# Slow Entropy and Variational Dynamical Systems

Minhua Cheng  DEPARTMENT OF MATHEMATICS, UNIVERSITY OF UTAH, SALT LAKE CITY,
UT 84112<EMAIL_ADDRESS>,  Carlos Ospina  DEPARTMENT OF MATHEMATICS,
UNIVERSITY OF UTAH, SALT LAKE CITY, UT 84112<EMAIL_ADDRESS>,  Kurt
Vinhage  DEPARTMENT OF MATHEMATICS, UNIVERSITY OF UTAH, SALT LAKE CITY, UT
84112<EMAIL_ADDRESS>and  Yibo Zhai  DEPARTMENT OF MATHEMATICS,
UNIVERSITY OF UTAH, SALT LAKE CITY, UT 84112<EMAIL_ADDRESS>
###### Abstract.

We define variational properties for dynamical systems with subexponential
complexity, and study these properties in certain specific examples. By
computing the value of slow entropy directly, we show that Sturmian systems
are not variational, while a class of interval exchange transformations are
variational.

##  1\. Introduction

The metric and topological entropies for measure-preserving and topological
dynamical systems are often the first and most important invariants to study
when considering classification problems. Several structural theorems give
justification for studying these notions, crucially the structure theorems of
Ornstein and Sinai, which establish that metric entropy is the only invariant
of Bernoulli systems, and that every measure-preserving system has a Bernoulli
factor of the same entropy, respectively. We refer the reader to [KH95,
Sections 3.1, 4.3] for a review of the classical entropy definition, and
[Kat23] for a review of the history of the standard entropy theory.

The metric and topological entropies for continuous transformations are linked
via the variational principle:

(1) $h_{\operatorname{top}}(T)=\sup_{\mu\in\mathcal{M}^{T}}h_{\mu}(T),$

where $\mathcal{M}^{T}$ is the set of $T$-invariant Borel probability
measures. If this supremum is achieved, a measure for which
$h_{\mu}(T)=h_{\operatorname{top}}(T)$ is called a measure of maximal entropy.

Entropy as a tool for classifying and describing the complexity of dynamical
systems has proven very useful when the value is positive and finite. This
corresponds to the system having a number of distinguishable orbits which is
exponentially growing. However, when the system has sub-exponential
complexity, the classical entropy is always zero, and a new class of
invariants is needed.

###  1.1. Slow entropy invariants

The measure-theoretic complexity or slow entropy is a class of invariants
which can describe slower growth rates such as polynomial or logarithmic, and
indeed can be given specific numerical values which are dynamical invariants.
We discuss the definitions and basic properties in Section 2, but introduce
some notation here, where the notation $p_{\chi}(n)=n^{\chi}$ represents the
family of polynomial scales:

  * •

if $h_{\mu,p_{\chi}}=d$, then the number of orbit types of length $n$ which
can be distinguished by $\mu$ is approximately $n^{d}$, and

  * •

if $h_{\operatorname{top},p_{\chi}}=d$, then the number of orbit types of
length $n$ which are topologically distinguished is approximately $n^{d}$.

By analogy, the classical entropy can be seen as the slow entropy with respect
to the family of exponential scales $e_{\chi}(n)=e^{\chi n}$, and one may
consider in general entropies at a more general family of scales $a_{\chi}$.

Slow entropy or measure-theoretic complexity has many applications in
classification questions. One important characterization is those systems with
minimal complexity. This was first proved by Ferenczi in [Fer97], where the
result was phrased using measure-theoretic complexity. We present a proof of
this theorem in Section 4.

###  1.2. Variational principles

It is natural to ask whether Equation 1 holds for slow entropy. We will see
that the answer in general is no (1.2), motivating the following definition.
We refer the reader to Section 2 for any undefined terms.

######  Definition 1.1.

Let $(X,d)$ be a metric space and $T:X\to X$ be a transformation. We say that
$T$ is variational at the family of scales $\set{a_{\chi}}$ if

$h_{\operatorname{top},a_{\chi}}=\sup_{\mu\in\mathcal{M}^{T}}h_{\mu,a_{\chi}}.$

We say that $T$ is strongly variational at the family of scales
$\set{a_{\chi}}$ if there exists a unique measure $\mu_{0}$ for which

$h_{\operatorname{top},a_{\chi}}=h_{\mu_{0},a_{\chi}}.$

Variational properties of several known examples can be deduced from from
existing work:

  * •

Transitive translations on compact abelian groups are strongly variational at
all scales (Ferenczi’s Theorem [Fer97], 4.2).

  * •

All continuous transformations of compact spaces are variational at
exponential scale (This is the classical variational principle, see eg, [KH95,
Theorem 4.5.3]).

  * •

Uniformly hyperbolic dynamical systems are strongly variational at exponential
scale (Existence and uniqueness of MMEs for uniformly hyperbolic follows from
classical works of Bowen and Margulis).

  * •

Transitive unipotent flows are strongly variational at polynomial scale
([KVW19]).

  * •

Some smooth systems obtained from combinatorial constructions are not
variational ([BKW23a],[BKW23b]).

In this paper, we establish the following theorems towards understanding which
systems are variational:

######  Theorem 1.2.

Sturmian subshifts and Denjoy circle transformations are not variational at
polynomial scale.

######  Theorem 1.3.

There exists a full Hausdorff dimension subset of 3-IETs which are strongly
variational at polynomial scale.

1.2 is proved in Section 5.1 and 1.3 is proved in Section 6.5. We also provide
a precise description of a set 3-IETs which are variational at polynomial
scale.

###  1.3. Future directions

The proof of 1.3 requires strong Diophantine conditions to compute the slow
entropy of the 3-IETs. While it is likely that these conditions can be
relaxed, the proof suggests that intermediate behavior for slow entropy is
possible. In particular, we believe that some 3-IETs are not variational.

This is perhaps less shocking after noting that interval exchanges have
discontinuities. However, by adding roof functions with controlled
singularities, such transformations are realized as first return maps for
surface flows with stationary points. It is therefore natural to ask whether
the surface flows are variational. The slow entropy of some surface flows was
computed in [Kan18], so a computation of their topological slow entropy would
determine their variational properties.

In Section 5.3, we observe that slow entropy does not behave like exponential
entropy with respect to ergodic decompositions, and there is little hope to
obtain a universal formula. In fact, one may have a system with positive slow
entropy for which every ergodic component is Kronecker. It is natural to ask
how large the gap between the entropy of the ergodic components and the
entropy of the integrated system can become. By the usual formula for entropy
given an ergodic decomposition, we know that the gap cannot be exponential.

Finally, we note that in all known examples of non-variational systems, the
gap between $\sup_{\mu\in\mathcal{M}(X)}h_{\mu,a_{\chi}}$ and
$h_{\operatorname{top},a_{\chi}}$ is obtained when $a_{\chi}$ is at most
polynomial scale, and in this setting, the difference is at most 1. It would
be interesting to ask what sort of gaps can appear, and if they can appear in
other families of scales $\set{a_{\chi}}$.

###  1.4. Organization of the paper

In Section 2, we review three types of invariants for systems with zero
entropy at an exponential scale. The first is the topological slow entropy
which we denote $h_{\operatorname{top},a_{\chi}}$. This is an invariant of
dynamical systems under continuous conjugations. The second invariant is the
metric slow entropy $h_{\mu,a_{\chi}}$, which is an invariant of measure-
preserving systems. The third and last invariant of measure-preserving systems
is the semi-topological slow entropy $h_{\operatorname{semi},\mu,a_{\chi}}$.
We compare these definitions with the classical topological and metric
entropies. We conclude the section with 2.10 showing that these definitions
are the classical entropies when $a_{\chi}(n)=e^{\chi n}.$ In Section 3, we
present two results, although not new, they are generalizations to arbitrary
scales. Particularly, 3.1 was proved for first time in [Goo69] to show that
the topological entropy is larger or equal to the metric entropy. We prove
that this result is still true without requiring continuity of the system, and
also, to include in the inequality the semi-topological entropy. To summarize,
we are able to prove, for any scale $a_{\chi}$, the inequality

$h_{\mu,a_{\chi}}\leq h_{\operatorname{semi},\mu,a_{\chi}}\leq
h_{\operatorname{top},a_{\chi}}.$

Section 4 is mainly expository. We have parsed [Fer97, Proposition 3] in 4.2,
to rewrite it to our notation and for referencing in later results of this
paper. We remark that is crucial for proving 1.2. In Sections 5.1 and 5.2, we
prove 1.2. We compute the semi-topological slow entropy of Sturmian systems,
the main result is summarized in 5.4, and we add a discussion of Denjoy circle
transformations to show that these systems are not variational. Finally, we
discuss aspects of the ergodic decomposition, and how different it is in the
case of slow entropy, see Section 5.3, in 5.8 that the metric entropy of the
geodesic flow 1 at polynomial scales with respect to the Haar measure, but 0
with respect to any ergodic measure and for any arbitrary scale.

In Section 6, we introduce the background of interval exchange transformations
and prove 1.3.

Acknowledgements. The authors would like to thank Przemyslaw Berk, Jon Chaika,
Adam Kanigowski, and Daren Wei for discussions and recommendations on the
direction of this paper. The authors also acknowledge the NSF award #1840190,
the research training group Algebra, Geometry and Topology, which provided the
space and opportunity for this work to be done.

##  2\. Definitions

Let $f\colon X\to X$ be a measurable transformation on a locally compact
metric space. We denote $(X,d)$ the metric structure, and if $\mu$ is a Radon
measure, let $(X,\mu,\mathcal{B})$ denote the triple determining a measure
space and $\sigma$-algebra of Borel sets $\mathcal{B}$. If $\mu$ is
$f$-invariant, i.e. $\forall B\in\mathcal{B}$ $\mu(f^{-1}(B))=\mu(B)$, then we
say that $(X,\mu,f,\mathcal{B})$ is a measure preserving system. We do not
assume $\mu$ is ergodic, instead, we mention it whenever it is necessary.
However, unless otherwise noted, we will assume that $\mu$ is a probability
measure, i.e. $\mu(X)=1$.

###  2.1. Topological Slow Entropy

For $x\in X$, and $\epsilon>0$, let $B(x,\epsilon)=\\{y\in
X:d(x,y)<\epsilon\\}$ denote an open ball. For $n$ a non-negative integer and
$\epsilon$ a positive real number, it is not hard to check that the map

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

defines a new metric on $X$, called the $n$-Bowen metric, or simply Bowen
metric. When $f$ is uniformly continuous, the metric $d_{f}^{n}$ is equivalent
to $d$. The $(\epsilon,n)$-Bowen ball is the ball in $d_{f}^{n}$ centered at
$x$ of radius $\epsilon$, or equivalently the following set:

$B^{n}_{f}(x,\epsilon)=\cap_{i=0}^{n-1}f^{-i}\left(B(f^{i}(x),\epsilon)\right).$

Let $N_{f,K}(\epsilon,n)$ denote the minimal number of $(\epsilon,n)$-Bowen
balls that cover a compact set $K\subset X$. Note that
$N_{f,K}(\epsilon,n)<\infty$ when $f$ is uniformly continuous.

