# Feldman-Katok pseudo-orbits and topological pressure

Fangzhou Cai  School of Mathematics and Systems Science, Guangdong Polytechnic
Normal University, Guangzhou, 510665, PR China<EMAIL_ADDRESS>
###### Abstract.

In this paper we introduce the notion of Feldman-Katok pseudo-orbits and use
it to study topological pressure. We prove that the topological pressure of a
dynamical system can be computed by measuring the Feldman-Katok pseudo-orbits
complexity of the shift map on the potential function, which extends Barge and
Swanson’s result on entropy and pseudo-orbits [9].

##  1\. Introduction

By a topological dynamical system (TDS for short) we mean a pair $(X,T)$ where
$X$ is a compact metric space with metric $d$ and $T:X\to X$ is a continuous
map. Topological entropy was introduced originally by Adler, Konheim and
McAndrew [1] in 1965. Later, Dinaburg [2] and Bowen [3] gave several
equivalent definitions by using separated and spanning sets. Bowen also gave a
characterization of dimension type for topological entropy, which generalized
the definition on non-compact sets [4]. Topological entropy is an important
dynamical invariant of topological conjugacy. Roughly speaking, Topological
entropy measures the maximal exponential growth rate of different forward
orbits for a topological dynamical system.

In the years since their development by Bowen [5] and Conley [7], pseudo-
orbits have proved to be a powerful tool in dynamical systems. A remarkable
result by Misiurewicz [8] stated that the topological entropy can be computed
by measuring the exponential growth rate of the numbers of pseudo-orbits.
Later Barge and Swanson [9] gave a similar result for periodic pseudo-orbits.
In [10], Hurley considered pseudo-orbits for inverse images and showed that
the point entropy of pseudo-orbits is in fact equal to the topological
entropy. In [11], Yan and Zeng obtained similar results for continuous maps on
compact uniform spaces. Recently, Cheng and Li used pseudo-orbits to
characterize the notion of scaled pressure [23].

As a non-trivial and natural generalization of topological entropy, the notion
of topological pressure was introduced in dynamical systems by Ruelle [21]
inspired by the theory of Gibbs states in statistical mechanics. Later Walters
[22] extended it to general topological dynamical systems. From a viewpoint of
dimension theory, Pesin and Pitskel [18] generalized Bowen’s definition of
topological entropy and defined the topological pressure on non-compact sets.
Topological pressure is a fundamental notion in thermodynamic formalism and
constitute the main components of the thermodynamic formalism ([6, 17, 19,
20]).

In 2017, Kwietniak and Łacka [24] introduced the Feldman-Katok metric as the
topological counterpart of edit distance $\bar{f}$ which was introduced by
Feldman [12] to study loosely Bernoulli system (see also [13, 14]). In recent
years, the Feldman-Katok metric proved to be a useful tool in dynamical
systems. In [25], the authors used it to characterize zero entropy loosely
Bernoulli systems and presented a purely topological characterization of their
topological models. In [15], Downarowicz, Kwietniak and Łacka introduced the
idea of $\bar{f}$-pseudometric to finite-valued stationary stochastic
processes and used it to study the entropy rate. In [16], Cai and Li gave
entropy formulas defined by Feldman-Katok metric.

The advantage to use Feldman-Katok metric is that it allows time delay by
ignoring the synchronization of points in orbits with only order preserving
required. Inspired by previous nice works, in this paper, we bring the idea
from the definition of Feldman-Katok metric into pseudo-orbits and attempt to
use it to study topological pressure.

The paper is organized as follows: We first introduce the notion of Feldman-
Katok pseudo-orbits. Then we give a formula for topological pressure with
respect to Feldman-Katok pseudo-orbits. We also consider scaled pressure and
show that in [23, Theorem F], the Lipschitz continuity of $T$ is redundant.
Finally we introduce and study topological pressure with respect to Feldman-
Katok metric.

##  2\.  Feldman-Katok pseudo-orbits and topological pressure

In this section we introduce the notion of Feldman-Katok pseudo-orbits and use
it to study topological pressure. Our idea is from the definition of Feldman-
Katok metric, which allows time delay in orbits with only order preserving
required.

###  2.1. Feldman-Katok pseudo-orbits

Let $(X,T)$ be a TDS and $\alpha>0$. Recall that we say
$(x_{0},x_{1},\ldots)\in X^{\mathbb{N}}$ is an $\alpha$-pseudo-orbit if

$d(Tx_{i},x_{i+1})\leq\alpha,i=0,1\ldots,n\ldots.$

Denote by $PO_{\alpha}(X,T)$ or simply $PO_{\alpha}$ the set of
$\alpha$-pseudo-orbits.

We bring time delay and “jump” in classical pseudo-orbits and give the
definition of Feldman-Katok pseudo-orbits.

######  Definition 2.1.

Let $(X,T)$ be a TDS, $n\in\mathbb{N}$ and $\alpha,\delta>0.$ We say
$(x_{0},x_{1},\ldots,x_{n-1})\in X^{n}$ is a $FK$-$\alpha$-pseudo-chain of
density $1-\delta$ if there exist $(1-\delta)n<k\leq n$ and

$0\leq i_{1}<i_{2}<\ldots<i_{k}\leq n-1,$ $0\leq j_{1}<j_{2}<\ldots<j_{k}\leq
n-1$

such that

$d(T^{j_{t+1}-j_{t}}(x_{i_{t}}),x_{i_{t+1}})\leq\alpha,t=1,\ldots,k-1.$

######  Remark 2.2.

In the definition above, if we choose $k=n$ and
$i_{1}=j_{1}=0,\ldots,i_{n}=j_{n}=n-1$, we get the definition of
$\alpha$-pseudo-chain (see [9]).

######  Definition 2.3.

Let $(X,T)$ be a TDS, $\delta>0$ and $N_{\delta}\in\mathbb{N}$. We say
$(x_{0},x_{1},\ldots)\in X^{\mathbb{N}}$ is a $FK$-$\alpha$-pseudo-orbit of
density $1-\delta$ controlled by $N_{\delta}$ if there exist a sequence $0\leq
s_{0}<s_{1}<\ldots<s_{i}<\ldots$ in $\mathbb{N}$ with

$s_{0}\leq N_{\delta},s_{i+1}-s_{i}\leq\ N_{\delta},i=0,1,\ldots,$

such that for all $m>n$, $(x_{s_{n}},x_{s_{n}+1},\ldots,x_{s_{m}-1})$ is a
$FK$-$\alpha$-pseudo-chain of density $1-\delta$.

Denote by $FKPO_{\alpha,\delta}(X,T,N_{\delta})$ the set of
$FK$-$\alpha$-pseudo-orbit of density $1-\delta$ controlled by $N_{\delta}$.

######  Remark 2.4.

  1. (1)

$N_{\delta}$ is needed to control the length, for we must avoid the situation
that the first finite components of a point ${\bf x}$ can be chosen arbitrary,
since in this situation we obtain a dense set in $X^{\mathbb{N}}$.

  2. (2)

It is easy to see that $FKPO_{\alpha,\delta}(X,T,N_{\delta})$ contains
$PO_{\alpha}(X,T)$ by definition.

  3. (3)

We can see points with the following form

$(x,y_{1},T^{2}x,y_{3},\ldots,T^{2n}x,y_{2n+1},\ldots)$

are in $FKPO_{\alpha,\delta}(X,T,N_{\delta})$ for
$\alpha>0,\delta>\frac{1}{2}$ and $N_{\delta}\geq 4.$ Hence there is an
essential difference between $FK$-pseudo-orbits and pseudo-orbits.

  4. (4)

If $\delta N_{\delta}<1$, the choice of $k$ in the definition is only $n$,
hence to avoid triviality, we can make $\delta N_{\delta}$ large enough by
choosing appropriate $N_{\delta}$.

  5. (5)

It is easy to see that $FKPO_{\alpha,\delta}(X,T,N_{\delta})$ is monotonically
increasing with respect to $\alpha$. But
$FKPO_{\alpha,\delta}(X,T,N_{\delta})$ may not monotonous with respect to
$\delta$.

In the sequel we fix $\\{N_{\delta}:\delta>0\\}\subset\mathbb{N}$.

We simply write $FKPO_{\alpha,\delta}$ instead of
$FKPO_{\alpha,\delta}(X,T,N_{\delta})$.

The following proposition shows that $FKPO_{\alpha,\delta}$ has a good
structure.

######  Proposition 2.5.

Let $(X,T)$ be a TDS and $\alpha,\delta>0$. Then $FKPO_{\alpha,\delta}$ is
closed in $X^{\mathbb{N}}$ and invariant under the shift $\sigma$.

###### Proof.

Since there is no restriction on $(x_{0},\ldots,x_{s_{0}-1})$ in the
definition of $FKPO_{\alpha,\delta}$, it is easy to see that
$FKPO_{\alpha,\delta}$ is invariant under $\sigma$. Now we prove
$FKPO_{\alpha,\delta}$ is closed.

Let ${\bf x}^{h}\in FKPO_{\alpha,\delta}$ with ${\bf x}^{h}\rightarrow{\bf
x}$. By definition there exist a sequence
$s_{0}^{h}<s_{1}^{h}<\ldots<s_{i}^{h}<\ldots$ with

$s_{0}^{h}\leq\ N_{\delta},s_{i+1}^{h}-s_{i}^{h}\leq N_{\delta},\
i=0,1,\ldots,$

such that for all $m>n$,
$(x_{s_{n}^{h}}^{h},x_{s_{n}^{h}+1}^{h},\ldots,x_{s_{m}^{h}-1}^{h})$ is a
$FK$-$\alpha$-pseudo-chain of density $1-\delta$. We can find
$s_{0}<s_{1}<\ldots<s_{i}<\ldots$ with

$s_{0}\leq N_{\delta},s_{i+1}-s_{i}\leq N_{\delta},i=0,1,\ldots,$

such that for any $l\in\mathbb{N}$, there exist infinite $h$ satisfy

$(s_{0}^{h},s_{1}^{h},\ldots,s_{l}^{h})=(s_{0},s_{1},\ldots,s_{l}).$

Fix $m>n$. Now we prove $(x_{s_{n}},x_{s_{n}+1},\ldots,x_{s_{m}-1})$ is a
$FK$-$\alpha$-pseudo-chain of density $1-\delta$. For $m$, there exist
infinite $h$ with

$(s_{0}^{h},s_{1}^{h},\ldots,s_{m}^{h})=(s_{0},s_{1},\ldots,s_{m}).$

For such $h$, since

$(x_{s_{n}^{h}}^{h},x_{s_{n}^{h}+1}^{h},\ldots,x_{s_{m}^{h}-1}^{h})=(x_{s_{n}}^{h},x_{s_{n}+1}^{h},\ldots,x_{s_{m}-1}^{h})$

is a $FK$-$\alpha$-pseudo-chain of density $1-\delta$, there exist $k_{h}$
with

$(1-\delta)(s_{m}-s_{n})<k_{h}\leq(s_{m}-s_{n})$

and

$0\leq i_{1}^{h}<i_{2}^{h}<\ldots<i_{k_{h}}^{h}\leq s_{m}-s_{n}-1,$ $0\leq
j_{1}^{h}<j_{2}^{h}<\ldots<j_{k_{h}}^{h}\leq s_{m}-s_{n}-1,$

such that

$d(T^{j_{t+1}^{h}-j_{t}^{h}}x_{i_{t}^{h}+s_{n}}^{h},x_{i_{t+1}^{h}+s_{n}}^{h})\leq\alpha,t=1,\ldots,k_{h}-1.$

We can choose some $k$ with

$(1-\delta)(s_{m}-s_{n})<k\leq(s_{m}-s_{n})$

and

$0\leq i_{1}<i_{2}<\ldots<i_{k}\leq s_{m}-s_{n}-1,$ $0\leq
j_{1}<j_{2}<\ldots<j_{k}\leq s_{m}-s_{n}-1,$

such that there exist infinite $h$ satisfy $k_{h}=k$ and

$(i_{1}^{h},i_{2}^{h},\ldots,i_{k}^{h})=(i_{1},i_{2},\ldots,i_{k}),$
$(j_{1}^{h},j_{2}^{h},\ldots,j_{k}^{h})=(j_{1},j_{2},\ldots,j_{k}).$

For such $h$ we have

$d(T^{j_{t+1}-j_{t}}x_{i_{t}+s_{n}}^{h},x_{i_{t+1}+s_{n}}^{h})\leq\alpha,t=1,\ldots,k-1.$

Let $h\to\infty$ we have

$d(T^{j_{t+1}-j_{t}}x_{i_{t}+s_{n}},x_{i_{t+1}+s_{n}})\leq\alpha,t=1,\ldots,k-1.$

Hence $(x_{s_{n}},x_{s_{n}+1},\ldots,x_{s_{m}-1})$ is a $FK$-$\alpha$-pseudo-
chain of density $1-\delta$. ∎

###  2.2. Topological pressure with respect to Feldman-Katok pseudo-orbits

In this subsection we use Feldman-Katok pseudo-orbits to study topological
pressure.

First let us recall some basic concepts.

Let $(X,T)$ be a TDS. For $x,y\in X,$ $n\in\mathbb{N}$ and $\epsilon>0$, set

$d_{n}(x,y)=\max_{0\leq i\leq n-1}d(T^{i}x,T^{i}y)$

be the Bowen metric. We say $E\subset X$ is an $(n,\epsilon)$-separated set of
$X$ if for each $x\neq y\in E$, $d_{n}(x,y)>\epsilon$. We say $F\subset X$ is
an $(n,\epsilon)$-spanning set of $X$ if for every $x\in X$ there is $y\in F$
with $d_{n}(x,y)\leq\epsilon.$

Let $f\in C(X)$, define

$sp_{n,\epsilon}(X,T,f)=\inf\\{\sum_{x\in
F}e^{\sum_{i=0}^{n-1}f(T^{i}x)}:F\text{ is\ an }(n,\epsilon)\text{-spanning\
set \ of }X\\},$ $sr_{n,\epsilon}(X,T,f)=\sup\\{\sum_{x\in
E}e^{\sum_{i=0}^{n-1}f(T^{i}x)}:E\text{ is\ an }(n,\epsilon)\text{-separated\
set \ of }X\\}.$

It is known that the topological pressure of $(X,T)$ with potential $f$ is
defined by

$\begin{split}P(X,T,f)&=\lim_{\epsilon\to
0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log sp_{n,\epsilon}(X,T,f)\\\
&=\lim_{\epsilon\to 0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
sr_{n,\epsilon}(X,T,f).\end{split}$

For ${\bf x}=(x_{0},x_{1},\ldots),{\bf y}=(y_{0},y_{1},\ldots)$ in
$X^{\mathbb{N}},$ the metric $\tilde{d}$ on $X^{\mathbb{N}}$ is defined by

$\tilde{d}({\bf x},{\bf y})=\sum_{i=0}^{\infty}\frac{d(x_{i},y_{i})}{2^{i}}.$

For $Y\subset X^{\mathbb{N}}$, we say $E\subset Y$ is an
$(n,\epsilon)$-separated set of $Y$ if for each ${\bf x}\neq{\bf y}$ in $E$,
there exists $0\leq i\leq n-1$ such that $d(x_{i},y_{i})>\epsilon$. We say
$F\subset Y$ is an $(n,\epsilon)$-spanning set of $Y$ if for every ${\bf x}\in
Y,$ there is ${\bf y}\in F$ with $d(x_{i},y_{i})<\epsilon,0\leq i\leq n-1.$

For $\alpha,\delta>0$ and $f\in C(X)$, define

$\begin{split}&FKPOsp_{n,\epsilon;\alpha,\delta}(X,T,f)=\\\ &\inf\\{\sum_{{\bf
x}\in F}e^{\sum_{i=0}^{n-1}f(x_{i})}:F\text{ is\ an
}(n,\epsilon)\text{-spanning\ set \ of }FKPO_{\alpha,\delta}\\},\end{split}$
$\begin{split}&FKPOsr_{n,\epsilon;\alpha,\delta}(X,T,f)=\\\ &\sup\\{\sum_{{\bf
x}\in E}e^{\sum_{i=0}^{n-1}f(x_{i})}:E\text{ is\ an
}(n,\epsilon)\text{-separated\ set \ of }FKPO_{\alpha,\delta}\\}.\end{split}$

We simply write $FKPOsp_{n,\epsilon;\alpha,\delta}$ and
$FKPOsr_{n,\epsilon;\alpha,\delta}$ if there is no confusion.

