\alpha_{1,\eta},\overline{d}_{F})<\eta$, and
$\alpha_{1,\eta}(\epsilon):=\\{A:\ A\in\alpha_{1,\eta},\ A\cap
X_{\epsilon}\neq\emptyset\\}$. It is clear that
$\alpha_{1,\varepsilon}^{0}(\epsilon)$ is an open cover of $X_{\epsilon}$. For
any $\delta>0$, there exists $\eta_{0}>0$ such that for any
$\overline{x},\overline{y}\in X_{1}$ with
$\overline{d}(\overline{x},\overline{y})<\eta_{0}$, $i\in\mathbb{N}$,
$|\varphi\circ\pi_{1}(g_{i}(\overline{x}))-\varphi\circ\pi_{1}(g_{i}(\overline{y}))|<\frac{\delta}{|F|}$.
By Lemma 3.1, we can choose $\epsilon_{0}>0$ satisfying that for any
$0\leqslant\epsilon\leqslant\epsilon_{0}$ and $\overline{x}\in X_{\epsilon}$,
there exists $\overline{y}\in X_{0}$ with
$\overline{d}(\overline{x},\overline{y})<\eta_{0}.$

So for any $0<\eta<\eta_{0}$ and $0\leqslant\epsilon\leqslant\epsilon_{0}$, we
have

$\displaystyle\sum_{A\in\alpha_{1,\eta}(\epsilon)}\sup_{x\in
A}\exp(s(\eta)S_{F,\varphi\circ\pi_{\epsilon}}(x))$ $\displaystyle\leqslant$
$\displaystyle\sum_{A\in\alpha_{1,\eta}(\epsilon)}\sup_{x\in
A}e^{s(\eta)\delta}\exp(s(\eta)S_{F,\varphi\circ\pi_{0}}(x)).$

Therefore, $p_{\eta,F}(X_{\epsilon},G,\varphi\circ\pi_{\epsilon},s)\leqslant
p_{\eta,F}(X_{0},G,\varphi\circ\pi_{0},s)e^{s(\eta)\delta}.$

On the other hand, it is obvious that
$p_{\eta,F}(X_{\epsilon},G,\varphi\circ\pi_{\epsilon},s)\geqslant
p_{\eta,F}(X_{0},G,\varphi\circ\pi_{0},s).$ ∎

######  Lemma 3.3.

If $(G,X)$ is a Lipschitz $G$-system, $\varphi\in C(X,\mathbb{R})$ with
$\varphi\geqslant 0$, and for $\varepsilon\in(0,1)$, $s\in\mathcal{S}^{*}$,
then $Q(X,G,\varphi,s)=Q(X_{0},G,\varphi,s).$

###### Proof.

Since $\pi_{0}$ is a surjective Lipschitz continuous map with Lipschitz
constant $L(\pi_{0})\leqslant 1$ and for any $\varepsilon>0$, if
$\overline{d}(\overline{x},\overline{y})<\varepsilon$, then
$d(\pi_{0}(\overline{x}),\pi_{0}(\overline{y}))<\varepsilon$. Thus if $A$ is
an $(F,\varepsilon)$ spanning set of $X_{0}$ then $\pi_{0}(A)$ is an
$(n,\varepsilon)$ spanning set of $X$.

Therefore,

$Q(X,G,\varphi,s)\leqslant Q(X_{0},G,\varphi,s).$

On the other hand, for any $\varepsilon>0$, there is $N\in\mathbb{N}$ such
that $\sum\limits_{i=N}^{\infty}\frac{D}{2^{i}}<\frac{\varepsilon}{3}$, where
$D:=\max\\{d(x,y):x,y\in X\\}$. Choose $\delta=\frac{\varepsilon}{4Lip_{N}}$
so that $d(x,y)<\delta$ implies $d_{N}(x,y)<\frac{\varepsilon}{3}$, where
$Lip_{N}:=\max\limits_{0\leqslant i\leqslant N}Lip(g_{i})$. For any
$(F,\delta)$ spanning set $A$ of $X$ then $Orb(A):=\\{(x_{g})_{g\in G}\in
X_{0}:x_{e}\in A\\}$ is an $(F,\varepsilon)$ spanning set of $X_{0}$.