Let $S_{f,K}(\epsilon,n)$ be the maximal number of disjoint
$(\epsilon,n)$-Bowen balls that can be arranged with centers in $K$, where all
the centers of such a collection of Bowen balls form a maximal
$(\epsilon,n)$-(Bowen) separating set.

The following inequalities are important in establishing that a well-defined
invariant exists. They are used even in the usual definition of entropy (see,
e.g. [KH95, Section 3.1.b])

(2) $\displaystyle S_{f,K}(\epsilon,n)$ $\displaystyle\leq$ $\displaystyle
N_{f,K}(\epsilon,n)$ (3) $\displaystyle N_{f,K}(2\epsilon,n)$
$\displaystyle\leq$ $\displaystyle S_{f,K}(\epsilon,n).$

Note that inequalities (2) and (3) still hold when $f$ is not continuous,
since $d_{f}^{n}$ is still a metric in this case. Uniform continuity is
usually used to guarantee the $d_{f}^{n}$ is equivalent to $d$.

######  Definition 2.1.

A scale $\\{a_{\chi}\\}$ is a family of increasing functions,

$a_{\chi}\colon\mathbb{Z}_{>0}\to\mathbb{R}_{>0}$

indexed by $\chi\in\mathbb{R}_{\geq 0}$, such that if $\chi<\chi^{\prime}$
then $a_{\chi}=o(a_{\chi^{\prime}})$.

######  Notation.

In subscripts, $a_{\chi}$ is used to indicate the scale chosen beforehand, and
$\chi$ is used to indicate that the quantity depends on the value of the
parameter $\chi$ at the given scale $\\{a_{\chi}\\}$.

We think of a scale as a family of functions indexed by $\chi$ that describe
the orbit growth. The corresponding slow entropy $h$ means that the orbits
grow in time as the function $a_{h}$. If the slow entropy with respect to a
given scale is zero (resp. infinity) then, in time $n$ the orbits grow slower
(resp. faster) than the sequence $\\{a_{h}(n)\\}_{n\in\mathbb{Z}_{>0}}$ for
all $h\in\mathbb{R}_{+}$.

######  Example 2.2.

  1. (1)

At exponential scale $a_{\chi}(n)=e^{\chi n}$, we will show, in 2.10, that the
topological slow entropy (similarly, for metric slow entropy and
semitopological slow entropy) and the classical topological entropy are equal.

  2. (2)

At polynomial scale $a_{\chi}(n)=n^{\chi}$, the slow topological entropy is
often called polynomial topological entropy. Similar for other scales.

  3. (3)

Another example is the logarithmic scale $a_{\chi}(n)=n(\log n)^{\chi}$.

These scales have been used, for example, in [KVW19, Theorem 1.7], it was
shown that quasi-unipotent flows have positive polynomial entropy, and [Kan18,
Theorems 1.1 and 1.2] show that the Korchegin flow has positive polynomial
entropy, and the Arnol’d flow has positive logarithmic entropy.

We define

$\delta_{f,K,\chi}^{N}(\epsilon)=\limsup_{n\to\infty}\frac{N_{f,K}(\epsilon,n)}{a_{\chi}(n)},$

and

$\delta_{f,K,\chi}^{S}(\epsilon)=\limsup_{n\to\infty}\frac{S_{f,K}(\epsilon,n)}{a_{\chi}(n)}.$

######  Definition 2.3.

The slow topological entropy of $f$ for the scale $a_{\chi}$ is

$\displaystyle h_{\operatorname{top},a_{\chi}}(f)=\sup_{K}\lim_{\epsilon\to
0}\left(\sup\left\\{\chi:\delta_{f,K,\chi}^{N}(\epsilon)>0\right\\}\right)$
$\displaystyle=\sup_{K}\lim_{\epsilon\to
0}\left(\sup\left\\{\chi:\delta_{f,K,\chi}^{S}(\epsilon)>0\right\\}\right).$

Here, the supremum is taken over all compact sets $K$.

Note that this is well-defined by inequalities (2) and (3). Further, one may
show that the innermost supremum is decreasing in $\epsilon$, so the limit
exists as $\epsilon\to 0$.

######  Remark 2.4.

We index the family of scales $(a_{\chi})$ by the non-negative reals. When
taking the supremum over some property $P$ of $\chi$, it can happen that the
set $\\{\chi:P(\chi)\text{ holds}\\}$ is empty. In such a case, we use the
convention that the $\sup$ is zero. The set
$\left\\{\chi:\delta_{f,K,\chi}^{N}(\epsilon)>0\right\\}$ is an interval that
starts at zero. However, the point
$\hat{\chi}=\sup\left\\{\chi:\delta_{f,K,\chi}^{N}(\epsilon)>0\right\\}$ may
or may not belong to the interval. All this follows because if
$\chi>\hat{\chi}$, then by definition we have
$\chi\not\in\left\\{\chi:\delta_{f,K,\chi}^{N}(\epsilon)>0\right\\};$ if
$\chi<\hat{\chi},$ again by definition we have $\gamma\in(\chi,\hat{\chi})$
such that $\delta^{N}_{f,K,\gamma}>0,$ and thus

$\begin{split}\limsup_{n\to\infty}\frac{N_{f,K}(\epsilon,n)}{a_{\chi}(n)}&=\limsup_{n\to\infty}\frac{N_{f,K}(\epsilon,n)}{a_{\gamma}(n)}\cdot\frac{a_{\gamma}(n)}{a_{\chi}(n)}\\\
&\geq\limsup_{n\to\infty}\frac{N_{f,K}(\epsilon,n)}{a_{\gamma}(n)}>0\end{split}$

since $a_{\chi}(n)=o(a_{\gamma}(n))$.

In conclusion, $\left\\{\chi:\delta_{f,K,\chi}^{N}(\epsilon)>0\right\\}$ is of
the form $[0,\hat{\chi})$ or $[0,\hat{\chi}]$.

######  Remark 2.5.

While many systems on compact metric spaces are continuous, there are certain
natural systems which have discontinuities appearing. In this paper we treat
the case of 3-IETs, but discontinuities appear naturally when considering
first return maps for Poincaré sections of flows. Crucially, it is important
to note that it still makes sense to consider topological entropy for such
systems, but the topological entropy may now depend on the choice of metric on
$X$.

###  2.2. Metric Slow Entropy

For a probability measure-preserving system $(X,\mu,f,\mathcal{B})$, consider
a finite measurable partition $\mathcal{P}=\\{P_{1},\dots,P_{k}\\}$. We call
each set $P_{i}$ an atom of $\mathcal{P}$. Note that every $x\in X$ defines a
coding sequence $(x_{s})\mathrel{\mathop{:}}=(x_{s})_{s\in\mathbb{Z}_{\geq
0}}$, where $x_{s}=j$ if $f^{s}(x)\in P_{j}$. For any $x,y\in X$, the Hamming
distance with respect to the partition $\mathcal{P}$ is the quantity

$\overline{d}_{f,\mathcal{P}}^{n}(x,y)=1-\frac{|\\{0\leq s\leq
n-1:x_{s}=y_{s}\\}|}{n},$

where $|\cdot|$ is the counting measure. The number
$\overline{d}_{f,\mathcal{P}}^{n}(x,y)$ is the proportion of times for which
the orbits of $x$ and $y$ lie in different atoms of the partition
$\mathcal{P}$ up to time $n$.

For $n\geq 0$ and $\epsilon>0$, the $(\epsilon,n)$-Hamming ball centered at
$x\in X$ is the set

$B^{n}_{f,\mathcal{P}}(x,\epsilon)=\\{y\in
X:\overline{d}_{f,\mathcal{P}}^{n}(x,y)<\epsilon\\}.$

Next, $F$ represents a finite subset of $X$, and define the number

$S_{f,\mathcal{P}}(\epsilon,n)=\min\left\\{\operatorname{card}(F):\mu\left(\bigcup_{x\in
F}B^{n}_{f,\mathcal{P}}(x,\epsilon)\right)>1-\epsilon\right\\}.$

For a given scale $a_{\chi}$, we define

$\delta_{f,\mathcal{P},\chi}^{S}(\epsilon)=\limsup_{n\to\infty}\frac{S_{f,\mathcal{P}}(\epsilon,n)}{a_{\chi}(n)},$

and the slow metric entropy for the partition $\mathcal{P}$ is

$h_{\mu,a_{\chi},\mathcal{P}}(f)=\lim_{\epsilon\to
0}\left(\sup\left\\{\chi:\delta_{f,\mathcal{P},\chi}^{S}(\epsilon)>0\right\\}\right).$

######  Definition 2.6.

The slow metric entropy of $f$ with respect to the scale $a_{\chi}$ is defined
as

$h_{\mu,a_{\chi}}(f)=\sup_{\mathcal{P}}h_{\mu,a_{\chi},\mathcal{P}}(f).$

Here the $\sup$ is taken over all finite measurable partitions of $X$.

###  2.3. Semi-topological slow entropy

A notion between topological and metric entropy can be obtained when a natural
metric and measure are linked. Let

$S_{\operatorname{semi},f}(\epsilon,n)=\min\left\\{{\operatorname{card}(F):\mu\left(\bigcup_{x\in
F}B_{f}^{n}(x,\epsilon)\right)>1-\epsilon}\right\\}.$

We similarly let
$\delta_{\operatorname{semi},f,\chi}^{S}(\epsilon)=\displaystyle\limsup_{n\to\infty}\dfrac{S_{\operatorname{semi},f}(\epsilon,n)}{a_{\chi}(n)}$.

######  Definition 2.7.

The semi-topological slow entropy of $f$ with respect to $\mu$ is

$h_{\operatorname{semi},\mu,a_{\chi}}(f)=\limsup_{\epsilon\to
0}\left(\sup_{\chi}\left\\{\chi:\delta_{\operatorname{semi},f,\chi}^{S}(\epsilon)>0\right\\}\right).$

###  2.4. Classical Entropy as Slow Entropy

Throughout, we assume that $(X,d)$ is a metric space, $\mu$ is a probability
measure on $(X,\mathcal{B})$, where $\mathcal{B}$ is the Borel
$\sigma$-algebra of $(X,d)$, and the $f:X\to X$ is a $\mu$-preserving
transformation.