Similar as the definition in [9], we define

$\begin{split}PFKPO(X,T,f,\epsilon,\delta)&=\inf_{\alpha>0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
FKPOsp_{n,\epsilon;\alpha,\delta}(X,T,f)\\\ &=\lim_{\alpha\to
0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
FKPOsp_{n,\epsilon;\alpha,\delta}(X,T,f).\end{split}$

The limit exists since $FKPO_{\alpha,\delta}$ is monotonically increasing with
respect to $\alpha$. Now we define

$PFKPO(X,T,f,\epsilon)=\varlimsup_{\delta\to 0}PFKPO(X,T,f,\epsilon,\delta).$

Note that here $FKPO_{\alpha,\delta}$ may not monotonous with respect to
$\delta$, hence “$\varlimsup\limits_{\delta\to 0}$” can not be replaced by
“$\inf\limits_{\delta>0}$” or “$\lim\limits_{\delta\to 0}$”.

It is easy to see $PFKPO(X,T,f,\epsilon)$ is non-increasing with respect to
$\epsilon$. We define

$PFKPO(X,T,f)=\lim_{\epsilon\to 0}PFKPO(X,T,f,\epsilon)$

be the topological pressure with respect to Feldman-Katok pseudo-orbits.

The next lemma shows that we can use separated set in the definition above.

######  Lemma 2.6.

Let $(X,T)$ be a TDS and $f\in C(X).$ Then

$PFKPO(X,T,f)=\lim_{\epsilon\to 0}\varlimsup_{\delta\to
0}\inf_{\alpha>0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
FKPOsr_{n,\epsilon;\alpha,\delta}(X,T,f).$

###### Proof.

By definition it is easy to see

$PFKPO(X,T,f)\leq\lim_{\epsilon\to 0}\varlimsup_{\delta\to
0}\inf_{\alpha>0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
FKPOsr_{n,\epsilon;\alpha,\delta}.$

Now we prove the opposite direction. Let $\alpha,\delta>0.$ For $\epsilon>0,$
set

$L(\epsilon)=\sup_{|x-y|\leq\epsilon}|f(x)-f(y)|.$

Assume that $E$ be an $(n,\epsilon)$-separated set and $F$ be an
$(n,\frac{\epsilon}{2})$-spanning set of $FKPO_{\alpha,\delta}$. The map
$\phi:E\to F$ is defined by choosing for each ${\bf x}\in E,$ some point
$\phi({\bf x})\in F$ with

$d(x_{i},\phi({\bf x})_{i})\leq\frac{\epsilon}{2},i=0,\ldots,n-1.$

Then $\phi$ is injective and

$\begin{split}\sum_{{\bf y}\in F}e^{\sum_{i=0}^{n-1}f(y_{i})}&\geq\sum_{{\bf
x}\in E}e^{\sum_{i=0}^{n-1}f(\phi({\bf x})_{i})}\\\ &=\sum_{{\bf x}\in
E}e^{\sum_{i=0}^{n-1}f(x_{i})}e^{\sum_{i=0}^{n-1}f(\phi({\bf
x})_{i})-\sum_{i=0}^{n-1}f(x_{i})}\\\ &\geq
e^{-nL(\frac{\epsilon}{2})}\sum_{{\bf x}\in
E}e^{\sum_{i=0}^{n-1}f(x_{i})}.\end{split}$

Thus

$\begin{split}&\varlimsup_{\delta\to
0}\inf_{\alpha>0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
FKPOsr_{n,\epsilon;\alpha,\delta}\\\ &\leq\varlimsup_{\delta\to
0}\inf_{\alpha>0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
FKPOsp_{n,\frac{\epsilon}{2};\alpha,\delta}+L(\frac{\epsilon}{2}).\end{split}$

Let $\epsilon\to 0$, we have

$\lim_{\epsilon\to 0}\varlimsup_{\delta\to
0}\inf_{\alpha>0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
FKPOsr_{n,\epsilon;\alpha,\delta}\leq PFKPO(X,T,f).$

∎

Now we show that the topological pressure with respect to Feldman-Katok
pseudo-orbits is equal to the classical topological pressure. We have
following theorem:

######  Theorem 2.7.

Let $(X,T)$ be a TDS and $f\in C(X)$. Then

$PFKPO(X,T,f)=P(X,T,f).$

###### Proof.

By definition we have $PFKPO(X,T,f)\geq P(X,T,f).$ Now we prove the opposite
direction.

Fix $\epsilon>0.$ Let $X=\bigcup_{i=1}^{l}X_{i}$ with
$diamX_{i}<\frac{\epsilon}{4},i=1,\ldots,l.$ Choose some $z_{i}\in X_{i}$.

For
$\delta<\frac{\epsilon}{2},n>\max\\{\frac{N_{\delta}}{\delta},\frac{2diamX}{\delta}\\},m>3n,$
let $F$ be an $(n+m,\frac{\delta}{4})$-spanning set of $X$. We can find

$0<\delta_{0}<\delta_{1}<\ldots<\delta_{2(n+m)}=\frac{\delta}{4}$

such that for every $\delta_{k},k=0,1,\ldots,2(n+m)-1$,

$d(x,y)<\delta_{k}\Rightarrow
d(T^{i}x,T^{i}y)<\delta_{k+1},i=0,\ldots,2(n+m)-1.$

Let
$\alpha<\min\\{\delta_{0},\delta_{k+1}-\delta_{k}:k=0,1,\ldots,2(n+m)-1\\}.$

For $x\in F,$ let $F_{x}$ be the collection of points with following form:

$(y_{0},y_{1},\ldots,y_{i_{1}-1},T^{j_{1}}x,y_{i_{1}+1},\ldots,y_{i_{k}-1},T^{j_{k}}x,y_{i_{k}+1},\ldots,y_{m+n-1},z_{1},z_{1},\ldots),$

where

$(1-3\delta)(n+m)<k\leq(n+m),$ $0\leq i_{1}<\ldots<i_{k}\leq n+m-1,$ $0\leq
j_{1}<\ldots<j_{k}\leq n+m-1$

and $y_{i}\in\\{z_{1},\ldots,z_{l}\\}.$

We have

$|F_{x}|\leq\sum_{(1-3\delta)(n+m)<k\leq n+m}(C_{n+m}^{k})^{2}l^{n+m-k}.$

Claim:

$FKPO_{\alpha,\delta}\subset\bigcup_{x\in F}\bigcup_{{\bf y}\in
F_{x}}B_{\tilde{d}_{m}}({\bf y},\epsilon),$

where $B_{\tilde{d}_{m}}({\bf y},\epsilon)$ denotes the open $\epsilon$-ball
of ${\bf y}$ with respect to $\tilde{d}_{m}$.

proof of claim: For ${\bf x}\in FKPO_{\alpha,\delta},$ by definition there
exist $0<s_{0}<s_{1}<\ldots<s_{i}<\ldots$ with

$s_{0}\leq N_{\delta},s_{i+1}-s_{i}\leq N_{\delta},i=0,1,\ldots,$

such that for all $b>a$, $(x_{s_{a}},x_{s_{a}+1},\ldots,x_{s_{b}-1})$ is a
$FK$-$\alpha$-pseudo-chain of density $1-\delta$. Let $b$ satisfy $s_{b}\leq
n+m$ and $s_{b+1}>m+n$. It is easy to check

$s_{b}-s_{0}>(1-2\delta)(n+m).$

Since $(x_{s_{0}},x_{s_{0}+1},\ldots,x_{s_{b}-1})$ is a $FK$-$\alpha$-pseudo
chain of density $1-\delta$, we can find $k$ with

$(1-3\delta)(n+m)\leq(1-\delta)(s_{b}-s_{0})<k\leq s_{b}-s_{0}\leq n+m$

and

$s_{0}\leq i_{1}<i_{2}<\ldots<i_{k}\leq s_{b}-1\leq n+m-1,$ $0\leq
j_{1}<j_{2}<\ldots<j_{k}\leq s_{b}-s_{0}-1\leq n+m-1$

such that

$d(T^{j_{t+1}-j_{t}}(x_{i_{t}}),x_{i_{t+1}})\leq\alpha,t=1,\ldots,k-1.$

Now we prove

(1)
$d(T^{j_{t}-j_{1}}(x_{i_{1}}),x_{i_{t}})<\delta_{2(t-2)}<\frac{\delta}{4},t=2,\ldots,k.$

We prove by induction. When $t=2$ it is clear. If (1) holds for $t$, we have

$d(T^{j_{t+1}-j_{1}}(x_{i_{1}}),T^{j_{t+1}-j_{t}}x_{i_{t}})<\delta_{2(t-2)+1},$

hence

$d(T^{j_{t+1}-j_{1}}(x_{i_{1}}),x_{i_{t+1}})<\delta_{2(t-2)+1}+\alpha\leq\delta_{2(t-1)}.$

By induction (1) is proved.

Since $F$ is an $(n+m,\frac{\delta}{4})$-spanning set of $X$, there exists
$x\in F$ such that

$d(T^{h}x,T^{h}x_{i_{1}})<\frac{\delta}{4},h=0,\ldots,n+m-1.$

Hence

$d(T^{j_{t}-j_{1}}x,x_{i_{t}})<\frac{\delta}{2},t=1,\ldots,k.$

We assume $x_{i}\in X_{h_{i}},i=0,1,\ldots,m+n-1.$ Note that
$\frac{1}{2^{n-1}}<\frac{\epsilon}{2diamX}$. We have ${\bf x}\in
B_{\tilde{d}_{m}}({\bf y},\epsilon),$ where

${\bf
y}=(z_{h_{0}},z_{h_{1}},\ldots,z_{h_{i_{1}-1}},x,z_{i_{1}+1},\ldots,z_{h{{}_{i_{k}-1}}},T^{j_{k}-j_{1}}x,z_{h_{i_{k}+1}},\ldots,z_{h_{m+n-1}},z_{1},z_{1},\ldots).$

The claim is proved.

Denote

$\begin{split}&Cov_{n,\epsilon}(\beta,\delta)=\\\
&\inf\\{\sum_{A\in\mathcal{A}}\sup_{{\bf x}\in
A}e^{\sum_{i=0}^{n-1}f(x_{i})}:\mathcal{A}\text{ is\ a\ finite\ subcover\ of\
}\vee_{i=0}^{n-1}\sigma^{-i}\mathcal{A}_{\epsilon}\text{ covers
}FKPO_{\beta,\delta}\\},\end{split}$

where $\mathcal{A}_{\epsilon}$ denote the cover of
$(X^{\mathbb{N}},\tilde{d})$ consists of all open $\epsilon$-balls.

By claim we have

$Cov_{m,\epsilon}(\alpha,\delta)\leq\sum_{x\in F}\sum_{{\bf y}\in
F_{x}}\sup_{{\bf z}\in B_{\tilde{d}_{m}}({\bf
y},\epsilon)}e^{\sum_{i=0}^{m-1}f(z_{i})}.$

Note that $\alpha$ dependents on $m$ and $\delta$.

For $x\in F$, if ${\bf y}\in F_{x}$ and ${\bf z}\in B_{\tilde{d}_{m}}({\bf
y},\epsilon)$ we have $d(y_{i},z_{i})<\epsilon,i=0,\ldots,m-1.$ Note that
$m>3n$, we have

$(1-3\delta)(n+m)-n>(1-4\delta)m$

and

$\sum_{i=0}^{m-1}f(z_{i})-\sum_{i=0}^{m-1}f(T^{i}x)\leq
mL(\epsilon)+2||f||4\delta m,$

where

$L(\epsilon)=\sup_{|x-y|\leq\epsilon}|f(x)-f(y)|.$

Hence

$\begin{split}\sum_{{\bf y}\in F_{x}}\sup_{{\bf z}\in B_{\tilde{d}_{m}}({\bf
y},\epsilon)}e^{\sum_{i=0}^{m-1}f(z_{i})}\leq|F_{x}|e^{\sum_{i=0}^{m-1}f(T^{i}x)}e^{(mL(\epsilon)+8||f||\delta
m)}.\end{split}$

Note that

$\begin{split}|F_{x}|&\leq\sum_{(1-3\delta)(n+m)<k\leq
n+m}(C_{n+m}^{k})^{2}l^{n+m-k}\\\ &\leq
3\delta(n+m)(C_{n+m}^{[3\delta(n+m)]})^{2}l^{3\delta(n+m)}.\end{split}$

We have

$\begin{split}Cov_{m,\epsilon}(\alpha,\delta)&\leq
3\delta(n+m)(C_{n+m}^{[3\delta(n+m)]})^{2}l^{3\delta(n+m)}\\\ &\cdot
e^{(mL(\epsilon)+8||f||\delta m)}\sum_{x\in F}e^{\sum_{i=0}^{m-1}f(T^{i}x)}\\\
&\leq 3\delta(n+m)(C_{n+m}^{[3\delta(n+m)]})^{2}l^{3\delta(n+m)}\sum_{x\in
F}e^{\sum_{i=0}^{m+n-1}f(T^{i}x)}\\\ &\cdot e^{(mL(\epsilon)+8||f||\delta
m)+n||f||}.\end{split}$

Hence

$\begin{split}&\inf_{m}\inf_{\alpha>0}\frac{1}{m}\log
Cov_{m,\epsilon}(\alpha,\delta)\\\
&\leq(L(\epsilon)+8||f||\delta)+\lim_{m\to\infty}\frac{1}{m}\log(C_{n+m}^{[3\delta(n+m)]})^{2}+3\delta\log
l+\varlimsup_{m\to\infty}\frac{1}{m}\log sp_{m,\frac{\delta}{4}}.\end{split}$

It is easy to see that for all $m,n\in\mathbb{N}$ and $\beta>0$, we have

$Cov_{m+n,\epsilon}(\beta,\delta)\leq
Cov_{m,\epsilon}(\beta,\delta)Cov_{n,\epsilon}(\beta,\delta).$

Hence

$\lim_{n\to\infty}\frac{1}{n}\log
Cov_{n,\epsilon}(\beta,\delta)=\inf_{n}\frac{1}{n}\log
Cov_{n,\epsilon}(\beta,\delta).$

It is easy to see that

$FKPOsr_{m,2\epsilon;\beta,\delta}\leq Cov_{m,\epsilon}(\beta,\delta).$

Hence

$\begin{split}\varlimsup_{\delta\to
0}\inf_{\alpha>0}\varlimsup_{m\to\infty}\frac{1}{m}\log
FKPOsr_{m,2\epsilon;\alpha,\delta}&\leq\varlimsup_{\delta\to
0}\inf_{\alpha>0}\varlimsup_{m\to\infty}\frac{1}{m}\log
Cov_{m,\epsilon}(\alpha,\delta)\\\ &\leq\varlimsup_{\delta\to
0}\inf_{\alpha>0}\inf_{m}\frac{1}{m}\log Cov_{m,\epsilon}(\alpha,\delta)\\\
&\leq\varlimsup_{\delta\to 0}\varlimsup_{m\to\infty}\frac{1}{m}\log
sp_{m,\frac{\delta}{4}}+L(\epsilon)\\\ &=P(X,T,f)+L(\epsilon).\end{split}$

Let $\epsilon\to 0,$ we have $PFKPO(X,T,f)\leq P(X,T,f).$ ∎

######  Remark 2.8.

  1. (1)

It is easy to see that

$\begin{split}P(X,T,f)&\leq\lim_{\epsilon\to
0}\inf_{\delta>0}\inf_{\alpha>0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
FKPOsp_{n,\epsilon;\alpha,\delta}\\\ &\leq\lim_{\epsilon\to
0}\varlimsup_{\delta\to
0}\inf_{\alpha>0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
FKPOsp_{n,\epsilon;\alpha,\delta},\end{split}$

we can easily establish a similar result if we replace
“$\varlimsup\limits_{\delta\to 0}$” by “$\inf\limits_{\delta>0}$” in the
definition of $PFKPO(X,T,f).$

  2. (2)

The set of periodic Feldman-Katok pseudo-orbits contains the set of periodic
pseudo-orbits and is contained in the set of Feldman-Katok pseudo-orbits,
hence the result with respect to periodic Feldman-Katok pseudo-orbits can be
directly obtained.