Therefore,

$Q(X,G,\varphi,s)\geqslant Q(X_{0},G,\varphi,s).$

∎

######  Theorem B.

Let $(X,G)$ be a Lipschitz $G$-system, if $G$ is a finitely generated
countable discrete amenable group, and for $s\in\mathcal{S}^{*}$, then
$SP(\varphi,s)=PSP(\varphi,s).$

###### Proof.

The inequality $SP(\varphi,s)\leqslant PSP(\varphi,s)$ follows directly from
definitions. By Lemma 3.3, it is sufficient to show that

$SP(\varphi,s)\geqslant PSP(\varphi,s).$

For any $0<\eta<\eta_{0}$ and $0\leqslant\epsilon\leqslant\epsilon_{0}$ as in
Lemma 3.2, if $E$ is an $(F,2\eta)$-separated set, $\alpha$ is an open cover
with mesh$(\alpha,\overline{d}_{F_{n}})<\eta$, then

$\sum_{(x_{g})_{g\in G}\in
E}\exp(s(2\eta)\sum_{g\in{F_{n}}}\varphi(x_{g}))\leqslant\sum_{A\in\alpha}\sup_{x\in
A}e^{s(\eta)\delta}\exp(s(\eta)S_{F_{n},\varphi\circ\pi_{\epsilon}}(x)).$

Therefore,
$\limsup\limits_{n\rightarrow\infty}\frac{1}{|F_{n}|}\log(POP_{2\eta,F_{n}}(PO_{\epsilon}(\mathfrak{G}),G,\varphi,s))\leqslant
p_{\eta}(X_{\epsilon},G,\varphi\circ\pi_{\epsilon},s).$

By Lemma 3.2, $\lim\limits_{\epsilon\rightarrow
0}p_{\eta}(X_{\epsilon},G,\varphi\circ\pi_{\epsilon},s)\leqslant
p_{\eta}(X_{0},G,\varphi\circ\pi_{0},s).$

Thus
$\limsup\limits_{n\rightarrow\infty}\frac{1}{|F_{n}|}\log(POP_{2\eta,F_{n}}(PO_{\epsilon}(\mathfrak{G}),G,\varphi,s))\leqslant
p_{\eta}(X_{0},G,\varphi\circ\pi_{0},s).$

Dividing by $s(\varepsilon)$, by Lemma 3.3, letting $\eta\rightarrow 0$,

$PSP(\varphi,s)\leqslant SP(\varphi,s).$

∎

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Figure 21: Left: The sample selection of TNG QGs. The selection is done with
the Illustris-TNG100-3 run and at its Snapshot#33 corresponding to $z=2$.
Galaxies marked with red squares are selected as our TNG QGs sample for the
SED fitting robustness tests. Right: The upper panel shows the distribution of
mass-weighted age for the TNG QGs sample. The vertical red solid line marks
the median. The whole sample is divided into two groups, the younger one with
mass-weighted ages less than the median and the old one larger than the
median. The bottom panels show the comparison between true SFHs and recovered
SFHs using the Dirichlet prior, with the younger group shown on the left and
the old one shown on the right. The thick red dashed line and the red shaded
region show the meidan and 16th-84th percentile range of the true SFHs, whlie
the black step plots with solid and dashed line styles show the recovered
median and 16th-84th percentile range of SFHs. Figure 22: Observed signal-to-
noise ratio (SNR) of each photometric band in the CANDELS/GOODS-South. From
left to right show the bands from blue to red. Circles are the medians, while
the error bars are the 16th-84th percentile ranges.