######  Definition 2.8.

The (classical) topological entropy $h_{\operatorname{top}}(f)$ is

(4) $h_{\operatorname{top}}(f)=\sup_{K}\lim_{\epsilon\to
0}\limsup_{n\to\infty}\frac{\log(N_{f,K}(\epsilon,n))}{n}$

We refer to the reader to [KH95, Section 3.1.b] or [Bow71] for an alternative
definition using $S_{f,K}(\epsilon,n)$, and a discussion on how the classical
topological entropy does not depend on the choice of metric $d$ determining
the topology when $X$ is compact and $f$ is continuous.

######  Definition 2.9.

The (classical) metric entropy of $h_{\mu}(f)$ is defined as

(5) $h_{\mu}(f)=\sup_{\mathcal{P}}h_{\mu,\mathcal{P}}(f),$

where the supremum is taken over all finite measurable partitions
$\mathcal{P}$ and

(6)
$h_{\mu,\mathcal{P}}(f)=\lim_{n\to\infty}\frac{1}{n}\text{H}_{\mu}(\vee_{i=0}^{n-1}f^{-i}(\mathcal{P})).$

In Equation 6, for any measurable partition $\mathcal{P}$,

$\text{H}_{\mu}(\mathcal{P})\mathrel{\mathop{:}}=\sum_{P\in\mathcal{P}}-\mu(P)\log(\mu(P)).$

We have the following well-known theorem in the literature that very few
authors proved.

######  Theorem 2.10 (Exponential scales in slow entropy).

Let $a_{\chi}(n)=e^{\chi n}$ and $\mu$ be an ergodic probability measure. Then

$h_{\mu,a_{\chi}}(f)=h_{\operatorname{semi},\mu,a_{\chi}}=h_{\mu}(f)\quad\text{and}\quad
h_{\operatorname{top},a_{\chi}}(f)=h_{\operatorname{top}}(f).$

###### Proof.

To prove that $h_{\operatorname{top},a_{\chi}}=h_{\operatorname{top}}$, we
claim that for any compact set $K$ and $\hat{\epsilon}>0$,

(7)
$\sup\\{\chi:\delta^{N}_{f,K,\chi}(\hat{\epsilon})>0\\}=\limsup_{n\to\infty}\frac{\log(N_{f,K}(\hat{\epsilon},n))}{n}.$

First of all, $\limsup\frac{\log(N_{f,K}(\hat{\epsilon},n))}{n}=\infty$, is
equivalent to: there exists $n_{i}\to\infty$ such that for all $M,\gamma>0$,
there exists $n_{*},$ such that if $n_{i}>n_{*}$, then
$\frac{\log(N_{f,K}(\hat{\epsilon},n_{i}))}{n_{i}}>M+\log(1+\gamma)$. This is
equivalent to
$\frac{N_{f,K}(\hat{\epsilon},n_{i})}{e^{Mn_{i}}}>(1+\gamma)^{n_{i}}$ for all
$n_{i}>n_{*}.$ Remember that

(8)
$\delta^{N}_{f,K,\chi}(\hat{\epsilon})=\limsup_{n\to\infty}\frac{N_{f,K}(\hat{\epsilon},n)}{e^{\chi
n}}.$

So, $\delta_{f,K,M}^{N}(\hat{\epsilon})=\infty.$ Since $M>0$ is arbitrary; we
conclude that it is equivalent that the left side in Equation 7 is equal to
$\infty.$

Now, assume that
$\hat{\chi}=\limsup_{n\to\infty}\frac{\log(N_{f,K}(\hat{\epsilon},n))}{n}<\infty$.

If $\chi>\hat{\chi}$, then the expression in Equation 8 is zero, because

(9) $\limsup_{n\to\infty}\frac{N_{f,K}(\hat{\epsilon},n)}{e^{\chi
n}}=\limsup_{n\to\infty}\frac{N_{f,K}(\hat{\epsilon},n)}{e^{\hat{\chi}n}}\limsup_{n\to\infty}e^{(\hat{\chi}-\chi)n}=0.$

This implies that

(10) $\hat{\chi}\geq\sup\\{\chi:\delta^{N}_{f,K,\chi}(\hat{\epsilon})>0\\}.$

If $\hat{\chi}=0$, Equation 10 is an equality, and it proves Equation 7. If
$\hat{\chi}>0$, let $\chi<\hat{\chi}$, and $n_{i}\to\infty$ any sequence of
positive integers such that
$\frac{N_{f,K}(\hat{\epsilon},n_{i})}{e^{\hat{\chi}n_{i}}}\geq c>0,$ for a
positive constant $c$. Then, we have that

(11) $\frac{N_{f,K}(\hat{\epsilon},n_{i})}{e^{\chi
n_{i}}}e^{(\chi-\hat{\chi})n_{i}}\geq c.$

Since $e^{(\chi-\hat{\chi})n_{i}}\to 0$, then
$\frac{N_{f,K}(\hat{\epsilon},n_{i})}{e^{\chi n_{i}}}\to\infty.$ And then
Equation 8 with $\chi<\hat{\chi}$ is equal to $\infty$. This implies that

(12) $\hat{\chi}=\sup\\{\chi:\delta^{N}_{f,K,\chi}(\hat{\epsilon})>0\\}.$

Finally, to prove that
$h_{\operatorname{top},a_{\chi}}=h_{\operatorname{top}}$, assume that
$h_{\operatorname{top},a_{\chi}}$ and $h_{\operatorname{top}}$ are finite. We
focus only on the case when both entropies are finite and leave the case of
infinite entropy to the reader.

It is enough to prove that for arbitrary $\epsilon>0$, then
$\left|h_{\operatorname{top},a_{\chi}}-h_{\operatorname{top}}\right|<\epsilon.$

By definition of the entropies, and by triangle inequality, there exists
$K\subset X$ compact and $\hat{\epsilon}>0$ sufficiently small, such that

$\begin{split}\left|h_{\operatorname{top},a_{\chi}}-h_{\operatorname{top}}\right|&\leq\left|h_{\operatorname{top},a_{\chi}}-\sup\\{\chi:\delta^{N}_{f,K,\chi}(\hat{\epsilon})>0\\}\right|\\\
+&\left|h_{\operatorname{top}}-\limsup_{n\to\infty}\frac{\log(N_{f,K}(\hat{\epsilon},n))}{n}\right|\\\
+&\left|\sup\\{\chi:\delta^{N}_{f,K,\chi}(\hat{\epsilon})>0\\}-\limsup_{n\to\infty}\frac{\log(N_{f,K}(\hat{\epsilon},n))}{n}\right|\\\
<&\frac{2}{3}\epsilon+\left|\sup\\{\chi:\delta^{N}_{f,K,\chi}(\hat{\epsilon})>0\\}-\limsup_{n\to\infty}\frac{\log(N_{f,K}(\hat{\epsilon},n))}{n}\right|.\end{split}$

By Equation 7,

$\left|h_{\operatorname{top},a_{\chi}}-h_{\operatorname{top}}\right|<\frac{2}{3}\epsilon.$

To prove that $h_{\mu,a_{\chi}}(f)=h_{\mu}(f)$. First, assume that
$h_{\mu}(f)$ is finite. Let $\mathcal{P}=\\{P_{1},\dots,P_{k}\\}$ be a finite
measurable partition with $h_{\mu}(f,\mathcal{P})=h$. By Shannon-McMillan-
Breiman Theorem [VO16, Theorem 9.3.1], for the partition $\mathcal{P}$ we have
that

$-\frac{1}{n}\log\mu([x]_{0}^{n-1})\rightarrow{h}\quad\mu{\text{-}a.e.}$

We use $[x]_{0}^{n-1}$ to denote the atom in
$\displaystyle\vee_{i=0}^{n-1}T^{-i}\mathcal{P}$ that contains $x$. Let
$\epsilon>0$, for $n$ be large enough, we get

$\mu\Big{(}x:\left|-\frac{1}{n}\log\mu([x]_{0}^{n-1})-h\right|<\epsilon\Big{)}>1-\epsilon.$

This implies:

(13)
$\mu\Big{(}x:e^{-n(h+\epsilon)}<\mu([x]_{0}^{n-1})<e^{-n(h-\epsilon)}\Big{)}>1-\epsilon.$

Note for any $y\in[x]_{0}^{n-1}$, $y_{j}=x_{j}$ for any $0\leq j\leq n-1$,
therefore $S_{f,\mathcal{P}}(\epsilon,n)\leq e^{n(h+\epsilon)}$. Hence, when
$a_{\chi}(n)=e^{\chi n}$, $h_{\mu,a_{\chi},\mathcal{P}}(f)\leq h$. Moreover,
$B_{f,\mathcal{P}}^{n}(x,\epsilon)$ is covered by at most $\dbinom{n}{\lfloor
n\epsilon\rfloor}(\lfloor n\epsilon\rfloor^{k})$ atoms in
$\displaystyle\bigvee_{i=0}^{n-1}T^{-i}\mathcal{P}$. This is because for any
$y\in B^{n}_{f,\mathcal{P}}(x,\epsilon)$, $\\{x_{i}\\}_{i=0}^{n-1}$ and
$\\{y_{i}\\}_{i=0}^{n-1}$ differ in at most $\lfloor n\epsilon\rfloor$
positions, each one has at most $k$ choices. From Equation 13, each atom is of
measure at least $e^{-n(h+\epsilon)}$, and they cover space of measure at
least $1-\epsilon$. Hence,
$S_{f,\mathcal{P}}(\epsilon,n)\geq(1-\epsilon)e^{n(h-\mathcal{O}(\epsilon))}$
since $k$ is a fixed constant and

$\dbinom{n}{n\epsilon}\approx
e^{n(\epsilon\ln(\epsilon)+(1-\epsilon)\log(1-\epsilon))}/\sqrt{2\pi
n\epsilon(1-\epsilon)}$

when $n$ is large enough. Therefore, $h_{\mu,a_{\chi},\mathcal{P}}(f)\geq h$,
proving the equality $h_{\mu,\mathcal{P}}(f)=h_{\mu,a_{\chi},\mathcal{P}}(f)$.
By definition,
$\displaystyle\sup_{\mathcal{P}}h_{\mu,\mathcal{P}}(f)=h_{\mu}(f)$ and
$\displaystyle
h_{\mu,a_{\chi}}(f)=\sup_{\mathcal{P}}h_{\mu,a_{\chi},\mathcal{P}}(f)$, these
give $h_{\mu}(f)=h_{\mu,a_{\chi}}(f)$. If $h_{\mu}(f)=\infty$, then for any
finite partition $\mathcal{P}$ with finite entropy, we still have
$h_{\mu,\mathcal{P}}(f)=h_{\mu,a_{\chi},\mathcal{P}}(f)$, and the result
follows by taking the $\sup$ over a sequence of partitions with finite entropy
that go to infinity. Therefore, $h_{\mu}(f)=h_{\mu,a_{\chi}}(f)=\infty$.