###  2.3. Pseudo-orbits and scaled pressure

In [23], the authors studied scaled pressure with respect to pseudo-orbits. In
this subsection we show that in [23, Theorem F], the condition that $T$ is
Lipschitz continuous is redundant.

First we recall some concepts.

A function $S:(0,1)\to(0,\infty)$ is called a scale function if for any
$\lambda\in(0,\infty),$

$\lim\limits_{x\to 0}\frac{S(\lambda x)}{S(x)}=1.$

Let $f\in C(X)$ and $S$ be a scale function, define

$Spa_{n,\epsilon}(X,T,f,S)=\inf\\{\sum_{x\in
F}e^{S(\epsilon)\sum_{i=0}^{n-1}f(T^{i}x)}:F\text{ is\ an
}(n,\epsilon)\text{-spanning\ set \ of }X\\},$
$Sep_{n,\epsilon}(X,T,f,S)=\sup\\{\sum_{x\in
E}e^{S(\epsilon)\sum_{i=0}^{n-1}f(T^{i}x)}:E\text{ is\ an
}(n,\epsilon)\text{-separated\ set \ of }X\\}.$

Scaled pressure of $(X,T)$ with respect to $f$ and $S$ is defined by

$\begin{split}Sdim(X,T,f,S)&=\varlimsup\limits_{\epsilon\to
0}\frac{\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
Spa_{n,\epsilon}}{S(\epsilon)}\\\ &=\varlimsup\limits_{\epsilon\to
0}\frac{\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
Sep_{n,\epsilon}}{S(\epsilon)}.\end{split}$

Denote by

$PPO_{n,\alpha}(X,T)=\\{{\bf x}\in
PO_{\alpha}(X,T):x_{n+k}=x_{k},k\in\mathbb{N}\\}$

the $n$-period $\alpha$ pseudo-orbits.

Define

$\begin{split}&POSpa_{n,\epsilon,\alpha}(X,T,f,S)(resp.\
PPOSpa_{n,\epsilon,\alpha}(X,T,f,S))=\\\ &\inf\\{\sum_{{\bf x}\in
F}e^{S(\epsilon)\sum_{i=0}^{n-1}f(x_{i})}:F\text{ is\ an
}(n,\epsilon)\text{-spanning\ set \ of }PO_{\alpha}(resp.\
PPO_{n,\alpha})\\},\end{split}$
$\begin{split}&POSep_{n,\epsilon,\alpha}(X,T,f,S)(resp.\
PPOSep_{n,\epsilon,\alpha}(X,T,f,S))=\\\ &\sup\\{\sum_{{\bf x}\in
E}e^{S(\epsilon)\sum_{i=0}^{n-1}f(x_{i})}:E\text{ is\ an
}(n,\epsilon)\text{-separated\ set \ of }PO_{\alpha}(resp.\
PPO_{n,\alpha})\\}.\end{split}$

Define scaled pressure with respect to pseudo-orbits (resp. periodic pseudo-
orbits) by

$\begin{split}&PSdim(X,T,f,S)(resp.\ PPSdim(X,T,f,S))\\\
&=\varlimsup\limits_{\epsilon\to 0}\frac{\lim\limits_{\alpha\to
0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
POSpa_{n,\epsilon,\alpha}(resp.\ PPOSpa_{n,\epsilon,\alpha})}{S(\epsilon)}\\\
&=\varlimsup\limits_{\epsilon\to 0}\frac{\lim\limits_{\alpha\to
0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
POSep_{n,\epsilon,\alpha}(resp.\
PPOSep_{n,\epsilon,\alpha})}{S(\epsilon)}.\end{split}$

Now we show that in [23, Theorem F], the condition that $T$ is Lipschitz
continuous is redundant.

######  Theorem 2.9.

Let $(X,T)$ be a TDS, $f\in C(X)$ and $S$ be a scale function. Then

$Sdim(X,T,f,S)=PSdim(X,T,f,S).$

If in addition $f\geq 0$ and $S$ is non-increasing, then

$Sdim(X,T,f,S)=PPSdim(X,T,f,S).$

###### Proof.

By definition we have $Sdim(X,T,f,S)\leq PSdim(X,T,f,S).$ Now we prove the
opposite direction.

Fix $\epsilon>0.$ Let $X=\bigcup_{i=1}^{l}X_{i}$ with
$diamX_{i}<\frac{\epsilon}{4},i=1,\ldots,l.$ Choose some $z_{i}\in X_{i}$.

For $n>\frac{2diamX}{\epsilon}$ and $m$, let $F$ be an
$(n+m,\frac{\epsilon}{4})$-spanning set of $X$. We can find

$0<\delta_{0}<\delta_{1}<\ldots<\delta_{2(n+m)}=\frac{\epsilon}{4}$

such that for every $\delta_{k},k=0,1,\ldots,2(n+m)-1$,

$d(x,y)<\delta_{k}\Rightarrow
d(T^{i}x,T^{i}y)<\delta_{k+1},i=0,\ldots,2(n+m)-1.$

Let

$\alpha<\min\\{\delta_{0},\delta_{k+1}-\delta_{k}:k=0,1,\ldots,2(n+m)-1\\}.$

Let

$\tilde{F}=\\{(x,Tx,\ldots,T^{i}x,\ldots,T^{m+n-1}x,z_{1},z_{1},\ldots):x\in
F\\}.$

We have $\bigcup_{{\bf y}\in\tilde{F}}B_{\tilde{d}_{m}}({\bf y},\epsilon)$
covers $PO_{\alpha}$. Denote

$\begin{split}&Cov_{n,\epsilon}(\beta)=\\\
&\inf\\{\sum_{A\in\mathcal{A}}\sup_{{\bf x}\in
A}e^{S(\epsilon)\sum_{i=0}^{n-1}f(x_{i})}:\mathcal{A}\text{ is\ a\ finite\
subcover\ of\ }\vee_{i=0}^{n-1}\sigma^{-i}\mathcal{A}_{\epsilon}\text{ covers
}PO_{\beta}\\},\end{split}$

where $\mathcal{A}_{\epsilon}$ denote the cover of
$(X^{\mathbb{N}},\tilde{d})$ consists of all open $\epsilon$-balls. We have

$Cov_{m,\epsilon}(\alpha)\leq\sum_{{\bf y}\in\tilde{F}}\sup_{{\bf z}\in
B_{\tilde{d}_{m}}({\bf y},\epsilon)}e^{S(\epsilon)\sum_{i=0}^{m-1}f(z_{i})}.$

Here $\alpha$ dependents on $m$.

For $x\in F,{\bf
y}=(x,Tx,\ldots,T^{i}x,\ldots,T^{m+n-1}x,z_{1},z_{1},\ldots)\in\tilde{F}$ and
${\bf z}\in B_{\tilde{d}_{m}}({\bf y},\epsilon)$, we have
$d(T^{i}x,z_{i})<\epsilon,i=0,\ldots,m-1.$ It follows that

$\sum_{i=0}^{m-1}f(z_{i})-\sum_{i=0}^{m-1}f(T^{i}x)\leq mL(\epsilon),$

where

$L(\epsilon)=\sup_{|x-y|\leq\epsilon}|f(x)-f(y)|.$

Hence

$\begin{split}\sum_{{\bf y}\in\tilde{F}}\sup_{{\bf z}\in
B_{\tilde{d}_{m}}({\bf
y},\epsilon)}e^{S(\epsilon)\sum_{i=0}^{m-1}f(z_{i})}\leq\sum_{x\in
F}e^{S(\epsilon)\sum_{i=0}^{m-1}f(T^{i}x)}e^{S(\epsilon)mL(\epsilon)}.\end{split}$

We have

$\begin{split}Cov_{m,\epsilon}(\alpha)&\leq
e^{S(\epsilon)mL(\epsilon)}\sum_{x\in
F}e^{S(\epsilon)\sum_{i=0}^{m-1}f(T^{i}x)}\\\ &\leq
e^{S(\epsilon)(mL(\epsilon)+n||f||)}\sum_{x\in
F}e^{S(\epsilon)\sum_{i=0}^{m+n-1}f(T^{i}x)}.\end{split}$

Hence

$\begin{split}\inf_{m}\inf_{\alpha>0}\frac{1}{m}\log
Cov_{m,\epsilon}(\alpha)\leq
S(\epsilon)L(\epsilon)+\varlimsup_{m\to\infty}\frac{1}{m}\log
Spa_{m,\frac{\epsilon}{4}}.\end{split}$

It is easy to see that

$\lim_{n\to\infty}\frac{1}{n}\log
Cov_{n,\epsilon}(\beta)=\inf_{n}\frac{1}{n}\log Cov_{n,\epsilon}(\beta)$

and

$POSep_{m,2\epsilon,\beta}\leq Cov_{m,\epsilon}(\beta).$

Hence

$\begin{split}\inf_{\alpha>0}\varlimsup_{m\to\infty}\frac{1}{m}\log
POSep_{m,2\epsilon,\alpha}&\leq\inf_{\alpha>0}\varlimsup_{m\to\infty}\frac{1}{m}\log
Cov_{m,\epsilon}(\alpha)\\\ &\leq\inf_{\alpha>0}\inf_{m}\frac{1}{m}\log
Cov_{m,\epsilon}(\alpha)\\\ &\leq\varlimsup_{m\to\infty}\frac{1}{m}\log
Spa_{m,\frac{\epsilon}{4}}+S(\epsilon)L(\epsilon).\end{split}$

It follows that

$\frac{\inf\limits_{\alpha>0}\varlimsup\limits_{m\to\infty}\frac{1}{m}\log
POSep_{m,2\epsilon,\alpha}}{S(2\epsilon)}\leq\frac{S(\frac{\epsilon}{4})}{S(2\epsilon)}\frac{\varlimsup\limits_{m\to\infty}\frac{1}{m}\log
Spa_{m,\frac{\epsilon}{4}}}{S(\frac{\epsilon}{4})}+\frac{S(\epsilon)}{S(2\epsilon)}L(\epsilon).$

Let $\epsilon\to 0,$ we have $Sdim(X,T,f,S)\geq PSdim(X,T,f,S).$

If in addition $f\geq 0$ and $S$ is non-increasing, then

$Sdim(X,T,f,S)=PPSdim(X,T,f,S)$

just follows from the discussion in [23, Theorem F]. ∎

##  3\. Topological pressure for Feldman-Katok metric

In this section we study topological pressure for Feldman-Katok metric.

First we recall the definition of Feldman-Katok metric.

Let $(X,T)$ be a TDS, $n\in\mathbb{N}$ and $\delta>0$. For $x,y\in X$, we
define an $(n,\delta)$-match of $x$ and $y$ to be an order preserving (i.e.
$\pi(i)<\pi(j)$ whenever $i<j$) bijection $\pi:D(\pi)\rightarrow R(\pi)$ such
that $D(\pi),R(\pi)\subset\\{0,1,\ldots,n-1\\}$ and for every $i\in D(\pi)$ we
have $d(T^{i}x,T^{\pi(i)}y)<\delta$. Let $|\pi|$ be the cardinality of
$D(\pi)$. We set

$\bar{f}_{n,\delta}(x,y)=1-\frac{\max\\{|\pi|:\pi\text{ is an
}(n,\delta)\text{-match of }x\text{ and }y\\}}{n}$

Define the Feldman-Katok metric by

$d_{FK_{n}}(x,y)=\inf\\{\delta>0:\bar{f}_{n,\delta}(x,y)<\delta\\}.$

For $n\in\mathbb{N}$ and $\epsilon>0$, We say a subset $E\subset X$ is a
$FK$-$(n,\epsilon)$-spanning set of $X$ if for any $x\in X$, there exists
$y\in E$ with $d_{FK_{n}}(x,y)\leq\epsilon.$ We say a subset $F\subset X$ is a
$FK$-$(n,\epsilon)$-separated set of $X$ if for $x\neq y\in F,$
$d_{FK_{n}}(x,y)>\epsilon.$

For $f\in C(X)$, define

$FKsp_{n,\epsilon}(X,T,f)=\inf\\{\sum_{x\in
F}e^{\sum_{i=0}^{n-1}f(T^{i}x)}:F\text{\ is\ a\
}FK\text{-}(n,\epsilon)\text{-spanning\ set\ of}\ X\\},$
$FKsr_{n,\epsilon}(X,T,f)=\sup\\{\sum_{x\in
E}e^{\sum_{i=0}^{n-1}f(T^{i}x)}:E\text{\ is\ a\
}FK\text{-}(n,\epsilon)\text{-separated\ set\ of}\ X\\}.$

Define the topological pressure with respect to Feldman-Katok metric by

$PFK(X,T,f)=\lim_{\epsilon\to 0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
FKsp_{n,\epsilon}(X,T,f).$

Similar as Lemma 2.6, we can use separated set in the definition above.

######  Lemma 3.1.

Let $(X,T)$ be a TDS and $f\in C(X).$ Then

$\begin{split}PFK(X,T,f)=\lim_{\epsilon\to
0}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
FKsr_{n,\epsilon}(X,T,f).\end{split}$

Now we prove the topological pressure with respect to Feldman-Katok metric is
equal to the classical topological pressure.

######  Theorem 3.2.

Let $(X,T)$ be a TDS and $f\in C(X)$. Then

$PFK(X,T,f)=P(X,T,f).$

###### Proof.

It is easy to see that $PFK(X,T,f)\leq P(X,T,f)$, we prove the opposite
direction.

Fix $\epsilon>0.$ Let $X=\cup_{i=1}^{l}X_{i}$ with
$diamX_{i}<\epsilon,i=1,\ldots,l.$ For $\delta\leq\epsilon$, let $F$ be a
$FK$-$(n,\frac{\delta}{2})$-spanning set of $X$. Let $x\in F$ and consider
sets with following form

$A_{0}\cap T^{-1}A_{1}\cap\ldots\cap T^{-(j_{1}-1)}A_{j_{1}-1}\cap
T^{-j_{1}}B_{\frac{\delta}{2}}(T^{i_{1}}x)\cap
T^{-(j_{1}+1)}A_{j_{1}+1}\cap\ldots$ $\cap T^{-(j_{k}-1)}A_{j_{k}-1}\cap
T^{-j_{k}}B_{\frac{\delta}{2}}(T^{i_{k}}x)\cap
T^{-(j_{k}+1)}A_{j_{k}+1}\ldots\cap T^{-(n-1)}A_{n-1},$

where

$(1-\frac{\delta}{2})n<k\leq n,$ $0\leq i_{1}<\ldots<i_{k}\leq n-1,$ $0\leq
j_{1}<\ldots<j_{k}\leq n-1$

and $A_{i}\in\\{X_{1},\ldots,X_{l}\\}.$

For every non-empty set with the form above, choose some point $y$ in it. Let
$F^{\prime}_{x}$ be the set consists of such $y$ and let
$F^{\prime}=\cup_{x\in F}F^{\prime}_{x}.$ We will prove $F^{\prime}$ is an
$(n,\epsilon)$-spanning set. For every $z\in X$, since $F$ is a
$FK$-$(n,\frac{\delta}{2})$-spanning set, there is some $x\in F$ with
$d_{FK_{n}}(x,z)<\frac{\delta}{2}$. Hence there exist

$(1-\frac{\delta}{2})n<k\leq n,$ $0\leq i_{1}<\ldots<i_{k}\leq n-1,$ $0\leq
j_{1}<\ldots<j_{k}\leq n-1$

such that

$d(T^{i_{h}}x,T^{j_{h}}z)<\frac{\delta}{2},h=1,\ldots,k.$

Assume $T^{i}z\in X_{t_{i}},i=0,\ldots,n-1.$ Then

$X_{t_{0}}\cap\ldots\cap T^{-(j_{1}-1)}X_{t_{j_{1}-1}}\cap
T^{-j_{1}}B_{\frac{\delta}{2}}(T^{i_{1}}x)\cap
T^{-(j_{1}+1)}X_{t_{j_{1}+1}}\ldots\cap T^{-(n-1)}X_{t_{n-1}}\neq\emptyset$

since $z$ is in it. By the construction of $F^{\prime}$, there is $y\in
F^{\prime}$ in the set above. It is easy to see that $d_{n}(y,z)<\epsilon.$
Hence $F^{\prime}$ is an $(n,\epsilon)$-spanning set.