In addition to the Dirichlet prior, we have also run Prospector for the TNG
QGs using the Continuity prior, which models changes of SFR in adjacent time
bins by means of the variate $\rm{x=log(SFR_{t_{i}}/SFR_{t_{i+1}})}$ and,
importantly, strongly disfavours sharp changes of x. We refer readers to Leja
et al. (2019a) for the details of the model, but, briefly, the prior in the
Continuity non-parametric SFH is assumed to be a Student’s-t distribution
$\rm{P(x,\nu,\sigma)}$666$\rm{P(x,\nu,\sigma)=\frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu\pi}\Gamma(\frac{1}{2}\nu)}(1+\frac{(x/\sigma)^{2}}{\nu})^{-\frac{\nu+1}{2}}}$,
where $\Gamma$ is the complete Gamma function., where $\sigma$ controls the
width of distribution and $\nu$ is the number of degrees of freedom that
controls the tail of the distribution. Calibrated using $\approx 30,000$
galaxies with $M_{*}\geq 10^{9}M_{\sun}$ at $z=0$ (the vast majority are star-
forming galaxies) from the Illustris simulations (Vogelsberger et al., 2014),
Leja et al. 2019a suggested $\sigma=0.3$ and $\nu=2$ which are used here. We
used the Continuity prior to fit the synthetic galaxies and compared the
recovered SFHs with the true ones, using the same procedures adopted for the
Dirichlet prior. As the second row of Figure 23 shows, while strong
correlations between parameters related to the recovered and the true SFH are
observed overall, systematic offsets exist. Specifically, compared with the
Dirichlet prior (the third row of Figure 23), the Continuity prior in general
over-estimates the mass-weighted age by $\approx 0.5$ Gyr. It also over-
estimates $z_{50}$, $z_{70}$ and $z_{90}$. These findings are consistent with
what reported by Leja et al. (2019a) for galaxies with the “Sudden Quench”
SFHs (see the right-most column of their Figure 5). Also noticed from the
third row of Figure 23 is the fact that the dispersion of the measurements is
larger when the Continuity prior is used, again showing that the Dirichlet
prior generally works better for the TNG QGs.

The reason behind the over-estimation of stellar ages of the TNG QGs with the
Continuity prior can be clearly seen in Figure 24. Because the Continuity
prior strongly opposes sharp changes in SFR, the reconstructed SFHs are forced
to be smooth. Specifically, rather than putting more mass in fewer time bins,
the Continuity prior tends to disperse the mass into more time bins. This is
particularly seen in the old time bins because young and bright stars outshine
older stars in the rest-frame optical-NIR, meaning that the goodness-of-fit
does not change too much when slightly altering the SFH of the older stellar
populations (Papovich et al., 2001) given the data coverage. The similar was
also found in Johnson et al. (2021) (see the bottom panel of their Figure 3).
However, because the TNG QGs indeed show very sudden changes in SFRs at early
epochs (i.e. old time bins), this leads to the obvious mismatch between the
assumption of the Continuity prior and the true shape of SFHs of the TNG QGs,
and consequentially the discrepancies between the recovered and the true SFHs.

Figure 23: Comparisons between the recovered SFH-related parameters and the
true values. The 1st row: The comparisons for the measurements assuming the
Dirichlet prior. From left to right show the results of mass-weighted age,
$z_{90}$, $z_{70}$ and $z_{50}$. Each galaxy is color-coded with its
metallicity. The black dashed line shows the one-to-one relation. The 2nd row:
The comparisons for the measurements assuming the Continuity prior. The 3rd
row: The distributions for the deviations of the measurements from the true
values. Black histograms are for the Dirichlet prior, the grey ones are for
the Continuity prior. Also marked in each histogram as vertical lines are the
medians. Figure 24: Similar to the bottom-right two panels of Figure 21, but
we now compare the true SFHs with the recovered SFHs assuming the Continuity
prior. The left panel shows the results for the younger TNG QGs, i.e. those
have mass-weighted ages less than the median value of the whole TNG QGs
sample, while the right panel shows the results for the older TNG QGs.