We omit the proof of $h_{\operatorname{semi},\mu,a_{\chi}}=h_{\mu}$ and refer
the reader to [Kat80, Theorem (I.I)]. ∎

##  3\. Structural Theorems

###  3.1. Slow Goodwyn’s theorem

The following result states a relation among the different entropies that we
defined in Section 2. In the setting of classical entropy theory, this is part
of the variational principle, due to Goodwyn [Goo69]. It states that under
general circumstances, the metric entropy is bounded above by the semi-
topological entropy and that the topological entropy is the largest of the
previous two.

######  Theorem 3.1 (Goodwyn).

Let $f\colon X\to X$ be a measurable transformation of a compact metric space,
and $\mathcal{M}^{f}(X)$ denote the space of $f$-invariant Borel probability
measures on $X$. Then for any measure $\mu\in\mathcal{M}^{f}(X)$

(14) $h_{\mu,a_{\chi}}(f)\leq h_{\operatorname{semi},\mu,a\chi}(f)\leq
h_{\operatorname{top},a_{\chi}}(f).$

######  Remark 3.2.

As in 2.5, we note that continuity is not required for this theorem.

To prove this theorem, we need an auxiliary result on the existence of certain
partitions. In this result, the remarkable part is that the atoms of the
partitions are small and their boundary is of measure zero.

######  Lemma 3.3.

Let $(X,d)$ be a compact metric space with a probability measure $\mu$. Then
for every $\delta>0$, there exists a partition
$\mathcal{P}=\\{P_{1},\dots,P_{k}\\}$ such that
$\operatorname{diam}(P_{i})<\delta$ for every $i=1,\dots,k$ and if
$X_{\epsilon}=\\{x\in X:d(x,\partial\mathcal{P})<\epsilon\\}$,
$\lim_{\epsilon\to 0}\mu(X_{\epsilon})=0$, where

$\displaystyle
d(x,\partial\mathcal{P})\mathrel{\mathop{:}}=\max_{P\in\mathcal{P}}d(x,\partial
P).$

Furthermore, if $x,y\in X$ belong to different atoms of the partition and
$x,y\not\in X_{\epsilon}$, then $d(x,y)\geq\epsilon$.

###### Proof of 3.1.

We will start by proving that $h_{\mu,a_{\chi}}\leq
h_{\operatorname{semi},\mu,a_{\chi}}$. Let $\epsilon>0$. Let
$\mathcal{P}_{\epsilon}$ be the partition in 3.3 for $\delta=\epsilon$, so
each set in $\mathcal{P}_{\epsilon}$ has diameter at most $\epsilon$. From
3.3, we conclude that the open set $X_{\hat{\epsilon}}$ has $\mu$ measure at
most $\epsilon$ for some $\hat{\epsilon}<\epsilon$, and if $x,y\in
X_{\hat{\epsilon}}$, $d(x,y)<\hat{\epsilon}$, then
$[x]_{\mathcal{P}_{\epsilon}}=[y]_{\mathcal{P}_{\epsilon}}$. We have that the
compact set $K_{\hat{\epsilon}}\mathrel{\mathop{:}}=X\backslash
X_{\hat{\epsilon}}$ has $\mu$ measure at least $1-\epsilon$, and satisfies
that for any $x\in K_{\hat{\epsilon}}$, we must have

(15) $B^{n}_{f,\mathcal{P}_{\epsilon}}(x,\epsilon)\supset
B^{n}_{f}(x,\hat{\epsilon}).$

By definition of $S_{f,\mathcal{P}_{\epsilon}}(\epsilon,n)$, for every $n$, we
may choose $F_{n}\subset K_{\epsilon}$, such that
$\operatorname{card}(F_{n})=S_{f,\mathcal{P}_{\epsilon}}(\epsilon,n)$ and

$\mu\left(\cup_{x\in
F_{n}}B_{f,\mathcal{P}_{\epsilon}}^{n}(x,\epsilon)\right)>1-\epsilon.$

Using Equation 15, we see that

$S_{f,\mathcal{P}_{\epsilon}}(\epsilon,n)\leq
S_{\operatorname{semi},f}(\hat{\epsilon},n)=\min\\{\operatorname{card}(H):\mu\left(\cup_{x\in
H}B^{n}_{f}(x,\hat{\epsilon})\right)>1-\hat{\epsilon}\\},$

and

$\delta_{f,\mathcal{P},\chi}^{S}(\epsilon)\leq\delta_{\operatorname{semi},f,\chi}^{S}(\hat{\epsilon}).$

Since $\epsilon\to 0$ implies that $\hat{\epsilon}\to 0$, it follows that

(16) $\lim_{\epsilon\to
0}\left(\sup_{\chi}\left\\{\delta_{f,\mathcal{P},\chi}^{S}(\epsilon)>0\right\\}\right)\leq\lim_{\hat{\epsilon}\to
0}\left(\sup_{\chi}\left\\{\delta_{\operatorname{semi},f,\chi}^{S}(\hat{\epsilon})>0\right\\}\right).$

What we have accomplished with the LABEL:{eq:AlmostMeasureSmallerThanSemi} is
that $h_{\mu,a_{\chi},\mathcal{P}_{\delta}}\leq
h_{\operatorname{semi},\mu,a_{\chi}},$ for a generating partition
$\mathcal{P}_{\delta}$ with atoms of diameter at most $\delta>0.$ The result
follows from [KT97, Proposition 1] where the authors proved that the $\sup$ in
$h_{\mu,a_{\chi}}=\sup_{\mathcal{P}}h_{\mu,a_{\chi},\mathcal{P}}$ can be
replaced by a limit $\lim_{m\to\infty}h_{\mu,a_{\chi},\mathcal{P}_{m}}$ over a
sequence of generating partitions $\\{\mathcal{P}_{m}\\}$.

Now we prove the inequality $h_{\operatorname{semi},\mu,a_{\chi}}\leq
h_{\operatorname{top},a_{\chi}}$, this part does not require continuity and
follows directly from definitions. Notice that in 2.3, we can drop the $\sup$
over compact subsets of $X$ and substitute $K=X$. Observe that for all
$\epsilon>0$, and all $n$ sufficiently large:

$S_{\operatorname{semi},f}(\epsilon,n)\leq N_{f,\chi}(\epsilon,n).$

This completes the proof of the inequalities in Equation 14. ∎

###### Proof of 3.3.

Since $\mu$ is finite, the set $X^{\prime}$ of atoms of $\mu$ is at most
countable. For $\delta>0$, let $0<\delta^{\prime}<\delta/2$ be such that if
$B$ is a ball with center in an atom and radius $\delta^{\prime}$, then
$\mu(\partial B)=0$. Similarly, compact set $X\backslash\cup_{x\in
X^{\prime}}B(x,\delta^{\prime})$ has a finite covering of balls of radius at
most $\delta/2$, and the boundary of these balls has measure zero. Hence,
there exists a finite covering of balls $B_{1},\dots,B_{k}$ with radius at
most $\delta/2$ and $\mu(\partial B_{j})=0$, for $1\leq j\leq k$. The
partition $\mathcal{P}$ is constructed by recursion: We put
$P_{1}=\overline{B_{1}}$, and
$P_{j}=\overline{B_{j}}\backslash\cup_{i=1}^{j-1}\overline{B_{i}}$ for $2\leq
j\leq k$.

It follows that $\lim_{\epsilon\to
0}\mu(X_{\epsilon})=\mu(\cup_{i=1}^{k}\partial P_{i})=0$, because the
probability measure is outer regular and $\cup_{i=1}^{k}\partial
P_{i}\subset\cup_{i=1}^{k}\partial B_{i}.$

To prove the final observation, let $x,y\not\in X_{\epsilon}$, i.e.
$d(x,\cup_{i=1}^{k}\partial P_{i})>\epsilon$ and $d(y,\cup_{i=1}^{k}\partial
P_{i})>\epsilon.$ If there are $1\leq i<j\leq k$, with $x\in P_{i}$, and $y\in
P_{j}$, then $d(x,y)>d(x,\partial B_{i})+d(y,\partial B_{i})\geq 2\epsilon.$ ∎

###  3.2. Homogeneous Measures and Slow Entropy

The following result is of a classification type. It states that for a class
of systems, both the semi-topological and topological entropies are the same.
In other words, the inequality on the right in Equation 14 is in fact, an
equality.

######  Definition 3.4.

Let $f\colon X\to X$ be a measurable transformation. A measure $\mu$ is called
homogeneous with respect to $f$, if for every $\epsilon>0$ there exists $c>0$
such that for any $x,y\in X$, and every $n\geq n_{0}$, for some $n_{0}$
sufficiently large,

$\frac{1}{c}\leq\frac{\mu(B_{f}^{n}(x,\epsilon))}{\mu(B_{f}^{n}(y,\epsilon))}<c.$

######  Theorem 3.5.

Suppose that $f\colon X\to X$ is a measurable transformation, $X$ is compact,
and $\mu$ is homogeneous. Then

$h_{\operatorname{top},a_{\chi}}(f)=h_{\operatorname{semi},\mu,a_{\chi}}(f)=\lim_{\epsilon\to
0}\left(\sup\\{\chi:C_{f,\chi}(\epsilon)>0\\}\right)$

where

$C_{f,\chi}(\epsilon)=\limsup_{n\to\infty}\dfrac{1}{a_{\chi}(n)\cdot\mu(B_{f}^{n}(x,\epsilon))}.$

###### Proof.

Let $\displaystyle k_{\mu,a_{\chi}}(f)$ be the quantity
$\displaystyle\lim_{\epsilon\to
0}\left(\sup\\{\chi:C_{f,\chi}(\epsilon)>0\\}\right).$

First we show that $h_{\operatorname{top},a_{\chi}}(f)=k_{\mu,a_{\chi}}(f).$
Given a compact set $K$, by locally compactness of $X$, we can choose a
sufficiently small $\epsilon$ and an open set $U$ such that
$B(K,\epsilon)\subset U.$ Denote
$\hat{\chi}:=\sup\left\\{\chi:\delta^{S}_{f,K,\chi}(\epsilon)>0\right\\}$.