We have

$\begin{split}sp_{n,\epsilon}(X,T,f)&\leq\sum_{y\in
F^{\prime}}e^{\sum_{i=0}^{n-1}f(T^{i}y)}\leq\sum_{x\in F}\sum_{y\in
F^{\prime}_{x}}e^{\sum_{i=0}^{n-1}f(T^{i}y)}\\\ &\leq\sum_{x\in F}\sum_{y\in
F^{\prime}_{x}}e^{\sum_{i=0}^{n-1}f(T^{i}x)+\sum_{i=0}^{n-1}f(T^{i}y)-\sum_{i=0}^{n-1}f(T^{i}x)}\\\
&\leq\sum_{x\in F}\sum_{y\in
F^{\prime}_{x}}e^{\sum_{i=0}^{n-1}f(T^{i}x)}e^{nL(\frac{\delta}{2})+n\delta||f||},\end{split}$

where $L(\epsilon)=\sup\limits_{|x-y|\leq\epsilon}|f(x)-f(y)|.$ Since

$\begin{split}|F^{\prime}_{x}|\leq\sum_{(1-\frac{\delta}{2})n<k\leq
n}(C^{k}_{n})^{2}l^{n-k}\leq\frac{\delta}{2}n(C^{[\frac{\delta}{2}n]}_{n})^{2}l^{\frac{\delta}{2}n},\end{split}$

we have

$sp_{n,\epsilon}\leq
e^{(nL(\frac{\delta}{2})+n\delta||f||)}\frac{\delta}{2}n(C^{[\frac{\delta}{2}n]}_{n})^{2}l^{\frac{\delta}{2}n}\sum_{x\in
F}e^{\sum_{i=0}^{n-1}f(T^{i}x)}.$

Hence we have

$sp_{n,\epsilon}\leq
e^{(nL(\frac{\delta}{2})+n\delta||f||)}\frac{\delta}{2}n(C^{[\frac{\delta}{2}n]}_{n})^{2}l^{\frac{\delta}{2}n}FKsp_{n,\frac{\delta}{2}}.$

It follows that

$\begin{split}\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
sp_{n,\epsilon}\leq L(\frac{\delta}{2})+\delta||f||+\frac{\delta}{2}\log
l+\lim\limits_{n\to\infty}\frac{1}{n}2\log
C_{n}^{[\frac{\delta}{2}n]}+\varlimsup\limits_{n\to\infty}\frac{1}{n}\log
FKsp_{n,\frac{\delta}{2}}.\end{split}$

Let $\delta\to 0$, we have

$\varlimsup\limits_{n\to\infty}\frac{1}{n}\log sp_{n,\epsilon}\leq
PFK(X,T,f).$

Let $\epsilon\to 0$, we get $P(X,T,f)\leq PFK(X,T,f).$ ∎

######  Remark 3.3.

Both Feldman-Katok pseudo-orbits and Feldman-Katok metric allow appropriate
time delay and error when computing topological pressure. Feldman-Katok
pseudo-orbits contain pseudo-orbits hence contain real orbits, it approximate
topological pressure from upper bound. On the other hand, Feldman-Katok metric
is less than Bowen metric, it approximate topological pressure from lower
bound.

## References

  * [1] R. Adler, A. Konheim, M. McAndrew,  Topological entropy.  Trans. Amer. Math. Soc., 114, (1965), 309–319.
  * [2] E. Dinaburg,  On the relations among various entropy characteristic of dynamical systems.  Izv. Akad. Nauk SSSR, Ser. Mat., 35, (1971), 324–366.
  * [3] R. Bowen,  Entropy for group endomorphisms and homogeneous spaces.  Trans. Amer. Math. Soc., 153, (1971), 401–414.
  * [4] R. Bowen, Topological entropy for noncompact sets. Trans. Amer. Math. Soc., 184, (1973), 125–136.
  * [5] R. Bowen, Equilibrium States and the Ergodic Theory of Axiom A Diffeomorphisms. Lecture Notes in Math., vol. 470, Springer-Verlag, New York, 1975.
  * [6] R. Bowen, Hausdorff dimension of quasicircles. Publ. Math. IHÉS., 50, (1979), 11–25.
  * [7] C. Conley, Isolated Invariant Sets and the Morse Index. CBMS Conference Series, vol. 38, 1978.
  * [8] M. Misiurewicz, Remark on the definition of topological entropy. in: Dynamical Systems and Partial Differential Equations, 1986, pp. 65–68.
  * [9] M. Barge, R. Swanson, Pseudo-orbits and topological entropy. Proc. Amer. Math. Soc., 109, (1990), 559–566.
  * [10] M. Hurley, On topological entropy of maps. Ergod. Theory Dyn. Syst., 15, (1995), 557–568.
  * [11] K. Yan, F. Zeng, Topological entropy, pseudo-orbits and uniform spaces. Topology Appl. 210, (2016), 168–182.
  * [12] J. Feldman, New K-automorphisms and a problem of Kakutani. Israel J. Math., 24 (1), 16–38, 1976.
  * [13] A. Katok, Monotone equivalence in ergodic theory. Izv. Akad. Nauk SSSR Ser. Mat., 41 (1), 104–157, 231, 1977.
  * [14] D. Ornstein, D. Rudolph, B. Weiss, Equivalence of measure preserving transformations. Mem. Amer. Math. Soc., 37, (262), 1982.
  * [15] T. Downarowicz, D. Kwietniak, M.Łacka, Uniform continuity of entropy rate with respect to the $\bar{f}$-pseudometric. IEEE Trans. Inform. Theory, 67, (2021), no. 11, 7010–7018.
  * [16] F. Cai, J. Li, On Feldman-Katok metric and entropy formulae. arXiv: 2104.12104.
  * [17] Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.
  * [18] Y. Pesin, B. Pitskel, Topological pressure and the variational principle for noncompact sets. Funct. Anal. Appl., 18(4), (1984), 307–318.
  * [19] D. Ruelle, Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Encyclopedia of Mathematics and Its Applications, vol.5, Addison–Wesley Publishing Co., Reading, Mass., 1978.
  * [20] D. Ruelle, Repellers for real analytic maps. Ergod. Theory Dyn. Syst., 2, (1982), 99–107.
  * [21] D. Ruelle,  Statistical mechanics on a compact set with $Z^{v}$ action satisfying expansiveness and specification.  Trans. Amer. Math. Soc., 187, (1973), 237–251.
  * [22] P. Walters, A variational principle for the pressure of continuous transformations. Am. J. Math., 97, (1975), 937–971.
  * [23] D. Cheng, Z. Li,  Scaled pressure of dynamical systems.  J. Differential Equations., 342, (2023), 441–471.
  * [24] D. Kwietniak, M. Łacka,  Feldman-Katok pseudometric and the GIKN construction of nonhyperbolic ergodic measures.  arXiv: 1702.01962.
  * [25] F. García-Ramos, D. Kwietniak,  On topological models of zero entropy loosely Bernoulli systems.  Trans. Amer. Math. Soc., 375, (2022), no. 9, 6155–6178.


# Quasimode concentration on compact space forms

Xiaoqi Huang  Department of Mathematics, Louisiana State University, Baton
Rouge, LA 70803<EMAIL_ADDRESS>and  Christopher D. Sogge  Department of
Mathematics, Johns Hopkins University, Baltimore, MD 21218<EMAIL_ADDRESS>
###### Abstract.

We show that the upper bounds for the $L^{2}$-norms of $L^{1}$-normalized
quasimodes that we obtained in [9] are always sharp on any compact space form.
This allows us to characterize compact manifolds of constant sectional
curvature using the decay rates of lower bounds of $L^{1}$-norms of
$L^{2}$-normalized log-quasimodes fully resolving a problem initiated by the
second author and Zelditch [15]. We are also able to characterize such
manifolds by the concentration of quasimodes near periodic geodesics as
measured by $L^{2}$-norms over thin geodesic tubes.

###### Key words and phrases:

Eigenfunctions, quasimodes, curvature, space forms

###### 2010 Mathematics Subject Classification:

58J50, 35P15

Both authors were supported in part by the Simons Foundation, and the second
author was supported in part by the NSF (DMS-1665373).

In memoriam: Steve Zelditch (1953-2022)

##  1\. Introduction and main results

There are many ways of measuring the concentration of eigenfunctions and
quasimodes on compact manifolds $(M,g)$ some of which may be sensitive to the
underlying geometry, such as curvature assumptions. See, e.g., [13]. A
standard way is through the growth rate of the $L^{q}(M)$ norms of
$L^{p}$-normalized modes. Since the manifold is compact this is only of
interest when $q>p$. Another method, which has been studied by several authors
over the last decade or so, starting with the second author and Zelditch [16],
is through $L^{2}$-norms over small sets, especially geodesic tubes.

Typically one works with $L^{2}$-normalized modes and so, for the first way of
measurement, one often takes $p=2$. There now are sharp estimates in this case
for log-quasimodes through the works of Bérard [1], Hassell and Tacy [8] and
the authors [9].

To describe these results, if $\delta(\lambda)\in(0,1]$ let

(1.1) $V_{[\lambda,\lambda+\delta(\lambda)]}=\\{\Phi_{\lambda}:\,\text{Spec
}\Phi_{\lambda}\subset[\lambda,\lambda+\delta(\lambda)]\\},$

be the space of $\delta(\lambda)$-quasimodes, where Spec  refers to the
spectrum of the first order operator

$P=\sqrt{-\Delta_{g}},$

with $\Delta_{g}$ being the Laplace-Beltrami operator associated to the metric
$g$. Abusing notation a bit, we shall say that $\Phi_{\lambda}$ is a log-
quasimode if $\delta(\lambda)=(\log\lambda)^{-1}$. Also, we typically assume
that $\delta(\lambda)\searrow 0$ and also that
$\lambda\to\delta(\lambda)\cdot\lambda$ is non-decreasing.

Also recall that if

(1.2) $q_{c}=\tfrac{2(n+1)}{n-1}$

and

(1.3)
$\mu(q)=\begin{cases}n(\tfrac{1}{2}-\tfrac{1}{q})-\tfrac{1}{2},\,\,\,q>q_{c}\\\
\tfrac{n-1}{2}(\tfrac{1}{2}-\tfrac{1}{q}),\,\,\,2<q\leq q_{c},\end{cases}$

then the second author showed in [11] that one always has the universal
estimates for unit-band modes saying that for $\lambda\geq 2$

(1.4)
$\|\Phi_{\lambda}\|_{L^{q}(M)}\lesssim\lambda^{\mu(q)}\|\Phi_{\lambda}\|_{L^{2}(M)},\quad\text{if
}\,\,\Phi_{\lambda}\in V_{[\lambda,\lambda+1]}.$

These estimates are always optimal (see [14]) and they also are saturated by
eigenfunctions on the round sphere, $S^{n}$, (see [10]). On the other hand,
under certain curvature assumptions, one has improvements for
$\delta(\lambda)$-quasimodes if $\delta(\lambda)\searrow 0$.

For relatively large exponents, if $\mu(q)$ is as in (1.3) one has for
$\lambda\geq 2$

(1.5) $\|\Phi_{\lambda}\|_{q}\leq
C_{q}\lambda^{\mu(q)}\sqrt{\delta(\lambda)}\|\Phi_{\lambda}\|_{2},\,\,\text{for
}\,q>q_{c}\,\,\text{if }\,\Phi_{\lambda}\in
V_{[\lambda,\lambda+\delta(\lambda)]},\,\,\delta(\lambda)\in[(\log\lambda)^{-1},1],\\\
\text{and if all the sectional curvatures of }\,M\,\,\text{are nonpositive}.$

This result is due to Bérard [1] for $q=\infty$ and to Hassell and Tacy [8]
for all other exponents $q>q_{c}$. It is optimal in the sense that
$o(\lambda^{\mu(q)}\sqrt{\delta(\lambda)})$ bounds are not possible by an easy
argument if $\delta(\lambda)\searrow 0$. On the other hand, one expects the
bounds in (1.5) to hold for $\delta(\lambda)<(\log\lambda)^{-1}$, and this is
the case, for instance, for tori (see e.g. [4]). The estimates in (1.5) are
saturated by quasimodes concentrating near points, which accounts for the fact
that they involve relatively large exponents. Also, these bounds do not
distinguish between manifolds of strictly negative sectional curvatures from
flat ones.

Recent work of the authors [9] treats the complementary range $q\in(2,q_{c}]$
of relatively small exponents. In this case the estimates that were obtained
are sensitive to the sign of the curvature and are saturated by quasimodes
concentrating near periodic geodesics on compact space forms (manifolds of
constant sectional curvature). Specifically, in [9] it was shown that

(1.6) $\|\Phi_{\lambda}\|_{q}\leq
C(\lambda\delta(\lambda))^{\mu(q)}\|\Phi_{\lambda}\|_{2},\,\,\text{for
}\,q\in(2,q_{c}],\\\ \text{if all the sectional curvatures of
}\,M\,\,\text{are nonpositive},$

and, moreover,

(1.7) $\|\Phi_{\lambda}\|_{q}\leq
C_{q}\lambda^{\mu(q)}\sqrt{\delta(\lambda)}\|\Phi_{\lambda}\|_{2},\,\,\text{for
}\,q\in(2,q_{c}],\\\ \text{if all the sectional curvatures of
}\,M\,\,\text{are negative},$

assuming, as in (1.5)

(1.8) $\Phi_{\lambda}\in V_{[\lambda,\lambda+\delta(\lambda)]}\quad\text{with
}\,\,\delta(\lambda)\in[(\log\lambda)^{-1},1].$

Note that the bounds in (1.7) are stronger than those in (1.6) since
$\mu(q)<1/2$ for $q\in(2,q_{c}]$.

The authors in [9] were also able to show that these estimates are always
sharp on compact space forms, which provided a classification of compact
manifolds of constant sectional curvature $K$ in terms of the sign of the
curvature and the growth rate of log-quasimodes. Specifically, it was shown
that for $q\in(2,q_{c}]$

(1.9) $\sup\bigl{\\{}\|\Phi_{\lambda}\|_{L^{q}(M)}:\,\Phi_{\lambda}\in
V_{[\lambda,\lambda+(\log)^{-1}]},\,\|\Phi_{\lambda}\|_{L^{2}(M)}=1\bigr{\\}}\\\
=\begin{cases}\Theta(\lambda^{\mu(q)}(\log\lambda)^{-1/2})\,\,\iff\,\,K<0\\\
\Theta(\lambda^{\mu(q)}(\log\lambda)^{-\mu(q)})\,\,\iff\,\,K=0\\\
\Theta(\lambda^{\mu(q)})\,\,\iff\,\,K>0,\end{cases}$

if $(M,g)$ is a compact space form all of whose sectional curvatures equal
$K$. Here we are taking the left side of (1.9) to be zero if
$V_{[\lambda,\lambda+(\log)^{-1}]}=\emptyset$. It was also shown that
analogous results are valid if $(\log\lambda)^{-1}$ is replaced with
$\delta(\lambda)\in(0,1]$ satisfying $\delta(\lambda)\searrow 0$. Recall that
if $f,g$ are nonnegative then we say that $f(\lambda)=\Theta(g(\lambda))$ if
$f(\lambda)=O(g(\lambda))$ and also $f(\lambda)=\Omega(g(\lambda))$, with the
latter being the negation of $f(\lambda)=o(g(\lambda))$.

The $O$-bounds which are implied by (1.9) follow from (1.4) ($K>0$), (1.6)
($K=0$) and (1.7) ($K<0$). The $\Omega$-bounds for $K<0$ are elementary, and
for $K>0$ they just follow from the fact that there are gaps of length one in
the spectrum of $\sqrt{-\Delta_{S^{n}}}$. The flat case, $K=0$, was more
difficult to handle. It was done in a somewhat circuitous manner by a Knapp-
type construction, which was reminiscent of arguments in Brooks [5] and Sogge
and Zeldtich [16]. We shall give more direct arguments that can be used to
prove the $\Omega$-bounds in (1.9) when $K>0$ or $K=0$ that we shall use to
obtain sharp lower bounds of $L^{1}$-norms of $L^{2}$-normalized spectrally
localized log-quasimodes on flat or positively curved compact space forms.