We conclude that our Prospector SED fitting procedure with the Dirichlet prior
can robustly recover the assembly histories for the TNG QGs. Our tests also
put in evidence the footprint of the assumed priors on the SFH reconstruction.
While the Continuity prior may be a very good choice for star-forming
galaxies, it is not as good as the Dirichlet prior for the QGs. However, we
stress the limitations of our test with the TNG simulations. First and
foremost, our conclusion on the best prior to use with non-parametric SFHs
relies on the assumptions that 1) any difference between the SFHs of simulated
TNG QGs and the real ones, regardless of how correctly the former capture the
latter, does not affect the performance of Prospector in carrying out the
reconstruction of the SFH itself, and 2) the systematic error from the
mismatch between the real stellar populations at high redshift and the
synthesis model (Fsps here) used to generate synthetic photometry for the TNG
QGs is minor. Second, we emphasize that we are not making the statement that
the Continuity prior should not be used for the QGs. Recall that the Student’s
distribution used by the Continuity prior is governed by $\sigma$ and $\nu$.
As already mentioned above, the values used here are from the calibration with
the Illustris simulations at $z=0$ where the vast majority of the sample are
star-forming galaxies. It is entirely plausible that, if we re-calibrate
$\sigma$ and $\nu$ using QGs, the Continuity prior may still work well. Such
calibration is beyond the scope of this work. The fact that using the
Dirichlet prior can reproduce the overall shapes of true SFHs, and
quantitatively recover the true values of parameters including mass-weighted
age, $z_{50}$, $z_{70}$ and $z_{90}$, makes us believe our reconstructed SFHs
of the QGs are statistically robust.

##  Appendix B Physical parameters derived with the Continuity prior for the
sample QGs

While tests with simulations provide very valuable guidance to assess the
performance of the SED fitting procedure, considering the limitations that we
detailed in the end of Appendix A, we decide to additionally do an empirical
test, namely to run a new set of Prospector fittings with the Continuity prior
for our sample QGs (section 2.1), to check if any significant systematic error
introduced by the prior could exist in the measures. In any case, as the
direct comparison of key parameters obtained using the two priors illustrates
(see Figure 25), the strong correlations between the two sets of measurements
are seen, meaning that switching from the Dirichlet prior to the Continuity
one will not qualitatively change any conclusion of this work.

Figure 25: Comparisons of physical parameters derived using different priors
of non-parametric SFHs for our sample QGs (section 2.1). The y-axis shows the
measurements using the Continuity prior, and the x-axis shows the measurements
using the Dirichlet prior. The black dashed line marks the one-to-one
relation. The inset of each panel shows the distribution of relative
differences between the two measurements where the 1$\sigma$ range is filled
with blue.

##  Appendix C Physical parameters derived by fixing metallicity during the
SED fitting

We also empirically test that if our results are sensitive to the assumption
on metallicity during the SED fitting. Unlike Appendix B where the entire
sample was used, here we only use the sample from the GOODS-South field for
the test. We run a new set of SED fittings using the same assumptions as our
fiducial model (Dirichlet) except that metallicity is fixed to the solar
metallicity during the fittings. We compare the measurements in Figure 26.
Compared to the systematic errors introduced by the priors (Figure 25), the
errors introduced by the assumptions on metallicity are much smaller, and they
cannot change any conclusion of this work.

Figure 26: Similar to Figure 25, but for the comparisons of physical
parameters derived by fixing (y-axis) or freeing (x-axis) metallicity during
the SED fitting.

##  Appendix D Comparing the Skewness of SFHs with
$\tau_{\rm{SF}}$/$\tau_{\rm{Q}}$

In section 2.4, we introduced two ways to characterize the asymmetry of SFHs,
namely the Skewness and $\tau_{\rm{SF}}$/$\tau_{\rm{Q}}$. We compare them in
the left panel of Figure 27 where we see great agreement between the two
metrics. We also find some outliers where clear discrepancies exist between
the two metrics. In the right panel of the Figure, we plot SFHs of the
individual outliers. We immediately see the complex shape of SFHs of the
outliers, showing the reason of the discrepancies between the Skewness and
$\tau_{\rm{SF}}$/$\tau_{\rm{Q}}$.

Figure 27: Left: The comparison between the Skewness of SFHs and
$\tau_{\rm{SF}}$/$\tau_{\rm{Q}}$ (see the definitions in section 2.4). The two
metrics describing the asymmetry of SFHs agree with other other very well,
except some outliers which are labelled with red squares. Right: SFHs of the
individual outliers.