On the one hand, fix a set $E$ such that for all distinct $x,y\in E$ we have
$B^{n}_{f}(x,\epsilon)\cap B^{n}_{f}(y,\epsilon)=\varnothing,$ and thus
$\cup_{y\in E}B^{n}_{f}(y,\epsilon)\subset U$ is a disjoint union. Since $\mu$
is homogeneous, we can choose $c>0$ such that for any $x,y\in X,$

$\displaystyle\mu\left(B^{n}_{f}(x,\epsilon)\right)\leq
c\mu\left(B^{n}_{f}(y,\epsilon)\right).$

Summing over $y\in E$ we obtain

$\displaystyle\mu\left(B^{n}_{f}(x,\epsilon)\right)\operatorname{card}(E)\leq
c\sum_{y\in E}\mu\left(B^{n}_{f}(y,\epsilon)\right)\leq c\mu(U)\leq c,$

which implies that for any $x\in X$ we have

$\displaystyle\mu\left(B^{n}_{f}(x,\epsilon)\right)\cdot
S_{f,K}(\epsilon,n)\leq c,$

therefore

$0<\frac{1}{c}\leq\limsup_{n\to\infty}\dfrac{1}{\mu(B_{f}^{n}(x,\epsilon))\cdot
S_{f,K}(\epsilon,n)}.$

We will use the following, although we do not prove it. This follows from the
definitions of $\lim$ and $\limsup$

######  Claim.

For two sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$, if $\lim_{n\to\infty}a_{n}$
exists and it is positive and $\limsup_{n\to\infty}b_{n}$ is positive, then

$\limsup_{n\to\infty}a_{n}\cdot
b_{n}=\lim_{n\to\infty}a_{n}\limsup_{n\to\infty}b_{n}.$

For $\chi<\hat{\chi},$ take $\chi<\chi^{\prime}<\hat{\chi}$, using the claim,
we have that

$\displaystyle C_{f,\chi}(\epsilon)$
$\displaystyle=\limsup_{n\to\infty}\dfrac{1}{a_{\chi}(n)\cdot\mu(B_{f}^{n}(x,\epsilon))}$
$\displaystyle=\limsup_{n\to\infty}\dfrac{S_{f,K}(\epsilon,n)}{a_{\chi^{\prime}}(n)}\cdot\dfrac{a_{\chi^{\prime}}(n)}{a_{\chi}(n)}\cdot\dfrac{1}{\mu(B_{f}^{n}(x,\epsilon))\cdot
S_{f,K}(\epsilon,n)}$ $\displaystyle=\infty.$

Denote $k_{\mu,a_{\chi},\epsilon}(f):=\sup\\{\chi:C_{f,\chi}(\epsilon)>0\\}$
and

$h_{\operatorname{top},a_{\chi},K,\epsilon}(f):=\sup\left\\{\chi:\delta_{f,K,\chi}^{S}(\epsilon)>0\right\\}.$

It follows that for any $\chi<\hat{\chi}$ we have

$\displaystyle k_{\mu,a_{\chi},\epsilon}(f)\geq\chi.$

By taking supremum over $\chi<\hat{\chi}$ on both sides we obtain

$\displaystyle
k_{\mu,a_{\chi},\epsilon}(f)\geq\hat{\chi}=h_{\operatorname{top},a_{\chi},K,\epsilon}(f).$

Since both $K$ and $\epsilon$ are arbitrary, it is clear that

$\displaystyle k_{\mu,a_{\chi}}(f)\geq h_{\operatorname{top},a_{\chi}}(f).$

On the other hand, assume $\mu(K)>0$. Since $\mu$ is homogeneous, we have

(17)
$\displaystyle\mu\left(B^{n}_{f}(x,\epsilon)\right)\geq\frac{1}{c}\mu\left(B^{n}_{f}(y,\epsilon)\right).$

Fix a finite covering $\left\\{B^{n}_{f}(x,\epsilon):x\in F\right\\}$ of $K$
consisting of $(n,\epsilon)$-Bowen balls so that $\cup_{y\in
F}B^{n}_{f}(y,\epsilon)\supset K$. By summing Equation 17 over $y\in F$, we
obtain

$\displaystyle\mu\left(B^{n}_{f}(x,\epsilon)\right)\operatorname{card}(F)\geq\frac{1}{c}\cdot\sum_{y\in
F}\mu\left(B^{n}_{f}(y,\epsilon)\right)\geq\frac{1}{c}\mu(K),$

which implies that for any $x\in X$ we have

$\displaystyle\mu\left(B^{n}_{f}(x,\epsilon)\right)N_{f,K}(\epsilon,n)\geq\frac{1}{c}\mu(K).$

For $\chi>\hat{\chi}$ we have

$\displaystyle C_{f,\chi}(\epsilon)$
$\displaystyle=\limsup_{n\to\infty}\dfrac{1}{a_{\chi}(n)\cdot\mu(B_{f}^{n}(x,\epsilon))}$
$\displaystyle=\limsup_{n\to\infty}\dfrac{N_{f,K}(\epsilon,n)}{a_{\hat{\chi}(n)}}\cdot\dfrac{a_{\hat{\chi}}(n)}{a_{\chi}(n)}\cdot\dfrac{1}{\mu(B_{f}^{n}(x,\epsilon))N_{f,K}(\epsilon,n)}$
$\displaystyle=0,$

and thus $k_{\mu,a_{\chi},\epsilon}(f)\leq\chi$ for any $\chi>\hat{\chi}.$ It
follows that
$k_{\mu,a_{\chi},\epsilon}(f)\leq\hat{\chi}=h_{\operatorname{top},a_{\chi},K,\epsilon}(f)$
and thus $k_{\mu,a_{\chi}}(f)\leq h_{\operatorname{top},a_{\chi}}(f).$ Now we
can conclude that $h_{\operatorname{top},a_{\chi}}(f)=k_{\mu,a_{\chi}}(f).$

Secondly, we show that $h_{\operatorname{semi},a_{\chi}}(f)\geq
k_{\mu,a_{\chi}}(f).$ Given $\epsilon>0,$ define

$\displaystyle\tilde{\chi}$
$\displaystyle\mathrel{\mathop{:}}=\sup\left\\{\chi:\delta_{\operatorname{semi},f,\chi}^{S}(\epsilon)>0\right\\},$

in which case we have
$\displaystyle\delta_{\operatorname{semi},f,\tilde{\chi}}^{S}(\epsilon)=\limsup_{n\to\infty}\frac{S_{\operatorname{semi},f}(\epsilon,n)}{a_{\tilde{\chi}}(n)},$
where

$\displaystyle
S_{\operatorname{semi},f}(\epsilon,n)=\min\left\\{\operatorname{card}(F):\mu\left(\cup_{x\in
F}B^{n}_{f}(x,\epsilon)\right)>1-\epsilon\right\\}.$

Consider a finite measurable set $F$ such that $\mu\left(\cup_{x\in
F}B^{n}_{f}(x,\epsilon)\right)>1-\epsilon.$ Since $\mu$ is homogeneous, then
there exists $c>0$ such that
$\mu\left(B^{n}_{f}(x,\epsilon)\right)\geq\frac{1}{c}\mu\left(B^{n}_{f}(y,\epsilon)\right).$
Summing over $y\in F$ we obtain

$\displaystyle\mu\left(B^{n}_{f}(x,\epsilon)\right)\operatorname{card}(F)\geq\sum_{y\in
F}\frac{1}{c}\mu\left(B^{n}_{f}(y,\epsilon)\right)\geq\frac{1}{c}\mu\left(\cup_{y\in
F}B^{n}_{f}(y,\epsilon)\right)>\frac{1-\epsilon}{c}.$

Since $F$ is arbitrary, we have
$\mu\left(B^{n}_{f}(x,\epsilon)\right)S_{\operatorname{semi},f}(\epsilon,n)\geq\frac{1-\epsilon}{c}.$
Hence for $\chi>\tilde{\chi}$

$\displaystyle C_{f,\chi}(\epsilon)$
$\displaystyle=\limsup_{n\to\infty}\frac{1}{a_{\chi}(n)\cdot\mu\left(B^{n}_{f}(x,\epsilon)\right)}$
$\displaystyle=\limsup_{n\to\infty}\frac{a_{\tilde{\chi}}(n)}{a_{\chi}(n)}\cdot\frac{S_{\operatorname{semi},f}(\epsilon,n)}{a_{\tilde{\chi}}(n)}\cdot\frac{1}{\mu\left(B^{n}_{f}(x,\epsilon)\right)\cdot
S_{\operatorname{semi},f}(\epsilon,n)}$
$\displaystyle\leq\frac{c}{1-\epsilon}\lim_{n\to\infty}\frac{a_{\tilde{\chi}}(n)}{a_{\chi}(n)}\limsup_{n\to\infty}\frac{S_{\operatorname{semi},f}(\epsilon,n)}{a_{\tilde{\chi}}(n)}=0.$

Letting $\epsilon\to 0$, as in the first part, we obtain that
$k_{\mu,a_{\chi}}(f)\leq h_{\operatorname{semi},a_{\chi}}(f).$

Finally, by 3.1 we have $h_{\operatorname{semi},a_{\chi}}(f)\leq
h_{\operatorname{top},a_{\chi}}$, we proved that
$h_{\operatorname{top},a_{\chi}}(f)=k_{\mu,a_{\chi}}(f)$ in the first part.
Thus the 3.5 follows. ∎

##  4\. Ferenczi’s Theorem

Here we will present the proof of [Fer97, Proposition 3], which provides a
characterization of Kronecker systems via slow entropy. This section is purely
expository, we include it to provide a complete account of the current state
of the theory.

######  Definition 4.1.

A topological dynamical system is called a Kronecker system if it is
isomorphic to a group rotation on a compact abelian metrizable group.

######  Theorem 4.2 ([Fer97]).

Let $\overline{X}=(X,\mu,f,\mathcal{M})$ be a probability measure preserving
system. Then, $\overline{X}$ is isomorphic to the Kronecker system if and only
if $h_{\mu,a_{\chi}}(f)=0$ for all scales $a_{\chi}.$

For the second part of the proof of 4.2, we need the following result.

######  Lemma 4.3.

If $\overline{X}=(X,\mu,f,\mathcal{M})$ is not isomorphic to a Kronecker
system, then we can find a partition $\mathcal{P}=\\{P_{1},P_{2}...,P_{l}\\}$
and $\epsilon_{0}>0$ such that,
$d(f^{n}(\mathcal{P}),\mathcal{P})>\epsilon_{0}$ for every $n$ in a density 1
subset $D\subseteq\mathbb{N}$.