Specifically, these new Knapp-type constructions will be the main step in
proving the following results which provide another way of characterizing
compact space forms.

######  Theorem 1.1.

Let $(M,g)$ be an $n$-dimensional compact Riemannian manifold and assume that
$\delta(\lambda)\searrow 0$ satisfies
$\delta(\lambda)\in[\log\lambda)^{-1},1]$ and
$\lambda\to\lambda\cdot\delta(\lambda)$ is nondecreasing. Then for
$\lambda\geq 2$

(1.10) $\sup\bigl{\\{}\|\Phi_{\lambda}\|_{L^{2}(M)}:\,\Phi_{\lambda}\in
V_{[\lambda,\lambda+\delta(\lambda)]},\,\|\Phi_{\lambda}\|_{L^{1}(M)}=1\bigr{\\}}\\\
=\begin{cases}O(\lambda^{\frac{n-1}{4}}(\delta(\lambda))^{N})\,\forall\,N,\,\,\text{if
all the sectional curvatures of }M\,\text{are negative}\\\
O(\lambda^{\frac{n-1}{4}}(\delta(\lambda))^{\frac{n-1}{4}})\,\,\text{if all
the sectional curvatures of }M\,\text{are nonpositive}\\\
O(\lambda^{\frac{n-1}{2}})\,\,\text{for any }\,M.\end{cases}$

Moreover, if $(M,g)$ is a compact space form with sectional curvatures equal
to $K$ then

(1.11) $\sup\bigl{\\{}\|\Phi_{\lambda}\|_{L^{2}(M)}:\,\Phi_{\lambda}\in
V_{[\lambda,\lambda+\delta(\lambda)]},\,\|\Phi_{\lambda}\|_{L^{1}(M)}=1\bigr{\\}}\\\
=\begin{cases}O(\lambda^{\frac{n-1}{4}}(\delta(\lambda))^{N})\,\forall
N\,\iff\,K<0\\\
\Theta(\lambda^{\frac{n-1}{4}}(\delta(\lambda))^{\frac{n-1}{4}})\,\iff\,K=0\\\
\Theta(\lambda^{\frac{n-1}{4}})\,\iff\,K>0.\end{cases}$

Earlier we mentioned the problem of detecting geometric properties of compact
space forms using the growth rate of $L^{q}$-norms of $L^{p}$ normalized
quasimodes. In our earlier work [9] we showed that this was possible for $p=2$
provided that, as in (1.9), $q$ belongs to the intervals
$(2,\tfrac{2(n+1)}{n-1}]$ which shrink to the empty set as $n\to\infty$. On
the other hand, we note that (1.11) allows us to detect the sign of the
curvature of a compact space form using the same pair $(p,q)=(1,2)$ in all
dimensions. As we shall see, this pair of Lebesgue exponents captures very
different types of concentration near periodic geodesics for the three
geometries.

The upper bounds (1.10) follow from Hölder’s inequality along with (1.4),
(1.6) and (1.7), as was shown in [9] and [15]. For the sake of completeness,
though, let us present the simple argument here.

Suppose that $0\neq\Phi_{\lambda}\in V_{[\lambda,\lambda+\delta(\lambda)]}$.
Then by Hölder’s inequality

$\|\Phi_{\lambda}\|_{2}\leq\|\Phi_{\lambda}\|_{1}^{\theta_{q}}\,\|\Phi_{\lambda}\|_{q}^{1-\theta_{q}},\quad\theta_{q}=\tfrac{q-2}{2(q-1)}.$

Thus, by (1.7), if all of the sectional curvatures of $(M,g)$ are negative

$\displaystyle\|\Phi_{\lambda}\|_{2}$
$\displaystyle\lesssim\|\Phi_{\lambda}\|_{1}^{\theta_{q}}\,\bigl{(}\,\lambda^{\frac{n-1}{2}(\frac{1}{2}-\frac{1}{q})}\,(\delta(\lambda))^{\frac{1}{2}}\|\Phi_{\lambda}\|_{2}\,\bigr{)}^{1-\theta_{q}}$
$\displaystyle=\bigl{(}\lambda^{\frac{n-1}{4}}\|\Phi_{\lambda}\|_{1}\bigr{)}^{\theta_{q}}\,(\delta(\lambda))^{\frac{(1-\theta_{q})}{2}}\|\Phi_{\lambda}\|_{2}^{1-\theta_{q}},$

since

$(1-\theta_{q})\cdot\tfrac{n-1}{2}(\tfrac{1}{2}-\tfrac{1}{q})=\tfrac{q}{2(q-1)}\cdot\tfrac{n-1}{4}(\tfrac{q-2}{q})=\tfrac{n-1}{4}\cdot\tfrac{q-2}{2(q-1)}=\tfrac{n-1}{4}\theta_{q}.$

Thus, if all the sectional curvatures of $(M,g)$ are negative, we have for
$q\in(2,q_{c}]$

$\|\Phi_{\lambda}\|_{2}\lesssim\lambda^{\frac{n-1}{4}}\,(\delta(\lambda))^{\frac{(1-\theta_{q})}{2\theta_{q}}}\,\|\Phi_{\lambda}\|_{1},$

leading to the first part of (1.11) since $\theta_{q}\searrow 0$ as $q\searrow
2$.

We obtain the second part of (1.10) from this argument if we use (1.6). The
last inequality in (1.10), which is due to Sogge and Zelditch [15], similarly
follows from the universal bounds (1.4).

Since we have established (1.10), in order to complete the proof of Theorem
1.1, it suffices to prove the $\Omega$-lower bounds for $K=0$ and $K>0$, which
are implicit in (1.11). We cannot use the simpler arguments from our earlier
work [9] that were used to prove the $\Omega$-bounds implicit in (1.9).
Instead, we shall have to construct appropriate Knapp-type spectrally
localized quasimodes for these two different types of constant curvature
geometries yielding the missing lower bounds. In the next section, we shall
give the constructions for compact space forms of positive curvature, while in
§3, we shall use somewhat more involved arguments to handle flat compact
manifolds. We shall use classical results about the structure of such space
forms that can be found, for instance, in Charlap [6] and Wolf [18].

We have defined here quasimodes as in (1.1) in terms of the spectrum of the
functions involved. Another common (but weaker) definition is to require that

(1.12)
$\|\Phi\|_{2}+(\lambda\delta(\lambda))^{-1}\|(\Delta_{g}+\lambda^{2})\Phi\|_{2}\leq
1.$

Clearly if $\Phi_{\lambda}\in V_{[\lambda,\lambda+\delta(\lambda)]}$ satisfies
$\|\Phi_{\lambda}\|_{2}\leq 1/4$, then $\Phi_{\lambda}$ satisfies the
condition in (1.12). Also, it is a simple exercise (see, e.g. [17]) to see
that the bounds in (1.4), (1.6) and (1.7) yield

(1.13)
$\|\Phi\|_{q}\lesssim\lambda^{\mu(q)}\bigl{(}\,\|\Phi\|_{2}+\lambda^{-1}\|(\Delta_{g}+\lambda^{2})\Phi\|_{2}\,\bigr{)},\,\,q\in(2,q_{c}],\quad\text{any
}\,M,$ (1.14)
$\|\Phi\|_{q}\lesssim(\lambda/\log\lambda)^{\mu(q)}\bigl{(}\,\|\Phi\|_{2}+(\lambda/\log\lambda)^{-1}\|(\Delta_{g}+\lambda^{2})\Phi\|_{2}\,\bigr{)},\,\,q\in(2,q_{c}],\\\
\text{if all the sectional curvatures of }\,M\,\text{are nonpositive},$

and

(1.15)
$\|\Phi\|_{q}\lesssim\lambda^{\mu(q)}(\log\lambda)^{-1/2}\,\bigl{(}\,\|\Phi\|_{2}+(\lambda/\log\lambda)^{-1}\|(\Delta_{g}+\lambda^{2})\Phi\|_{2}\,\bigr{)},\,\,q\in(2,q_{c}],\\\
\text{if all the sectional curvatures of }\,M\,\text{are negative},$

respectively.

To use these bounds to obtain another classification of compact space forms,
as we mentioned before, we shall use the fact that another way of measuring
concentration properties of quasimodes is using the potential decay rates of
$L^{2}$-norms over shrinking geodesic tubes. See e.g., [3], [13] and [16] for
earlier results. To state our results let $\varPi$ be the space of unit length
geodesics in our compact manifold $(M,g)$, and, for $\gamma\in\varPi$ and
$r\ll 1$, ${\mathcal{T}}_{\gamma}(r)$ a geodesic $r$-tube about $\gamma$.
Using (1.13), (1.14), (1.15) and the Knapp-type constructions to follow we
shall be able to prove the following.

######  Theorem 1.2.

Let

(1.16)
$\widetilde{V}_{\lambda}=\\{\Phi:\,\|\Phi\|_{L^{2}(M)}+(\lambda/\log\lambda)^{-1}\|(\Delta_{g}+\lambda^{2})\Phi\|_{L^{2}(M)}\leq
1\\}.$

Then for $\lambda\geq 2$ if all the sectional curvatures of $M$ are
nonpositive

(1.17)
$\sup_{\gamma\in\varPi,\,\Phi\in\widetilde{V}_{\lambda}}\|\Phi\|_{L^{2}({\mathcal{T}}_{\gamma}(R\lambda^{-1/2}))}=O(R^{\frac{n-1}{n+1}}(\log\lambda)^{-\frac{n-1}{2(n+1)}}),\,1\leq
R\leq(\log\lambda)^{1/2}.$

while if all of the sectional curvatures are negative

(1.18)
$\sup_{\gamma\in\varPi,\,\Phi\in\widetilde{V}_{\lambda}}\|\Phi\|_{L^{2}({\mathcal{T}}_{\gamma}(\lambda^{-1/2}))}=O((\log\lambda)^{-1/2}),$

and if $N\in{\mathbb{N}}$ is fixed

(1.19)
$\sup_{\gamma\in\varPi,\,\Phi\in\widetilde{V}_{\lambda}}\|\Phi\|_{L^{2}({\mathcal{T}}_{\gamma}((\log\lambda)^{N}\lambda^{-1/2}))}=O_{\varepsilon,N}((\log\lambda)^{-1/2+\varepsilon}),\,\,\forall\,\varepsilon>0.$

Furthermore, if $(M,g)$ is a compact space form of curvature $K$ then

(1.20)
$\sup_{\gamma\in\varPi,\,\Phi\in\widetilde{V}_{\lambda}}\|\Phi\|_{L^{2}({\mathcal{T}}_{\gamma}(R\lambda^{-1/2}))}\\\
=\begin{cases}\Theta(1),\,\text{for }R=1\quad\iff\,K>0\\\
\Theta((R^{\frac{n-1}{n+1}}(\log\lambda)^{-\frac{n-1}{2(n+1)}}),\,\text{for
}1\leq R\leq(\log\lambda)^{1/2}\quad\iff\,K=0\\\
O_{\varepsilon,N}((\log\lambda)^{-1/2+\varepsilon}),\,\,R=(\log\lambda)^{N},\,\forall\varepsilon>0,\,N\in{\mathbb{N}},\,\,\quad\iff\,K<0.\end{cases}$

For simplicity, we have only stated things here for functions satisfying the
quasimode condition (1.12) with $\delta(\lambda)=(\log\lambda)^{-1}$, however,
there are analogous results for $\delta(\lambda)\searrow 0$ as above.

By arguing as in the proof of (1.9), it is a simple exercise using Hölder’s
inequality to see that (1.14) and (1.15) yield (1.17), (1.18) and (1.19),
respectively. Also, trivially, the left side of (1.20) is bounded by one. As a
result, in order to complete the proof of Theorem 1.2, it suffices to prove
the $\Omega$-lower bounds implicit in (1.20) for compact space forms of
positive and zero curvature. As we shall see, the Knapp constructions in the
next two sections that will allow us to finish the proof of Theorem 1.1 will
also do the same for Theorem 1.2.

In the next two sections we shall complete the proofs of our theorems. Then,
in the final section, we shall state some problems related to our failure to
obtain $\Omega$-lower bounds in (1.11) and (1.20) for manifolds all of whose
sectional curvatures are negative.

##  2\. Knapp examples for $K>0$

To finish the proof of Theorem 1.1 and Theorem 1.2 we need to prove the
$\Omega$-lower bounds implicit in (1.11) and (1.20) for compact space forms
with sectional curvatures equal to $K>0$ as well as for the flat case where
$K=0$. In this section we shall treat the positive curvature case.

To establish the lower bounds in (1.11) and (1.20) for $K>0$ we shall appeal
to the following proposition saying that there are highly focused Knapp
examples for positively curved space forms.

######  Proposition 2.1.

Suppose that $(M,g)$ is a connected compact space form with constant sectional
curvature $K>0$. Then there is a periodic geodesic $\gamma_{0}\subset M$, a
point $x_{0}\in\gamma_{0}$ and a subsequence of eigenfunctions
$e_{\lambda_{k_{\ell}}}$ with eigenvalues $\lambda_{k_{\ell}}\to\infty$ so
that, for a uniform constant $C_{0}<\infty$,

(2.1) $\|e_{\lambda_{k_{\ell}}}\|_{L^{2}(M)}\leq C_{0}\quad\text{and
}\,\|e_{\lambda_{k_{\ell}}}\|_{L^{1}(M)}\leq
C_{0}\lambda^{-\frac{n-1}{4}}_{k_{\ell}},$

and, moreover,

(2.2) $|e_{\lambda_{k_{\ell}}}(x)|\geq
C_{0}^{-1}\lambda_{k_{\ell}}^{\frac{n-1}{4}},\quad
x\in\mathcal{T}_{\gamma_{0}}(C^{-1}_{0}\lambda^{-1/2}_{k_{\ell}})\cap{\mathcal{N}}_{0},$

with $\mathcal{N}_{0}$ being a fixed neighborhood of $x_{0}$.

Before proving the proposition, let us see how it leads to the aforementioned
lower bounds.

For the ones needed for (1.11) with $K>0$, we note that since

$|\mathcal{T}_{\gamma_{0}}(C^{-1}_{0}\lambda^{-1/2}_{k_{\ell}})\cap{\mathcal{N}}_{0}|\approx\lambda_{k_{\ell}}^{-\frac{n-1}{2}},$

by (2.2) we obtain

$\|e_{\lambda_{k_{\ell}}}\|_{L^{2}(M)}\geq\|e_{\lambda_{k_{\ell}}}\|_{L^{2}(\mathcal{T}_{\gamma_{0}}(C^{-1}_{0}\lambda^{-1/2}_{k_{\ell}})\cap{\mathcal{N}}_{0})}\geq
c_{0}$

for some uniform $c_{0}>0$. Using this and the second part of (2.1) yields the
uniform lower bounds

$\|e_{\lambda_{k_{\ell}}}\|_{L^{2}(M)}/\|e_{\lambda_{k_{\ell}}}\|_{L^{1}(M)}\geq
c_{0}^{\prime}\lambda_{k_{\ell}}^{\frac{n-1}{4}},\,\,c_{0}^{\prime}>0,$

which establishes the $\Omega$-bounds in (1.11) when $K>0$.

A similar argument yields the $\Omega$-bounds in (1.20) for $K>0$.

To prove the proposition we shall use the fact that the conclusions are valid
for the special case where $M$ is the round sphere $S^{n}$ of curvature $K=1$:

######  Lemma 2.2.

Let

(2.3)
$Q_{k}(x)=k^{\frac{n-1}{4}}(x_{1}+ix_{2})^{k},\,\,\,(x_{1},x_{2},x^{\prime})\in
S^{n},\,\,\,x^{\prime}=(x_{3},\dots,x_{n+1}).$

Then

(2.4)
$\sqrt{-\Delta_{S^{n}}}\,Q_{k}=\lambda_{k}Q_{k},\quad\lambda_{k}=\sqrt{k(k+(n-1)},$

and, moreover,

(2.5) $\|Q_{k}\|_{L^{2}(S^{n})}\approx 1\quad\text{and
}\,\,\|Q_{k}\|_{L^{1}(S^{n})}\approx
k^{-\frac{n-1}{4}}\approx\lambda_{k}^{-\frac{n-1}{4}},$

as well as, for fixed $\delta>0$,

(2.6) $|Q_{k}(x)|=O(k^{-\sigma})\,\forall\,\sigma\in{\mathbb{N}}\,\,\,\text{if
}x\in S^{n}\,\,\text{and }\,\,\text{dist }(x,\tilde{\gamma}_{0})\geq\delta,\\\
\text{with
}\,\,\tilde{\gamma}_{0}=\\{(\cos\theta,\sin\theta,0,\dots,0):\,\theta\in[0,2\pi)\\}\subset
S^{n}.$

Finally, if $\delta_{0}>0$ is small enough there is a uniform constant
$c_{0}>0$ so that

(2.7) $|Q_{k}(x)|\geq c_{0}k^{\frac{n-1}{4}}\quad\text{if }x\in
S^{n}\,\,\text{and }\,\,\text{dist
}(x,\tilde{\gamma}_{0})\leq\delta_{0}k^{-1/2},$

This result is well known (see, e.g., [12]). The identity (2.4) follows from
the fact that $Q_{k}$ is a spherical harmonic of degree $k$, and (2.5) -(2.6)
follow from the fact that we can write

(2.8)
$Q_{k}(x)=k^{\frac{n-1}{4}}\bigl{(}\sqrt{1-|x^{\prime}|^{2}}\bigr{)}^{k}e^{ik\theta}\quad\text{if
}\,\,x\in S^{n}\,\,\text{and }\,\,\theta=\text{arg }(x_{1}+ix_{2}).$

Let us now focus on the proof of Proposition 2.1. Without loss of generality
we may take $K=1$. It then follows that our space form $(M,g)$ is finitely
covered by $S^{n}$. Thus, we can identify all the eigenfunctions on $M$ with
eigenfunctions on $S^{n}$ (spherical harmonics) with certain symmetry
properties. The model is real projective space $RP^{n}$ which has half the
volume of $S^{n}$ and eigenfunctions corresponding to spherical harmonics of
even order. Recall that the distinct eigenvalues of $\sqrt{-\Delta_{S^{n}}}$
are $\sqrt{k(k+n-1)}$, $k=0,1,2,\dots$, and, so, those on $RP^{n}$ are the
above with $k=0,2,4,\dots$. Also the multiplicity of each of these eigenvalues
on $RP^{n}$ agrees with that on $S^{n}$, which is consistent with the fact
that, by the Weyl formula, we must have
$N_{RP^{n}}(\lambda)=\tfrac{1}{2}N_{S^{n}}(\lambda)+o(N_{S^{n}}(\lambda))$, by
the above volume considerations, if $N_{M}(\lambda)$ denotes the number of
eigenvalues of $\sqrt{-\Delta_{g}}$ counted with respect to multiplicity that
are $\leq\lambda$.

Returning to our compact space form of curvature $K=1$, since it is finitely
covered by $S^{n}$, we can write $M\simeq S^{n}/\Gamma$, where $\Gamma$ is
finite group of isometries each of which is the restriction to $S^{n}$ of an
element $\Gamma\in O(n+1)$ (abusing notation a bit), with $O(n+1)$ being the
orthogonal group for ${\mathbb{R}}^{n+1}$. So, if $\tilde{e}_{\lambda}$ is an
eigenfunction on $S^{n}$ with eigenvalue $\lambda$, and, if $p:S^{n}\to
M\simeq S^{n}/\Gamma$ is the covering map,

(2.9)
$e_{\lambda}(x)=\sum_{\alpha\in\Gamma}\tilde{e}_{\lambda}(\alpha(\tilde{x})),\quad\text{if
}\,x=p(\tilde{x})\in M$

(locally) defines an eigenfunction on $M$ with the same eigenvalue $\lambda$
whenever $e_{\lambda}$ does not vanish identically.. In the model case,
$RP^{n}$, $\Gamma$ consists of the identity map $I$ and reflections about the
origin.

If we take $\lambda_{k}$ as in (2.4) and

(2.10) $\tilde{e}_{\lambda_{k}}=Q_{k}$

then clearly, by (2.5), the resulting eigenfunctions on $M$ with eigenvalue
$\lambda_{k}$ satisfy the bounds in (2.1) with eigenvalue $\lambda_{k_{\ell}}$
being equal to $\lambda_{k}$. So the proof of Proposition 2.1 would be
complete if we could show that when for $\tilde{\gamma}_{0}\subset S^{n}$ as
in (2.6) and

(2.11) $\gamma_{0}=p(\tilde{\gamma}_{0})\subset M,$

then we have (2.2). Note that since $\tilde{\gamma}_{0}$ is a periodic
geodesic (the equator) in $S^{n}$ of period $2\pi$, its projection
$\gamma_{0}$ is a periodic geodesic in $M$ (with perhaps a different period).

If $\alpha_{0}=Id$ and $\Gamma=\\{\alpha_{j}\\}_{j=0}^{N}\subset O(n+1)$ are
the deck transformations above, it follows that

(2.12) $\tilde{\gamma}_{j}=\alpha^{-1}_{j}(\tilde{\gamma}_{0})\subset
S^{n},\quad j=1,2,\dots,N,$

like $\tilde{\gamma}_{0}$ are great circles. After possibly relabelling, we
may assume that for some $m\in\\{1,\dots,N\\}$ we have

$\tilde{\gamma}_{j}=\tilde{\gamma}_{0},\,\,j\leq m-1\quad\text{but
}\,\,\tilde{\gamma}_{j}\neq\tilde{\gamma}_{0}\quad\text{for }\,\,j\geq m.$

We may further more assume that

(2.13) ${\bf 1}=(1,0,\dots,0)\notin\tilde{\gamma}_{j},\quad j\geq m,$

and then we shall take the point $x_{0}\in M$ in Proposition 2.1 to be
$x_{0}=p({\bf 1})$.

It follows that
$\\{\alpha_{0},\dots,\alpha_{m-1}\\}=\Gamma_{0}\subset\Gamma\subset O(n+1)$
are isometries not only mapping $S^{n}$ to itself but $\tilde{\gamma}_{0}$
into itself. Thus, when acting on $\tilde{\gamma}_{0}$, $\Gamma_{0}$ is
isomorphic to a cyclic group ${\mathbb{Z}}/m$ for some $m>1$. So, after
relabelling, we may assume that

(2.14) $\alpha_{j}((\cos\theta,\sin\theta,0,\dots,0))=(\cos(\theta+\tfrac{2\pi
j}{m}),\sin(\theta+\tfrac{2\pi j}{m}),0,\dots,0),\\\
j=0,\dots,m-1,\,\,\text{if }\,m>1.$

Note that by (2.6), (2.10) and (2.13), we can choose $\delta>0$

(2.15)
$\tilde{e}_{\lambda_{k}}(\alpha_{j}(\tilde{x}))=O(\lambda_{k}^{-\sigma}),\,\,\forall\,\sigma,\,\text{if
}\,j\geq m,\,\,\text{and }\,\,\text{dist }(\tilde{x},{\bf 1})<\delta.$

Thus, since $x_{0}=p({\bf 1})$, if $\mathcal{N}_{0}$ is a $\delta$-ball about
$x_{0}$, the summands in (2.9) corresponding to $j\geq m$ are trivial here:

(2.16) $\sum_{j\geq
m}|\tilde{e}_{\lambda_{k}}(\alpha_{j}(\tilde{x}))|=O(\lambda_{j}^{-\sigma})\,\forall\,\sigma\quad\text{if
}\,\,\text{dist }(\tilde{x},{\bf 1})<\delta.$

Thus, if $m=1$, for all $\sigma$ we have

(2.17)
$\sum_{\alpha\in\Gamma}\tilde{e}_{\lambda_{k}}(\alpha(\tilde{x}))=Q_{k}(\tilde{x})+O(\lambda_{k}^{-\sigma}),\quad\text{if
}\,\,\text{dist }(\tilde{x},{\bf 1})<\delta.$

From this and (2.7) we deduce that if ${\mathcal{N}}_{0}$ is above then we
must have (2.2) if $C_{0}$ is large enough if $m=1$ and
$\lambda_{k_{\ell}}=\lambda_{\ell}$, $\ell=1,2,\dots$. So, when $m=1$, we do
not have to pass to a subsequence of eigenvalues.

If $m>1$ we do have to pass to a subsequence to ensure that there is no
cancellation over the sum corresponding to the cyclic subgroup
$\Gamma_{0}=\\{\alpha_{0},\dots,\alpha_{m-1}\\}$ for which we have (2.14).
Since $\alpha_{j}\in O(n+1)$, $j\leq m-1$, rotates the 2-plane $x^{\prime}=0$
in ${\mathbb{R}}^{n+1}$ by angle $2\pi j/m$, it follows that there must be
$m_{j}\in O(n-1)$ so that

$\alpha_{j}=\begin{pmatrix}R_{2\pi j/m}&0\\\ 0&m_{j}\end{pmatrix},$

with $R_{2\pi j/m}\in O(2)$ denoting rotating by this angle. Since
$|m_{j}y^{\prime}|=|y^{\prime}|$ it follows from (2.8) that we must have for
each $j\leq m-1$ and $\ell\in{\mathbb{N}}$

$Q_{\ell m}(\alpha_{j}(\tilde{x}))=(\ell
m)^{\frac{n-1}{4}}\bigl{(}\sqrt{1-|m_{j}\tilde{x}^{\prime}|^{2}}\bigr{)}^{\ell
m}e^{i\ell m(\theta+2\pi j/\ell m)}\\\ =(\ell
m)^{\frac{n-1}{4}}\bigl{(}\sqrt{1-|\tilde{x}^{\prime}|^{2}}\bigr{)}^{\ell
m}e^{ilm\theta}=Q_{k_{\ell}}(\tilde{x}),\quad k_{\ell}=\ell m.$

Thus, if $m>1$, we have the following variant of (2.17)

(2.18)
$\sum_{\alpha\in\Gamma}\tilde{e}_{\lambda_{k}}(\alpha(\tilde{x}))=mQ_{k}(\tilde{x})+O(\lambda_{k}^{-\sigma}),\quad\text{if
}\,\,\text{dist }(\tilde{x},{\bf 1})<\delta,$

which gives us (2.2) just as before. ∎

##  3\. Knapp examples for $K=0$

In this section we shall prove the $\Omega$-lower bounds implicit in (1.11)
and (1.20) for compact space forms with sectional curvatures equal to $K=0$.
By arguing as in the remarks below Proposition 2.1, it suffices to prove the
following

######  Proposition 3.1.

Suppose that $(M,g)$ is a connected compact space form with constant sectional
curvature $K=0$. Then there is a periodic geodesic $\gamma_{0}\subset M$, a
point $x_{0}\in\gamma_{0}$ and a sequence of quasimodes $\psi_{\lambda_{k}}$
with $\psi_{\lambda_{k}}\in
V_{[\lambda_{k}-\delta_{k}/2,\lambda_{k}+\delta_{k}/2]}$,
$\lambda_{k}\to\infty$ and $\delta_{k}\in[(\log\lambda_{k})^{-1},1]$ so that,
for a uniform constant $C_{0}<\infty$,

(3.1) $\|\psi_{\lambda_{k}}\|_{L^{2}(M)}\leq C_{0}\quad\text{and
}\,\|\psi_{\lambda_{k}}\|_{L^{1}(M)}\leq
C_{0}(\lambda_{k}\delta_{k})^{-\frac{n-1}{4}},$

and, moreover,

(3.2) $|\psi_{\lambda_{k}}(x)|\geq
C_{0}^{-1}(\lambda_{k}\delta_{k})^{\frac{n-1}{4}},\quad
x\in\mathcal{T}_{\gamma_{0}}(C^{-1}_{0}(\lambda_{k}\delta_{k})^{-1/2})\cap{\mathcal{N}}_{0},$

with $\mathcal{N}_{0}$ being a fixed neighborhood of $x_{0}$.

Here for $K=0$ the assumption $\delta_{k}\in[(\log\lambda_{k})^{-1},1]$ can be
weakened to $\delta_{k}\in[\lambda_{k}^{-1+\varepsilon},1]$ for any
$\varepsilon>0$ without changing the proof.

By a classical theorem of Cartan and Hadamard, if $(M,g)$ is a compact flat
manifold, it must be of the form ${\mathbb{R}}^{n}/\Gamma$, where $\Gamma$ is
the set of deck transformations. Also by a theorem of Bierbach [2] (see e.g.,
Corollary 5.1 and Theorem 5.3 in Chapter 2 in [6] or [18]), $\Gamma$ must be a
Bieberbach subgroup of the group rigid motions, $E(n)$, of ${\mathbb{R}}^{n}$.
In addition, if we fix an arbitrary point $x_{0}\in M$ and let
$p=\exp_{x_{0}}:{\mathbb{R}}^{n}\to M$, then $p$ is a covering map. If
$D\subset{\mathbb{R}}^{n}$ is a Dirichlet domain containing the origin, then
we can identify $D$ with $M$ by setting $p(\tilde{x})=x$ if $\tilde{x}\in D$.

A function $f$ in ${\mathbb{R}}^{n}$ is called periodic in $\Gamma$ if
$f(x)=f(\alpha(x))$ for all $x\in{\mathbb{R}}^{n}$ and $\alpha\in\Gamma$. For
a given $\tilde{f}\in C^{\infty}({\mathbb{R}}^{n})$ which is periodic in
$\Gamma$, we can define $f(x)=\tilde{f}(\tilde{x})$ if $p(\tilde{x})=x$, and
if $\Delta=\partial^{2}/\partial x_{1}^{2}+\cdots+\partial^{2}/\partial
x_{n}^{2}$ is the standard Laplacian we have

(3.3) $\Delta_{g}f(x)=\Delta\tilde{f}(\tilde{x})$

Similarly, for a given $f\in C^{\infty}(M)$, we can define
$\tilde{f}(\tilde{x})=f(x)$ if $p(\tilde{x})=x$ for $\tilde{x}\in D$, and
extend $\tilde{f}$ to a smooth periodic function in ${\mathbb{R}}^{n}$ which
satisfies (3.3).

Furthermore, by the first part of Bieberbach’s theorem (see e.g., Chapter 2 in
[6]), if we denote $\Lambda\subset\Gamma$ to be the subgroup of translations,
then $\Lambda$ has finite index. In other words, $\Lambda$ consists of
translations by a lattice of full rank, and
$\mathbb{T}^{n}\simeq{\mathbb{R}}^{n}/\Lambda$ is a flat torus, which may not
be the standard torus depending on the lattice. And as discussed above, any
function defined on $\mathbb{T}^{n}$ can be identified naturally with a
function in ${\mathbb{R}}^{n}$ which is periodic in $\Lambda$.

We shall need the following lemma about relations between periodic functions
in $\Gamma$ and $\Lambda$,

######  Lemma 3.2.

There exist finitely many $\alpha_{i}\in\Gamma$, which satisfy

(3.4)
$\alpha_{1}=\text{Identity},\,\,\alpha_{i}\notin\Lambda\,\,\,i=2,3,\dots,N,$

such that if $\tilde{f}$ is a function in ${\mathbb{R}}^{n}$ which is periodic
in $\Lambda$

(3.5) $f(x)=\sum_{i=1}^{N}\tilde{f}(\alpha_{i}(x))$

is a periodic function in $\Gamma$.

As an example, in the case of Klein bottle, $N=2$ and we can take $\alpha_{2}$
to be any element in $\Gamma$ that is not a translation.

###### Proof.

Assume the subgroup of translations $\Lambda$ has index $N$, then we can write
$\Gamma$ as $\Gamma=\cup_{i=1}^{N}\Lambda\alpha_{i}$, where
$\Lambda\alpha_{i}=\\{g\circ\alpha_{i}:g\in\Lambda\\}$ are mutually disjoint
right cosets of $\Lambda$. If we take $\alpha_{1}=\text{Identity}$, then it is
clear that $\alpha_{i}\notin\Lambda$ for $i\geq 2$ since $\Lambda\alpha_{i}$
are mutually disjoint.