For two partitions $\mathcal{P}=\\{P_{1},\dots,P_{n}\\}$ and
$\mathcal{Q}=\\{Q_{1},\dots,Q_{n}\\}$, we denote the partition distance,
between $\mathcal{P}$ and $\mathcal{Q}$ by

$\displaystyle
d(\mathcal{P},\mathcal{Q})\mathrel{\mathop{:}}=\displaystyle\sum_{i=1}^{n}\mu(P_{i}\Delta
Q_{i}).$

###### Proof of 4.2.

If $\overline{X}$ is isomorphic to the Kronecker system, we can assume $f$ is
an isometry and $X$ is a compact metric space with metric $d$. Let $\delta>0$
be a fixed constant, and let
$\mathcal{P}\mathrel{\mathop{:}}=\mathcal{P}_{\delta}$ be a measurable
partition of $X$ described in 3.3. Given $\epsilon>0$, we want to show
$S_{f,\mathcal{P}}(\epsilon,n)\leq C_{\epsilon}$ when $n$ is large enough
where $C_{\epsilon}$ is a constant independent of $n$. Let $\delta(\epsilon)$
be a constant depending only on $\epsilon$, define $X_{\epsilon}=\\{x\in
X:d(x,\partial{P})<\delta(\epsilon)\text{\;for some \;}P\in\mathcal{P}\\}$. By
taking $\epsilon$ small enough, $\mu(X_{\epsilon})<\epsilon$. Let
$C_{\epsilon}$ be the minimal number of balls of radius $\delta(\epsilon)/2$
covering $X$. Now, for every $n$ large enough

(18)
$\mu\left(\left\\{x:\left|\frac{1}{n}\sum_{i=0}^{n-1}\mathds{1}_{X_{\epsilon}}(f^{i}(x))\right|<\epsilon\right\\}\right)>1-\epsilon.$

Therefore, by Equation 18, the set of indices

$E_{x}^{n}=\left\\{0\leq j<n:f^{j}(x)\in X_{\epsilon}^{c}\right\\}$

satisfies that $|E_{x}^{n}|>n(1-\epsilon)$ for $x$ in a set with measure at
least $1-\epsilon$. For such $x$, suppose $y$ is in the same ball of radius
$\delta(\epsilon)/2$ containing $x$. Then, when $j\in E_{x}^{n}$, $f^{j}x$ and
$f^{j}y$ are in the same atom of partition $\mathcal{P}$ because $f$ is an
isometry. Therefore, $S_{f,\mathcal{P}}(\epsilon,n)\leq C_{\epsilon}$ because
each Hamming ball contains a Bowen ball of radius $\delta(\epsilon)/2$. Since
the Bowen balls are exactly the balls in the metric $d$,
$S_{f,\mathcal{P}}(\epsilon,n)$ is bounded by a constant depending only on
$\mathcal{P}$ and $\epsilon$. Consequently, any Kronecker system has zero
entropy at all scales.

To prove the other direction, we need 4.3. Assume $h_{\mu,a_{\chi}}(f)=0$ for
all scales $a_{\chi}$, we claim that

$\displaystyle\liminf_{n\rightarrow\infty}S_{f,\mathcal{P}}(\epsilon,n)\leq
C_{\epsilon}$

for any fixed measurable partition $\mathcal{P}$ and given $\epsilon$. Suppose
not, then there exists a measurable partition $\mathcal{P}_{0}$ and
$\epsilon_{0}$ $s.t.$ for any $k$ there exists $n_{k}$ satisfying
$S_{f,\mathcal{P}_{0}}(\epsilon,n)>k$ for every $\epsilon<\epsilon_{0}$ and
$n_{k}<n$. Choose $a_{\chi}(n)$ to be some value between $k$ and $k+1$ when
$n_{k}\leq n<n_{k+1}$. This implies $h_{\mu,a_{\chi},\mathcal{P}_{0}}(f)>0$,
which is a contradiction.

If the system $\overline{X}$ is not isomorphic to a Kronecker system, by
applying 4.3 we can find a measurable partition
$\mathcal{P}=\\{P_{1},...,P_{l}\\}$. Fixed an arbitrarily large $N$, there
exists a constant $C$ $s.t.$ $S_{f,\mathcal{P}}(\epsilon^{2},N)\leq C$. By
using $M\leq C$ Hamming balls
$B_{i}=B_{f,\mathcal{P}}^{N}(x^{i},\epsilon^{2})$ where $1\leq i\leq M$, one
can cover at least $1-\epsilon^{2}$ space of $X$. Consider space
$(X\times\\{0,1,\dots,N-1\\},\mu\times\nu)$, where
$\nu(E)=\frac{1}{N}\times\text{cardinality of }E$. Define a measurable
function on $X\times\\{0,1,\dots,N-1\\}$ by

$g(x,n)=\left\\{\begin{aligned} 1&\;\text{if there exists $i$ $s.t.$ $x\in
B_{i}$ and $f^{n}(x)$ and $f^{n}(x^{i})$ are }\\\ &\;\text{in the same
atom}\\\ 0&\;\text{others}\\\ \end{aligned}\right.$

By definition of Hamming balls,

$\int g\,d\mu\times\nu>(1-\epsilon^{2})(1-\epsilon^{2}).$

Now, there exists a subset $E$ of $\\{0,1,\dots,N-1\\}$ with cardinality
larger than $N(1-4\epsilon)$ satisfying for every $n\in E$, $\int
g\,d\mu>1-\epsilon/2$. This is because we can get $\int gd\nu<1-2\epsilon^{2}$
otherwise.

We claim that we can find a positive density subset $\Lambda$ of $\mathbb{N}$
$s.t.$ for every $n\in\Lambda$, $d(f^{n}(\mathcal{P}),\mathcal{P})<\epsilon$,
which is a contradiction to 4.3. Note that

$d(f^{n}(\mathcal{P}),f^{j}(\mathcal{P}))=\int_{X}\min\\{|x_{n}-x_{j}|,1\\}d\mu(x)$

for every $n$ and $j$. Moreover, for each $n$, $(x_{n}^{1},\dots,x_{n}^{M})$
has at most $l^{M}$ choices. Therefore, for every $n\in E$, there exists an
integer $m(n)$, $s.t.$ $x_{n}^{i}=x_{m(n)}^{i}$ for every $1\leq i\leq M$.
Hence, $d(f^{n}(\mathcal{P}),f^{m(n)}(\mathcal{P}))<\epsilon$ because the set
satisfying there exists $i$ $s.t.$ $x\in B_{i}$ and $x_{n}^{i}=x_{n}$ has
measure larger than $1-\epsilon$ for every $n\in E$. From the pigeonhole
principle, there exists a subset $\Lambda_{N}$ of $\\{0,1,\dots,N-1\\}$ with
cardinality at least $N(1-4\epsilon)/l^{M}$ such that for every
$n\in\Lambda_{N}$, $d(f^{n}(\mathcal{P}),f^{m}(\mathcal{P}))<\epsilon$ for
some $m\in\\{0,1,\dots,N-1\\}$. Since the metric is $f$-invariant, we can
conclude the result. ∎

######  Definition 4.4.

If $\overline{X}=(X,\mu,f,\mathcal{M})$ is a measure preserving system, we say
a function $g\in L^{2}(X,\mu)$ is almost periodic if
$\\{U_{f}^{n}g:n\in\mathbb{Z}\\}$ is a precompact set in $L^{2}(X,\mu)$.

###### Proof of 4.3.

We give a sketch of proof here. If the system is isomorphic to a Kronecker
system, then $g$ is almost periodic for any $g\in L^{2}(X)$. Suppose
$\overline{X}$ is not isomorphic to the Kronecker system, then there exists a
non-zero measurable function $g$ with mean $0$ such that $g$ is orthogonal to
every almost periodic function. Then, there is a density 1 subset $D$ of
$\mathbb{N}$ $s.t.$

$\displaystyle\lim_{n\rightarrow\infty,n\in D}\langle
U_{f}^{n}(g),g\rangle=0.$

Assume w.l.o.g. that $|\\!|g|\\!|=1$, given $\epsilon=\frac{1}{100},$ there is
a simple function $h=\sum\limits_{i=1}^{l}c_{i}\mathds{1}_{A_{i}}$ such that
$|\\!|g-h|\\!|\leq\epsilon/4.$ Therefore, when $n$ is large enough,

$|\langle U_{f}^{n}(h),h\rangle|\leq\epsilon.$

Choose partition $\mathcal{P}=\\{A_{1},...,A_{l}\\}$, then there exists
$\epsilon_{0}<\frac{\min\\{\mu(A_{1}),...,\mu(A_{l})\\}}{100}$, such that
$d(f^{n}(\mathcal{P}),\mathcal{P})>\epsilon_{0}$ for $n\in D.$ ∎

##  5\. Differences between Slow and Exponential Entropy

###  5.1. Sturmian subshifts

For more information about Sturmian systems, we refer the reader to [Fog02,
Chapter 6].

######  Definition 5.1.

A sequence $u\in\\{0,1\\}^{\mathbb{N}}$ $(\in\\{0,1\\}^{\mathbb{Z}})$ is a
(bi-infinite) Sturmian sequence if for every $n\geq 1$, the number of words of
length $n$ that appear in $u$ is equal to

$p_{n}(u)=n+1,$

and it is not eventually periodic.

Consider $f\colon\\{0,1\\}^{\mathbb{Z}}\to\\{0,1\\}^{\mathbb{Z}}$, the shift
map defined as $w=f(v)$, $w_{i}=v_{i+1}$.

######  Definition 5.2.

A Sturmian system $(X_{u},f)$ for a bi-infinite Sturmian sequence $u$ is the
shift map $f$ on $X_{u}=\overline{\\{f^{k}(u):k\in\mathbb{Z}\\}}$.

######  Theorem 5.3.

Any Sturmian system $(X_{u},f)$ is measurably equivalent to an irrational
rotation $(\mathbb{R}/\mathbb{Z},R_{\theta})$.

###### Sketch of the proof with ideas in [Fog02, Chapter 6] .

Any initial point $\beta\in\mathbb{R}/\mathbb{Z}$, generates a Sturmian
sequence $v=(v_{i})$, by taking $v_{i}=j$ if $R_{\theta}^{i}(\beta)\in I_{j}$,
$I_{0}=[0,1-\theta)$ and $I_{1}=[1-\theta,1)$.

The converse is very elaborate see [Fog02, Sections 6.3 and 6.4]. The idea is
that every $v\in X_{u}$ has a coding $(a_{n},b_{n})$. The coefficients
$(a_{n})$ are the partial quotients of the continued fraction expansion of
$\theta$. The initial point $\beta$ is determined by $b_{n}$; these are the
coefficients of its Ostrowski expansion. Things are very subtle; for instance,
see [Fog02, Exercise 6.2.13 item 5]. ∎

The 5.3 combined with 4.2 implies that Sturmian systems have 0 metric entropy
at all scales. The following immediately implies 1.2:

######  Theorem 5.4.

For any Sturmian system $(X_{u},f)$, we have that

$0=h_{\mu,a_{\chi}}<h_{\operatorname{semi},\mu,a_{\chi}}=h_{\operatorname{top},a_{\chi}}=1,$

for the polynomial scale $a_{\chi}(N)=N^{\chi}$, and the unique $f$-invariant
measure $\mu$.

###### Proof.

We can define a metric that generates the product topology on
$\\{0,1\\}^{\mathbb{Z}}$ as:

$d(v,w)=\frac{1}{2^{k}},$

for all $v,w\in\\{0,1\\}^{\mathbb{Z}}$ and

$k=\inf\\{n\geq 0:v_{n}\not=w_{n}\text{ or }v_{-n}\not=w_{-n}\\}.$

A basis element for the topology is of the form

$[k;b_{0},b_{1},\dots,b_{l-1}]=\\{v\in\\{0,1\\}^{\mathbb{Z}}:v_{i}=b_{i-k}\text{
for }i\in\\{k,\dots,k+l-1\\}\\}.$

Proof of $h_{\operatorname{top},a_{\chi}}=1$.

We will compute the number of separating $(\epsilon,N)-$Bowen balls.

Fix $\epsilon>0$, then let $n\geq 0$ be the smallest $n$ such that
$2^{-n}\leq\epsilon$. Then for every $N\geq 1$,

$B^{N}_{f}(x,\epsilon)=[-n;x_{-n},\dots,x_{n+N-1}].$

Remember that for every $0\leq i\leq N-1$, if $v\in B^{N}_{f}(x,\epsilon)$,
then

$d(f^{i}(x),f^{i}(v))<\epsilon.$

Since $X_{u}$ is a Sturmian system, there are $2n+N+1$ admissible words of
length $2n+N$. Hence,

$\delta_{f,X_{u},\chi}(\epsilon)=\limsup_{N\to\infty}\frac{N_{f,X_{u}}(\epsilon,N)}{N^{\chi}}=\lim_{N\to\infty}\frac{2n+N+1}{N^{\chi}}=\left\\{\begin{array}[]{cc}0,&\chi>1,\\\
1,&\chi=1,\\\ \infty,&\chi<1.\end{array}\right.$

This proves that for the linear scale $a_{\chi}(N)=N^{\chi}$,

$h_{\operatorname{top},a_{\chi}}=\lim_{\epsilon}(\sup\\{\chi:\delta_{f,X_{u},\chi}(\epsilon)>0\\})=1.$

Proof of $h_{\operatorname{semi},\mu,a_{\chi}}=1$.

Let $\mu$ be the invariant probability measure for $(X_{u},f)$. If the
sequence $u$ was coded by the partition
$\mathcal{P}=\\{P_{0}=[0,1-\theta),P_{1}=[1-\theta,1)\\}$, then for every
cylinder set $\omega\subset X_{u}$ defined by a word $b_{0}b_{1}\dots
b_{n-1}\in L_{n}$, we must have that $\mu(\omega)=\operatorname{Leb}(P)$ for
some
$P\in\mathcal{P}^{n}\mathrel{\mathop{:}}=\vee_{i=0}^{n-1}R^{-i}_{\theta}\mathcal{P}.$
The atoms in the partition $\mathcal{P}^{n}$ have endpoints in

$\\{0,R_{1-\theta}(0),\dots,R^{n}_{1-\theta}(0)\\}.$

Let $1-\theta=[0;a_{1},a_{2},a_{3},\dots]$ be the continued fraction
expansion, and $\\{\frac{p_{k}}{q_{k}}\\}$ the sequence of best approximants.
Writing $n=mq_{k}+q_{k-1}+r$ with $1\leq m\leq a_{k+1}$, and $0\leq r<q_{k}$.
By the Three Gap Theorem, see [AB98, Section 3], the measure of the cylinder
$\omega$ is

  1. (1)

$\eta_{k}\mathrel{\mathop{:}}=(-1)^{k}(q_{k}(1-\theta)-p_{k})$, in which case
there are $n+1-q_{k}$ cylinders of this measure. These are the smallest gaps.

  2. (2)

$\eta_{k-1}-m\eta_{k}$. There are $r+1$ cylinders of this measure.

  3. (3)

$\eta_{k-1}-(m-1)\eta_{k}$. There are $q_{k}-(r+1)$ cylinders of this measure.
These are the biggest gaps, and their length is the sum of the lengths of the
previous types.

When $n=(m+1)q_{k}+q_{k-1}-1$, for $1\leq m\leq a_{k+1}$ ($r=q_{k}-1$), then
there are no cylinders with the length in item 3.

If

$C(n)\mathrel{\mathop{:}}=\min\left\\{\operatorname{card}(F):\sum_{\omega\in
F}\mu(\omega)>1-\epsilon\right\\}$

where $F$ is a family of cylinders sets of size $n$, then for some $0\leq
l,s,\kappa\leq 1$ depending on $n$,

(19) $C(n)=n+1-\left[\kappa(n+1-q_{k})+s(r+1)+l(q_{k}-(r+1))\right]$

where

(20)
$\kappa(n+1-q_{k})\eta_{k}+s(r+1)(\eta_{k-1}-m\eta_{k})+l(q_{k}-(r+1))(\eta_{k-1}-(m-1)\eta_{k})<\epsilon.$

Specializing to the case where there are no cylinders of type item 3

$n=n_{k}=(a_{k+1}+1)q_{k}+q_{k-1}-1=q_{k+1}+q_{k}-1,$

we can substitute $l=0$ and $r=q_{k}-1$ into Equation 20 to obtain:

(21)
$\begin{split}-\kappa&>\frac{s(r+1)(\eta_{k-1}-a_{k+1}\eta_{k})-\epsilon}{(n+1-q_{k})\eta_{k}}\\\
&=\frac{sq_{k}\eta_{k+1}-\epsilon}{q_{k+1}}.\end{split}$

In this special case, combining Equation 19 and Equation 21, we obtain that

(22)
$\begin{split}\frac{C(n_{k})}{n_{k}}&=1+\frac{1}{n_{k}}-\kappa\frac{q_{k+1}}{n_{k}}-s\frac{q_{k}}{n_{k}}\\\
&>1+\frac{1}{n_{k}}+\frac{(sq_{k}\eta_{k+1}-\epsilon)}{q_{k+1}\eta_{k}}\frac{q_{k+1}}{n_{k}}-s\frac{q_{k}}{n_{k}}\\\
&=1+\frac{1}{n_{k}}+s\frac{q_{k}}{n_{k}}\left(\frac{\eta_{k+1}}{\eta_{k}}-1\right)-\frac{\epsilon}{n_{k}\eta_{k}}.\end{split}$

Using Khinchin’s inequality, see for instance [Khi97, Theorems 9 and 13]:

$\frac{1}{q_{k+1}+q_{k}}<\eta_{k}<\frac{1}{q_{k+1}}$

and that $n_{k}=q_{k+1}+q_{k}-1$, we obtain

(23)
$1=1-\lim_{k\to\infty}\frac{1}{q_{k+1}+q_{k}}\leq\limsup_{k\to\infty}n_{k}\eta_{k}\leq
1+\limsup_{k\to\infty}\frac{q_{k}}{q_{k+1}}<\infty.$

By Equation 22 and Equation 23, we have that

$\delta_{\operatorname{semi},f,1}^{S}(\epsilon)=\limsup_{n\to\infty}\frac{C(n)}{n}\geq\limsup_{k\to\infty}\frac{C(n_{k})}{n_{k}}\geq
1-o(\epsilon).$

We conclude that $h_{\operatorname{semi},\mu,a_{\chi}}=1$ by Equation 14 in
3.1 ∎

###  5.2. Denjoy circle transformations

The Sturmian systems considered above can be realized as invariant sets for
transformations of the circle. Indeed, one may build $C^{1,\alpha}$ circle
diffeomorphisms by starting with an irrational circle rotation and “blowing
up” an orbit by inserting an interval at each point of the orbit. Such
examples were first studied by Denjoy and their construction can be found in
[KH95, Section 12.2]. We characterize them here:

######  Definition 5.5.

We say that a circle homeomorphism $f:S^{1}\to S^{1}$ is Denjoy if the
rotation number $\theta$ of $f$ is irrational, and there is a semiconjugacy
$h:S^{1}\to S^{1}$ and a point $x_{0}\in S^{1}$ such that

  * •

$h\circ f=R_{\theta}\circ h$,

  * •

$h^{-1}(f^{n}(x_{0}))$ is a nontrivial closed interval for all
$n\in\mathbb{Z}$,

  * •

$h^{-1}(x)$ is a single point for all $x$ outside of the orbit of $x_{0}$.

######  Lemma 5.6.

If $f$ is a Denjoy circle transformation, then $f|_{NW(f)}$ is topologically
conjugated to a Sturmian subshift.

###### Sketch of proof.

Recall that Sturmian sequences can be obtained by looking at codes appearing
of the rotation $R_{\theta}$ using the intervals $[0,1-\theta)$ and
$[1-\theta,1)$. In the case of a circle rotation the map which sends the code
to the point is not one-to-one. However, in the case of a Denjoy
transformation, the coding intervals can be taken to cover only the
nonwandering set, and are therefore disjoint. This yields a conjugacy instead
of a semiconjugacy. ∎

######  Corollary 5.7.

Denjoy circle transformations are not variational.

###### Proof.

By Poincaré recurrence, any invariant measure for a Denjoy transformation must
be supported on its nonwandering set. Since restricted to this set, the system
is topologically conjugated to a Sturmian shift, we conclude that it is
uniquely ergodic and that the unique invariant measure is Kronecker. Thus, the
metric entropy has 0 entropy at all scales. However, since there is a compact
invariant set topologically conjugated to a Sturmian subshift, the semi-
topological and topological entropies are both linear. ∎

###  5.3. Geodesic flow on $\mathbb{T}^{2}$

Another unexpected feature of slow entropy is the failure of additivity over
ergodic decompositions. Let $\mu=\int_{\mathcal{E}}\nu d\hat{\mu}(\nu)$ be the
ergodic decomposition of $\mu$, where $\mathcal{E}$ is the space of ergodic
invariant measures and $\hat{\mu}$ is a probability measure on $\mathcal{E}$.
For the classical entropy at exponential scale [VO16, Theorem 9.6.2], we have
that

(24) $h_{\mu}(f)=\int_{\mathcal{E}}h_{\nu}(f)d\hat{\mu}(\nu).$

In this section, we explain that such a formula cannot hold for slow entropy,
even when restricting so a fixed scale such as the polynomial scale.

It is well-known that the geodesic flow on $T^{1}\mathbb{T}^{2}$, the unit
tangent bundle to $\mathbb{T}^{2}$, is not ergodic and has a smooth ergodic
decomposition. Each ergodic component is diffeomorphic to $\mathbb{T}^{2}$ and
corresponds to the unit speed linear flow in an irrational direction (the
rational directions have measure 0, so we may omit them from the ergodic
decomposition). Hence, at any scale, the ergodic components of the Haar
measure on $T^{1}\mathbb{T}^{2}$ all have 0 entropy at all scales. The
following Lemma shows that we can obtain a positive slow entropy by “gluing”
several copies of Kronecker systems together in an interesting way.