It remains to show that for the choice of $\alpha_{i}$ above, $f(x)$ defined
in (3.5) is periodic in $\Gamma$. To see this, we claim that, for any
$\alpha\in\Gamma$, $\alpha_{i}\alpha=g_{i}\alpha_{j(i)}$ for some
$g_{i}\in\Lambda$ and $j(i)\in\\{1,2,\dots N\\}$, and furthermore,
$j(i_{1})\neq j(i_{2})$ if $i_{1}\neq i_{2}$. The first fact just follows from
$\Gamma=\cup_{i=1}^{N}\Lambda\alpha_{i}$. To show that $j(i)$ is injective,
assume for $i_{1}\neq i_{2}$, there exist a $1\leq j\leq N$ such that
$\alpha_{i_{1}}\alpha=g_{i_{1}}\alpha_{j}$ and
$\alpha_{i_{2}}\alpha=g_{i_{2}}\alpha_{j}$, this implies that
$\alpha_{i_{2}}=g_{i_{2}}g^{-1}_{i_{1}}\alpha_{i_{1}}$, which would lead to
the contradiction that $\Lambda\alpha_{i_{1}}=\Lambda\alpha_{i_{2}}$, thus the
claim follows. Using this along with the fact that $\tilde{f}(x)$ is periodic
in $\Lambda$, we have for any $\alpha\in\Gamma$

(3.6) $\displaystyle f(\alpha(x))$
$\displaystyle=\sum_{i=1}^{N}\tilde{f}(\alpha_{i}\alpha(x))=\sum_{i=1}^{N}\tilde{f}(g_{i}\alpha_{j(i)}(x))$
$\displaystyle=\sum_{i=1}^{N}\tilde{f}(\alpha_{j(i)}(x))=f(x)$

∎

The proof of Proposition 3.1 follows the same lines as that of Proposition 2.1
for $K>0$. So, we shall first construct quasimodes that satisfy the desired
concentration properties on $\mathbb{T}^{n}$, and then use Lemma 3.2 to build
up associated quasimodes on the original manifold $M$.

To proceed, let us assume ${b}_{1},{b}_{2},\dots,{b}_{n}$ are linearly
independent vectors in ${\mathbb{R}}^{n\times 1}$, and

(3.7)
$\Lambda=\\{{Bx}:{x}\in{\mathbb{Z}}^{n}\\}=\\{\sum_{i=1}^{n}x_{i}{b}_{i}:x_{i}\in{\mathbb{Z}}\\},$

where ${B}=[{b}_{1},{b}_{2},\dots,{b}_{n}]$ is an $n\times n$ invertible
matrix. For simplicity, let us first assume ${b}_{n}=(0,0,\dots,0,s)^{T}$, we
shall discuss how to modify the arguments for general $b_{n}$ at the end of
this section by using a rotation matrix.

Any periodic functions in $\Lambda$ must be of form

(3.8) $\sum_{\xi\in\mathbb{Z}^{n}}a_{\xi}e^{2\pi i\xi\cdot{B}^{-1}x},$

where each $e^{2\pi i\xi\cdot{B}^{-1}x}$ is a periodic function in $\Lambda$,
which corresponds to an eigenfunction of $\Delta$ on $\mathbb{T}^{n}$ with
eigenvalue $|(B^{T})^{-1}\xi|^{2}$. By using a change of variable $x={B}y$,
each $e^{2\pi i\xi\cdot{B}^{-1}x}$ can also be identified with $e^{2\pi
i\xi\cdot y}$, which is an eigenfunction of $Q(\nabla)$ on
${\mathbb{R}}^{n}/\mathbb{Z}^{n}$ with the same eigenvalue, if
$Q(\xi)=\sum_{i,j}\beta_{ij}\xi_{i}\xi_{j}$ is a positive definite quadratic
form in ${\mathbb{R}}^{n}$ with $\beta_{ij}$ being the $i,j$th element of the
matrix ${(B^{T})^{-1}B^{-1}}$.

In [7], Germain and Myerson constructed the following example on
${\mathbb{R}}^{n}/\mathbb{Z}^{n}$

(3.9) $f(y)=e^{2\pi i\lambda(\xi_{0})_{n}\cdot
y_{n}}\sum_{\xi^{\prime}\in\mathbb{Z}^{n-1}}\phi(\frac{\xi^{\prime}-\lambda\xi_{0}^{\prime}}{\sqrt{\lambda\delta}})e^{2\pi
i\xi^{\prime}\cdot y^{\prime}},$

where $\phi\in C_{0}^{\infty}({\mathbb{R}}^{n-1})$ is such that
$\widehat{\phi}(x)\geq 1$ for $|x|\leq 1$, $y=(y^{\prime},y_{n})$, and
$\xi_{0}=(\xi_{0}^{\prime},(\xi_{0})_{n})$ is a point on the ellipse
$\\{\xi\in{\mathbb{R}}^{n},Q(\xi)=1\\}$ such that the normal vector to the
ellipse at $\xi_{0}$ is colinear to $(0,0,\dots,0,1)$ and
$\lambda(\xi_{0})_{n}\in\mathbb{N}$. Due to the choice of $\xi_{0}$ , it is
not hard to check that if we choose the support of $\phi$ to be small enough
centered around origin, we have

(3.10)
$\lambda-\delta/2\leq\sqrt{Q(\xi)}\leq\lambda+\delta/2,\,\,\,\text{if}\,\,\xi=(\xi^{\prime},\lambda(\xi_{0})_{n})\,\,\,\text{and}\,\,\,\phi(\frac{\xi^{\prime}-\lambda\xi_{0}^{\prime}}{\sqrt{\lambda\delta}})\neq
0.$

By the Poisson summation formula, $f$ can also be written as

(3.11) $e^{2\pi i\lambda(\xi_{0})_{n}\cdot
y_{n}}(\lambda\delta)^{\frac{n-1}{2}}\sum_{\eta\in\mathbb{Z}^{n-1}}\widehat{\phi}(\sqrt{\lambda\delta}(y^{\prime}-\eta))e^{2\pi
i\lambda\xi_{0}^{\prime}\cdot(y^{\prime}-\eta)}.$

Using (3.11), it is straightforward calculation to check that if
$D=[0,1]^{n}$, or any of its translations in ${\mathbb{R}}^{n}$,

(3.12)
$\|f\|_{L^{p}(D)}\sim(\lambda\delta)^{\frac{n-1}{2}(1-\frac{1}{p})},\,\,\text{if}\,\,\,\lambda\delta\gg
1.$

To prove Proposition 3.1, let us fix $\lambda=\lambda_{k},\delta=\delta_{k}$
in (3.9) and define

(3.13) $\displaystyle\varphi_{\lambda_{k}}(y)$
$\displaystyle=(\lambda_{k}\delta_{k})^{-\frac{n-1}{4}}f(y-y_{0})$
$\displaystyle=e^{2\pi i\lambda_{k}(\xi_{0})_{n}\cdot
y_{n}}(\lambda_{k}\delta_{k})^{-\frac{n-1}{4}}\sum_{\xi^{\prime}\in\mathbb{Z}^{n-1}}\phi(\frac{\xi^{\prime}-\lambda_{k}\xi_{0}^{\prime}}{\sqrt{\lambda_{k}\delta_{k}}})e^{2\pi
i\lambda_{k}\xi^{\prime}\cdot(y^{\prime}-y_{0})}$ $\displaystyle=e^{2\pi
i\lambda_{k}(\xi_{0})_{n}\cdot
y_{n}}(\lambda_{k}\delta_{k})^{\frac{n-1}{4}}\sum_{\eta\in\mathbb{Z}^{n-1}}\widehat{\phi}(\sqrt{\lambda_{k}\delta_{k}}(y^{\prime}-y^{\prime}_{0}-\eta))e^{2\pi
i\lambda_{k}\xi_{0}^{\prime}\cdot(y^{\prime}-y^{\prime}_{0}-\eta)},$

where $\lambda_{k}(\xi_{0})_{n}=k$ for $k\in\mathbb{Z}$ with $k\geq C$ for
some $C$ large enough, and $y_{0}=(y_{0}^{\prime},0)$ with
$|y_{0}^{\prime}|=c_{0}$ for some small constant $c_{0}$. The choice of
$y_{0}$ may depend on $\alpha_{i}$, but not on $\lambda_{k}$; we shall specify
the details on the choice of $y_{0}$ later.

By using the changes of variable $y={B}^{-1}x$ and Lemma 3.2, it is clear that

(3.14)
$\tilde{\psi}_{\lambda_{k}}(x)=\sum_{i=1}^{N}\varphi_{\lambda_{k}}({B}^{-1}\alpha_{i}(x))$

defines a smooth periodic function in $\Gamma$. And by (3.10), if we identify
$\tilde{\psi}_{\lambda_{k}}(x)$ with a function $\psi_{\lambda_{k}}\in
C^{\infty}(M)$ via the covering map, we have $\text{Spec
}\psi_{\lambda_{k}}\subset[\lambda_{k}-\delta_{k}/2,\lambda_{k}+\delta_{k}/2]$.
Next, let $D\subset{\mathbb{R}}^{n}$ be a Dirichlet domain for $M$ containing
the origin, and

(3.15)
$\mathcal{T}_{k,x_{0}}=\\{(x^{\prime},x_{n})\in{\mathbb{R}}^{n}:|x^{\prime}-x_{0}^{\prime}|\leq
c_{1}(\lambda_{k}\delta_{k})^{-1/2},\,\,|x_{n}-(x_{0})_{n}|\leq c_{2}\\},$

where $x_{0}^{\prime}=A^{-1}y_{0}^{\prime}$ for $A$ defined in (4.2) below,
and $c_{1},c_{2},(x_{0})_{n}>0$ are small constants to be specified later
which are independent of $\lambda_{k},\delta_{k}$. Since we are assuming
${b}_{n}=(0,0,\dots,0,s)^{T}$, by (3.7), the group $\Gamma$ contains the deck
transformation $\alpha$ with $\alpha(x)=(x^{\prime},x_{n}+s)$, thus the
straight line $\\{(x_{0}^{\prime},t):t\in{\mathbb{R}}\\}$ must be mapped to
some periodic geodesic $\gamma_{0}\in M$ by the covering map $p$, and
$\mathcal{T}_{k,x_{0}}$ must be mapped onto a
$c_{1}(\lambda_{k}\delta_{k})^{-\frac{1}{2}}$ neighborhood of $\gamma_{0}$
near the point $p(x_{0})$ if $x_{0}=(x_{0}^{\prime},\,\,(x_{0})_{n})$.

Thus, to prove Proposition 3.1, it suffices to show

(3.16) $\|\tilde{\psi}_{\lambda_{k}}\|_{L^{2}(D)}\leq C_{0}\quad\text{and
}\,\|\tilde{\psi}_{\lambda_{k}}\|_{L^{1}(D)}\leq
C_{0}(\lambda_{k}\delta_{k})^{-\frac{n-1}{4}},$

as well as

(3.17) $|\tilde{\psi}_{\lambda_{k}}(x)|\geq
C_{0}^{-1}(\lambda_{k}\delta_{k})^{\frac{n-1}{4}},\quad
x\in\mathcal{T}_{k,x_{0}}.$

The proof of (3.16) just follows from (3.12) and the definition of
$\varphi_{\lambda_{k}}$ in (3.13). To prove (3.17), let us first make some
simple reductions. Recall that since ${b}_{n}=(0,0,\dots,0,s)^{T}$, we have

(3.18) $B^{-1}=\left(\begin{array}[]{ccccc}&&0\\\ \quad\mbox{
\huge${A}$}&&\vdots\\\ &&0\\\
a_{1}\,\,\,\cdots\,\,\,a_{n-1}&&s^{-1}\end{array}\right),\quad\text{with
}\,{A}\in GL_{n-1}({\mathbb{R}}).$

Also, recall that $\alpha_{i}\in E(n)$, the group of rigid motions, and so we
may assume $\alpha_{i}(x)=m_{i}x+j_{i}$, where $m_{i}\in O(n)$ is an $n\times
n$ orthogonal matrix and
$j_{i}=(j_{i}^{\prime},(j_{i})_{n})\in{\mathbb{R}}^{n}$. Here, since
$\varphi_{\lambda_{k}}$ is periodic in $\mathbb{Z}^{n}$ and
$B^{-1}\alpha_{i}(x)=(B^{-1}m_{i})x+B^{-1}j_{i}$, without loss of generality,
we may assume $B^{-1}j_{i}\in[0,1)^{n}$, and furthermore, since the number of
choices of $i$ is finite, we can assume
$\alpha_{i}(x)=m_{i}x+j_{i}\,\,\text{with}\,\,B^{-1}j_{i}\in[0,1-\delta_{0})^{n}$
for some $0<\delta_{0}<1$ uniformly in $i$. Thus, by choosing the constants in
(3.15) small enough so that $\mathcal{T}_{k,x_{0}}$ is close enough to the
origin and $|y^{\prime}_{0}|=c_{0}\ll\delta_{0}$, we may assume that

(3.19)
$B^{-1}\alpha_{i}(x)-y_{0}\in[-1+\delta_{0}/2,1-\delta_{0}/2)^{n},\,\,\,i=1,2\cdots,N,\quad\,\,\text{if}\,\,\,x\in\mathcal{T}_{k,x_{0}}.$

If we let $\pi:\pi(x)=x^{\prime}$ be the projection map onto the first $n-1$
coordinates, (3.19) implies that

(3.20)
$\left|\widehat{\phi}(\sqrt{\lambda_{k}\delta_{k}}(A\pi\alpha_{i}(x)-y^{\prime}_{0}-\eta))\right|\leq
C_{\sigma}(\lambda_{k}\delta_{k}|\eta|)^{-\sigma}\,\,\forall\,\sigma\in{\mathbb{N}},\,\,\eta\in\mathbb{Z}^{n-1},\,\,|\eta|\neq
0.$

Consequently, if $\lambda_{k}\delta_{k}\to\infty$ as $k\to\infty$ and $k\geq
C$ is large enough, to prove (3.17), it suffices to assume $\eta=0$, and show
that

(3.21)
$|\sum_{i=1}^{N}\widehat{\phi}(\sqrt{\lambda_{k}\delta_{k}}(A\pi\alpha_{i}(x)-y^{\prime}_{0}))|\geq(2C_{0})^{-1}\,\,\,\,\text{if}\,\,x\in\mathcal{T}_{k,x_{0}}.$

To proceed, let us define
$\ell_{0}=\\{(x_{0}^{\prime},t):t\in{\mathbb{R}}\\}$, where
$x_{0}^{\prime}=A^{-1}y_{0}^{\prime}$, then since
$\widehat{\phi}\in\mathcal{S}$ and $\widehat{\phi}(x)\geq 1$ for $|x|\leq 1$,
it is straightforward to check that for fixed $\delta>0$

(3.22)
$|\widehat{\phi}(\sqrt{\lambda_{k}\delta_{k}}(A\pi(x)-y^{\prime}_{0}))|\leq
C_{\sigma}(\lambda_{k}\delta_{k})^{-\sigma}\,\,\forall\,\sigma\in{\mathbb{N}},\,\,\,\text{if}\,\,\,\text{dist
}(x,\ell_{0})\geq\delta,$

and for some $c_{1}>0$ small enough which may depend on $A$,

(3.23)
$|\widehat{\phi}(\sqrt{\lambda_{k}\delta_{k}}(A\pi(x)-y^{\prime}_{0}))|\geq
1\quad\text{if }\text{dist }(x,\ell_{0})\leq
c_{1}(\lambda_{k}\delta_{k})^{-1/2}.$

By the definition of $\mathcal{T}_{k,x_{0}}$ in (3.15), we certainly have
$\text{dist }(x,\ell_{0})\leq c_{1}(\lambda_{k}\delta_{k})^{-1/2}$ if
$x\in\mathcal{T}_{k,x_{0}}$. Thus, to prove (3.21), it suffices to show for
$x\in\mathcal{T}_{k,x_{0}}$

(3.24)
$|\sum_{i=2}^{N}\widehat{\phi}(\sqrt{\lambda_{k}\delta_{k}}(A\pi\alpha_{i}(x)-y^{\prime}_{0}))|\leq
C_{\sigma}(\lambda_{k}\delta_{k})^{-\sigma}\,\,\forall\,\sigma\in{\mathbb{N}}.$

If for each $\alpha\in\Gamma$, we also define
$\ell_{\alpha}=\\{x:\alpha(x)\in\ell_{0}\\}$, the preimage of $\ell_{0}$ under
$\alpha$, then we claim that (3.24) would be a consequence of the following
lemma.

######  Lemma 3.3.

There exist a $x_{0}^{\prime}\in{\mathbb{R}}^{n-1}$, where $|x_{0}^{\prime}|$
can be arbitrary small, such that if
$\ell_{0}=\\{(x_{0}^{\prime},t):t\in{\mathbb{R}}\\}$ ,
$\ell_{\alpha_{i}}\neq\ell_{0}$ for all $i=2,\dots,N$.

Let us first prove the claim. Note that for each $i$, (3.22) is equivalent to

(3.25)
$|\widehat{\phi}(\sqrt{\lambda_{k}\delta_{k}}(A(\pi\alpha_{i}(x))-y^{\prime}_{0}))|\leq
C_{\sigma}(\lambda_{k}\delta_{k})^{-\sigma}\,\,\forall\,\sigma\in{\mathbb{N}},\,\,\,\text{if}\,\,\,\text{dist
}(x,\ell_{\alpha_{i}})\geq\delta.$

Thus, we would have (3.24) if we can choose $\mathcal{T}_{k,x_{0}}$ such that
for some fixed $\delta>0$,

(3.26) $\text{dist
}(x,\ell_{\alpha_{i}})\geq\delta,\,\,i=2,3,\dots,N\quad\text{if}\,\,x\in\mathcal{T}_{k,x_{0}}.$

To prove this, by Lemma 3.3, each line $\ell_{\alpha_{i}}$ can intersect
$\ell_{0}$ at most once. Thus, there must be a point
$x_{0}=(x_{0}^{\prime},(x_{0})_{n})$ with $|x_{0}|\leq c$ such that
$x_{0}\notin\ell_{\alpha_{i}}$ for all $i\geq 2$. Here we emphasize that the
constant $c$ can be chosen small enough, which is necessary for (3.19) to
hold. Since $x_{0}\in\mathcal{T}_{k,x_{0}}$, if we define
$\delta_{1}=\min_{i}\,{{\rm dist}}(x_{0},\ell_{\alpha_{i}})$, by further
choosing the constants $c_{1},c_{2}$ in (3.15) to be small enough relative to
$\delta_{1}$, for $\lambda_{k}\delta_{k}\gg\delta_{1}^{-1}$, which holds as
long as $k\geq C_{0}$ is large enough, we must have (3.26) with
$\delta=\delta_{1}/2$, which finishes the proof of (3.24).

###### Proof of Lemma 3.3.

Note that $\ell_{\alpha_{i}}\neq\ell_{0}$ is equivalent to
$\alpha_{i}(\ell_{0})\neq\ell_{0}$, if $\alpha_{i}(\ell_{0})$ denotes the
image of $\ell_{0}$ under $\alpha_{i}$. For $\alpha_{i}(x)=m_{i}x+j_{i}$ with
$j_{i}=(j_{i}^{\prime},(j_{i})_{n})$, we shall divide our discussion into the
following cases.

(i) $j_{i}^{\prime}\neq 0$. In this case, since the number of choices of $i$
is finite, we may assume $|j_{i}^{\prime}|\geq\delta_{2}$ for some constant
$\delta_{2}$ independent of $i$, thus if we choose $x_{0}^{\prime}$ such that
$|x_{0}^{\prime}|\leq\frac{1}{2}\delta_{2}$, it is not hard to check that for
$(x_{0}^{\prime},0)\in\ell_{0}$,
$\alpha_{i}((x_{0}^{\prime},0))\notin\ell_{0}$ for all $i$ satisfying (i).

(ii) $m_{i}(0,0,\dots,0,1)^{T}\neq(0,0,\dots,0,\pm 1)^{T}$. In this case, the
directions of $\alpha_{i}(\ell_{0})$ and $\ell_{0}$ are transverse, thus we
certainly have $\alpha_{i}(\ell_{0})\neq\ell_{0}$ for all $i$ satisfying (ii).

(iii) $j_{i}^{\prime}=0$ and $m_{i}(0,0,\dots,0,1)^{T}=(0,0,\dots,0,\pm
1)^{T}$. In this case, we may assume
$\alpha_{i}(x)=m_{i}x+(0,\cdots,0,(j_{i})_{n})$, where

(3.27) $m_{i}=\left(\begin{array}[]{ccccc}&&0\\\ \quad\mbox{
\huge$\overline{m}_{i}$}&&\vdots\\\ &&0\\\
0\,\,\,\cdots\,\,\,0&&\mu_{i}\end{array}\right),\quad\text{with
}\,\overline{m}_{i}\in O(n-1),\,\,\mu_{i}=\pm 1.$

We claim that $\overline{m}_{i}\neq I_{n-1}$. To see this, note that if
$\overline{m}_{i}=I_{n-1}$ and $\mu_{i}=1$, we have
$\alpha_{i}(x)=x+(0,\cdots,(j_{i})_{n})$, which is impossible since by (3.4),
$\alpha_{i}$ is not a translation for $i\geq 2$. On the other hand, if
$\overline{m}_{i}=I_{n-1}$ and $\mu_{i}=-1$, we have $\alpha_{i}(x)=x$ for
$x=(0,\cdots,0,\frac{(j_{i})_{n}}{2})$, which contradicts with the fact that
the deck group $\Gamma$ acts freely on the covering space ${\mathbb{R}}^{n}$.

Since $\overline{m}_{i}\neq I_{n-1}$, the set of fixed points
$\\{x^{\prime}\in{\mathbb{R}}^{n-1}:\overline{m}_{i}x^{\prime}=x^{\prime}\\}$
must be a subset of some hyperplane containing origin. Thus, there must exist
a point $x^{\prime}_{0}\in{\mathbb{R}}^{n-1}$ that is outside these finitely
many hyperplanes, i.e., $\overline{m}_{i}x_{0}^{\prime}\neq x_{0}^{\prime}$
for all values of $i$ satisfying (iii), which implies
$\alpha_{i}(\ell_{0})\neq\ell_{0}$. Also, since $\overline{m}_{i}$ is a linear
matrix, the same results hold if we replace $x_{0}^{\prime}$ by the point
$cx_{0}^{\prime}$ for any $c<1$. ∎

Finally, we shall deal with the general case $b_{n}\neq(0,\dots,0,s)^{T}$. Let
us choose an orthogonal matrix $Q$ such that $Qb_{n}=(0,\dots,0,s)^{T}$ for
some $s$, and write $B=Q^{T}QB=Q^{T}\bar{B}$, where the last column of
$\bar{B}$ is $(0,\dots,0,s)^{T}$ . In this case, we can define

(3.28)
$\tilde{\psi}_{\lambda_{k}}(x)=\sum_{i=1}^{N}\varphi_{\lambda_{k}}({B}^{-1}\alpha_{i}(x))=\sum_{i=1}^{N}\varphi_{\lambda_{k}}({\bar{B}}^{-1}\bar{\alpha}_{i}(Qx)),$

where $\bar{\alpha}_{i}=Q{\alpha}_{i}Q^{T}$. Then $\bar{B}^{-1}$ satisfies
(4.2), and $\bar{\alpha}_{i}\in E(n)$ satisfies the two properties we used for
$\alpha_{i}$, i.e., $\bar{\alpha}_{i}$ is not a translation and does not have
a fixed point. Thus we can define $\bar{x}=Qx$ and repeat the above arguments
to prove Proposition 3.1 for the general case.

##  4\. Problems and comparison with earlier results

We stated Theorem 1.2 in terms of the log-quasimodes defined by (1.16), since
these were the ones considered by Brooks [5] for compact space forms of
negative sectional curvature. On the other hand, all of the conclusions remain
valid if we consider log-quasimodes defined by the spectral condition (1.1).
Specifically, we have the following.

######  Theorem 4.1.

Consider $\\{\Phi_{\lambda}\in
V_{[\lambda,\lambda+(\log\lambda)^{-1}]}:\,\|\Phi_{\lambda}\|_{2}=1\\}$. Then
for $\lambda\geq 2$ if all the sectional curvatures of $M$ are nonpositive

(4.1)
$\sup_{\gamma\in\varPi}\|\Phi_{\lambda}\|_{L^{2}({\mathcal{T}}_{\gamma}(R\lambda^{-1/2}))}=O(R^{\frac{n-1}{n+1}}(\log\lambda)^{-\frac{n-1}{2(n+1)}}),\,1\leq
R\leq(\log\lambda)^{1/2}.$

while, if all of the sectional curvatures are negative,

(4.2)
$\sup_{\gamma\in\varPi}\|\Phi_{\lambda}\|_{L^{2}({\mathcal{T}}_{\gamma}(\lambda^{-1/2}))}=O((\log\lambda)^{-1/2}),$

and $N\in{\mathbb{N}}$ is fixed

(4.3)
$\sup_{\gamma\in\varPi}\|\Phi_{\lambda}\|_{L^{2}({\mathcal{T}}_{\gamma}((\log\lambda)^{N}\lambda^{-1/2}))}=O_{\varepsilon,N}((\log\lambda)^{-1/2+\varepsilon}),\,\,\forall\,\varepsilon>0.$

Furthermore, if $(M,g)$ is a compact space form of curvature $K$ then

(4.4)
$\sup_{\gamma\in\varPi}\|\Phi_{\lambda}\|_{L^{2}({\mathcal{T}}_{\gamma}(R\lambda^{-1/2}))}\\\
=\begin{cases}\Theta(1),\,\text{for }R=1\quad\iff\,K>0\\\
\Theta((R^{\frac{n-1}{n+1}}(\log\lambda)^{-\frac{n-1}{2(n+1)}}),\,\text{for
}1\leq R\leq(\log\lambda)^{1/2}\quad\iff\,K=0\\\
O_{\varepsilon,N}((\log\lambda)^{-1/2+\varepsilon}),\,\,R=(\log\lambda)^{N},\,\forall\varepsilon>0,\,N\in{\mathbb{N}},\,\,\quad\iff\,K<0.\end{cases}$

The proof of this result is almost identical to that of Theorem 1.2. The upper
bounds (4.1), (4.2) and (4.3), just as before, follow via Hölder’s inequality
from (1.4), (1.6) and (1.7). Moreover, since the quasimodes in Propositions
2.1 and 3.1 are spectrally localized, we can repeat the earlier arguments to
obtain the $\Omega$-lower bounds implicit in the first two parts of (4.4),
which completes the proof.

It would be interesting to try to obtain the analog of Theorem 1.1 for the
log-quasimodes defined by (1.16). We are unable to prove the upper bounds
(1.10) for $\Phi\in\widetilde{V}_{\lambda}$ since, unlike (1.4), (1.6) and
(1.7), the bounds in (1.12) for $\widetilde{V}_{\lambda}$ do not only involve
$L^{2}$-norms in the left. As a result, we can not use Hölder’s inequality as
we did for the proof of the upper bounds in Theorem 1.1.

It would also be interesting to try to replace the $O$-upper bounds in Theorem
1.1, 1.2 or 4.1 with $\Theta$-bounds for compact space forms of negative
sectional curvature (i.e., $K<0$). This appears difficult. One thing that is
missing for this case is analog of Propositions 2.1 or 3.1 for compact space
forms of negative curvature. We could obtain the Knapp examples in these
propositions since there were model modes on covers of positively curved or
flat compact space forms, i.e., the round sphere or tori, respectively. We are
unaware of any simple model modes for compact space forms of negative
curvature.

It would also be interesting to try to bridge the gap between the
concentration results obtained here and in Brooks [5]. Brooks’ results imply
that if $\gamma_{0}$ is a periodic geodesic in a compact space form of
negative curvature, then there is a $\delta>0$ and a sequence of frequencies
$\lambda_{j}\to\infty$ and associated log-quasimodes $\Phi=\Phi_{\lambda}$ as
in (1.16) so that if ${\mathcal{N}}$ is any fixed neighborhood of $\gamma_{0}$
then $\liminf\|\Phi_{\lambda}\|_{L^{2}({\mathcal{N}})}>\delta$. On the other
hand, by Theorem 1.2, for any fixed $N\in{\mathbb{N}}$, if
${\mathcal{N}}={\mathcal{N}}(\lambda)={\mathcal{T}}_{\gamma_{0}}((\log\lambda)^{N}\lambda^{-1/2})$,
then we must have
$\liminf\|\Phi_{\lambda}\|_{L^{2}({\mathcal{N}(\lambda)})}=0$. These tubes
have volume tending to zero; however, are there analogs of the lower bounds of
Brooks for tubes of volume tending to zero slower than the ones above? Also,
for the case of compact space forms with $K>0$, as we have shown, tube radius
$\lambda^{-1/2}$ is a natural barrier for lower bounds of $L^{2}$-norms over
geodesic tubes, while for flat manifolds, as in (1.20), tube-radius
$(\lambda/\log\lambda)^{-1/2}$ is natural for flat compact space forms when
considering log-quasimodes. Is there a natural scale for compact space forms
of negative curvature? And, finally, do the analog of Brooks’ results hold for
the quasimodes defined by the spectral condition (1.1)?

## References

  * [1] P. H. Bérard.  On the wave equation on a compact Riemannian manifold without conjugate points.  Math. Z., 155(3):249–276, 1977.
  * [2] L. Bieberbach.  Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich.  Math. Ann., 72(3):400–412, 1912.
  * [3] M. D. Blair and C. D. Sogge.  Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions.  J. Differential Geom., 109(2):189–221, 2018.
  * [4] J. Bourgain, P. Shao, C. D. Sogge, and X. Yao.  On $L^{p}$-resolvent estimates and the density of eigenvalues for compact Riemannian manifolds.  Comm. Math. Phys., 333(3):1483–1527, 2015.
  * [5] S. Brooks.  Logarithmic-scale quasimodes that do not equidistribute.  Int. Math. Res. Not. IMRN, 22:11934–11960, 2015.
  * [6] L. S. Charlap.  Bieberbach groups and flat manifolds.  Universitext. Springer-Verlag, New York, 1986.
  * [7] P. Germain and S. L. Rydin Myerson.  Bounds for spectral projectors on tori.  Forum Math. Sigma, 10:Paper No. e24, 20, 2022.
  * [8] A. Hassell and M. Tacy.  Improvement of eigenfunction estimates on manifolds of nonpositive curvature.  Forum Mathematicum, 27(3):1435–1451, 2015.
  * [9] X. Huang and C. D. Sogge.  Curvature and sharp growth rates of log-quasimodes on compact manifolds.  preprint.
  * [10] C. D. Sogge.  Oscillatory integrals and spherical harmonics.  Duke Math. J., 53(1):43–65, 1986.
  * [11] C. D. Sogge.  Concerning the $L^{p}$ norm of spectral clusters for second-order elliptic operators on compact manifolds.  J. Funct. Anal., 77(1):123–138, 1988.
  * [12] C. D. Sogge.  Hangzhou lectures on eigenfunctions of the Laplacian, volume 188 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 2014.
  * [13] C. D. Sogge.  Problems related to the concentration of eigenfunctions.  Journés équations aux dérivées partielles, avalaible at http://www.numdam.org/, 2015.
  * [14] C. D. Sogge.  Fourier integrals in classical analysis, volume 210 of Cambridge Tracts in Mathematics.  Cambridge University Press, Cambridge, second edition, 2017.
  * [15] C. D. Sogge and S. Zelditch.  Lower bounds on the Hausdorff measure of nodal sets.  Math. Res. Lett., 18(1):25–37, 2011.
  * [16] C. D. Sogge and S. Zelditch.  On eigenfunction restriction estimates and $L^{4}$-bounds for compact surfaces with nonpositive curvature.  In Advances in analysis: the legacy of Elias M. Stein, volume 50 of Princeton Math. Ser., pages 447–461. Princeton Univ. Press, Princeton, NJ, 2014.
  * [17] C. D. Sogge and S. Zelditch.  A note on $L^{p}$-norms of quasi-modes.  In Some topics in harmonic analysis and applications, volume 34 of Adv. Lect. Math. (ALM), pages 385–397. Int. Press, Somerville, MA, 2016\.
  * [18] J. A. Wolf.  Spaces of constant curvature.  AMS Chelsea Publishing, Providence, RI, sixth edition, 2011.