######  Lemma 5.8.

If $\varphi_{t}:T^{1}\mathbb{T}^{2}\to T^{1}\mathbb{T}^{2}$ is the geodesic
flow on $\mathbb{T}^{2}$, then the topological and Haar slow entropy of
$\varphi_{t}$ is 1 at polynomial scales $n_{\chi}(t)=t^{\chi}$.

###### Proof.

Observe that $\operatorname{Isom}(\mathbb{R}^{2})\cong
S^{1}\ltimes\mathbb{R}^{2}$ acts simply transitively on $T^{1}\mathbb{R}^{2}$,
and that $T^{1}\mathbb{T}^{2}$ is the quotient of $T^{1}\mathbb{R}^{2}$ by
$\mathbb{Z}^{2}$. Furthermore, since the isometry group takes orbits of the
geodesic flow to orbits of the geodesic flow, it follows that the geodesic
flow is smoothly conjugated to homogeneous flow on
$\operatorname{Isom}(\mathbb{R}^{2})/\mathbb{Z}^{2}$ by a one-parameter
subgroup of $\mathbb{R}^{2}\subset\mathbb{T}^{1}\mathbb{R}^{2}$. If
$\Theta\in\operatorname{Lie}(\operatorname{Isom}(\mathbb{R}^{2}))$ represents
the generator of the subgroup $S^{1}$, and $X$ and $Y$ represent orthonormal
generators of $\mathbb{R}^{2}$, then (up to choice of orientation), we have
structure relations

$[\Theta,X]=Y\qquad[\Theta,Y]=-X\qquad[X,Y]=0.$

From this, one easily checks that in this basis $\operatorname{ad}(X)$ is

$\begin{pmatrix}0&1&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix},$

so by [KVW19], it follows that the polynomial slow entropy of $\varphi_{t}$ is
1. ∎

We remark that this geodesic flow is also conjugated to the suspension of the
affine map $(x,y)\mapsto(x,y+x)$, so we have the phenomenon for
transformations as well.

##  6\. The slow entropy of some interval exchanges

Interval exchange transformations or IETs are piecewise isometries, with a
finite number of discontinuities. Moreover, IETs preserve the orientation.
These maps can be regarded as generalizations of rotations. In this section,
we will compute the metric slow entropy of 3-IETs. The computations for the
metric entropy will occupy most of Section 6. We will prove that for a large
class of 3-IETs the metric slow entropy is 1 for the polynomial scale
$a_{\chi}(n)=n^{\chi}$, see 6.1.

######  Theorem 6.1.

Let $\Delta\subset\mathbb{R}^{3}$ be the set
$\\{(x,y,z):x+y+z=1,\,x,y,z>0\\}$. There exists a set $A\subset\Delta$ of
Hausdorff dimension 2 such that if $g$ is a 3-IET determined by $\lambda\in
A$, then $h_{\operatorname{Leb},a_{\chi}}(g)=1.$

We will easily see that the topological slow entropy of the corresponding
symbolic system is at most 1 with respect to the same scale, see 6.5. This
combined with 6.1 prove 1.3.

######  Remark 6.2.

We have used the convention that the limits appearing in the definitions of
the functions $\delta_{f,\cdot,\chi}^{S}$ slow entropy are $\limsup$’s. In
general these are not actual limits, and we rely on this choice several times
in the proof. It would be interesting to make similar computations for the
$\liminf$ definitions. It is already known that a gap may exist, and special
attention is paid to this subtlety in [BKW23b].

###  6.1. Preliminaries of IET’s

We refer the reader to [Yoc10, Via06] for more details about interval exchange
transformations (IET).

Let $\mathcal{A}$ be a collection of $d$ symbols and
$\lambda\in\mathbb{R}^{\mathcal{A}}_{>0}$ be a vector of positive entries.
Given two bijective functions $\pi_{t}:\mathcal{A}\to\\{1,\dots,d\\}$,
$\pi_{b}:\mathcal{A}\to\\{1,\dots,d\\}$, we obtain a permutation of the
symbols in $\mathcal{A}$ defined by

$\pi=\begin{pmatrix}\pi_{t}^{-1}(1)&\dots&\pi_{t}^{-1}(d)\\\
\pi_{b}^{-1}(1)&\dots&\pi_{b}^{-1}(d)\\\ \end{pmatrix}.$

Let $I\subset\mathbb{R}$ be a bounded interval, closed on the left and open on
the right, and denote the length of $I$ by $|I|$. From now on, we will assume
that the left endpoint of $I$ is 0. The vector $\lambda$ and the permutation
$\pi$ determine a partition $\\{I_{a}\\}_{a\in\mathcal{A}}$ where

$I_{a}=\left[\sum_{\\{b:\pi_{t}^{-1}(b)<\pi_{t}^{-1}(a)\\}}\lambda_{b},\sum_{\\{b:\pi_{t}^{-1}(b)\leq\pi_{t}^{-1}(a)\\}}\lambda_{b}\right),$

and the interval $I=\left[0,|I|\right)$, where
$|I|=\sum_{A\in\mathcal{A}}\lambda_{a}.$

######  Definition 6.3.

An interval exchange transformation on $d$ intervals ($d$-IET)
$g\mathrel{\mathop{:}}=g_{\lambda,\pi}\colon I\to I$ determined by a length
vector $\lambda$ and permutation $\pi$ is the bijective map defined by

(25)
$g(x)=x+\sum_{\\{b:\pi_{b}^{-1}(b)<\pi^{-1}_{b}(a)\\}}\lambda_{b}-\sum_{\\{b:\pi_{t}^{-1}(b)<\pi_{t}^{-1}(a)\\}}\lambda_{b},\quad\text{if
}x\in I_{a}.$

Any IET is a measure preserving transformation with respect to the Lebesgue
measure $\operatorname{Leb}$ on the interval $I.$ If for some $k<d$, the set
$\\{1,\dots,k\\}$ is $\pi$ invariant, the IET is a concatenation of IETs with
fewer intervals. If $\\{1,\dots,k\\}$ is not invariant for every $k<d$, we say
that the permutation (and the IET) is irreducible.

###  6.2. Topological and semi-topological slow entropy of a class of $d$-IETs

Let $g=g_{\lambda,\pi}$ be a $d$-IET, and denote
$D=\\{\beta_{1},\dots,\beta_{d-1}\\}$ the set of discontinuities of $g$. Here,
$\beta_{i}=\sum_{j=1}^{i}\lambda_{j}$. For convenience, denote $\beta_{0}=0$
and $\beta_{d}=|I|$. The map $g$ has the idoc property (infinite distinct
orbit condition) if for all $n>1$, $D\cap g^{-n}(D)=\varnothing.$ Keane in
[Kea75] proved that an idoc IET is minimal.

From now on, we will assume that $\pi$ is an irreducible permutation. The set
$I\backslash D$ has the $d$ continuity intervals of $g.$ By including the left
endpoint, let $\mathcal{P}\mathrel{\mathop{:}}=\\{I_{a}\\}_{a\in\mathcal{A}}$
be the natural partition of $g$.

We have the following results regarding the number of Bowen balls.

######  Lemma 6.4.

Let $\epsilon>0$ be sufficiently small. Suppose that $g$ has the idoc
property. There exists $n_{0}$ such that
$B_{g}^{n+1}(x,\epsilon)\subset\mathcal{P}^{n}(x)$ and
$\mathcal{P}^{n}(x)\subset B_{g}^{n}(x,\epsilon)$ for all $n\geq n_{0}$ and
$x\in I.$

###### Proof.

Assume that $\epsilon$ is small enough so that it has the following property:
if $d(x,y)<\epsilon$ but $y\not\in\mathcal{P}(x)$, then
$d(g(x),g(y))\geq\epsilon$. Equivalently, if $d(x,y)<\epsilon$ and
$d(g(x),g(y))<\epsilon$, then $y\in\mathcal{P}(x)$. If no such $\epsilon$
existed, then at least one of the points $\beta_{i}$ would be a removable
discontinuity. Since $g$ has the idoc property, then
$||\mathcal{P}^{n}||\mathrel{\mathop{:}}=\max_{P\in\mathcal{P}^{n}}|P|\downarrow
0$. Let $n_{0}$ be such that $||\mathcal{P}^{n_{0}}||<\frac{\epsilon}{d|I|}$.

Since $\mathcal{P}^{n}=\bigwedge_{i=0}^{n-1}g^{-i}(P)$, then for all $n\geq
n_{0}$ and $y\in\mathcal{P}^{n}(x)$, we have that
$d(g^{i}(x),g^{i}(y))<\epsilon$ for $0\leq i\leq n-1$. This proves that
$\mathcal{P}^{n}(x)\subset B^{n}_{g}(x,\epsilon).$

To see the other containment, suppose that $d(g^{i}(x),g^{i}(y))<\epsilon$ for
$i=0,\dots,n$. By the property establishing the smallness of $\epsilon$, it
follows that $g^{i}(x)\in\mathcal{P}(g^{i}(y))$ for $i=0,\dots,n-1$. ∎

With the above, we can conclude the following about the topological entropy.

######  Proposition 6.5.

Let $g_{\lambda,\pi}$ be $d$-IET. Suppose that it has the idoc property. Then
$|\mathcal{P}^{n}|=dn$ and $h_{top,a_{\chi}}=1$ with the polynomial scale
$a_{\chi}(n)=n^{\chi}$.

###### Proof.

The idoc property implies that
$\mathcal{P}^{n}=\bigwedge_{i=0}^{n-1}g^{-i}(\mathcal{P})$ has exactly $dn$
atoms. Applying 6.4,

$\delta_{g,I,\chi}(\epsilon)=\limsup_{n\to\infty}\frac{d}{n^{\chi-1}}.$

Then

$h_{top,a_{\chi}}(g)=\lim_{\epsilon\to
0}(\sup\\{\chi:\delta_{g,I,\chi}(\epsilon)>0\\})=1.$

∎

Define $\epsilon_{n}\mathrel{\mathop{:}}=\min_{P\in\mathcal{P}^{n}}|P|$, the
length of the smallest atom in the partition $\mathcal{P}^{n}.$ An IET is
linearly recurrent if there exists a constant $C>0$ such that for every $n\geq
1$, then $n\epsilon_{n}\geq C.$

We have the following characterization of a big class of $d$-IETs.

######  Proposition 6.6.

Suppose that $g$ is an idoc $d$-IET. The following are equivalent:

  1. (1)

The Lebesgue measure is homogeneous, see 3.4.

  2. (2)

$g$ is linearly recurrent.

###### Proof.

If we assume (1), fix $\epsilon>0$ and $c>0$ as in the definition of
homogeneous measure. Then

$\frac{1}{c}\leq\frac{\operatorname{Leb}(B^{n}_{g}(x,\epsilon))}{\operatorname{Leb}(B^{n}_{g}(y,\epsilon))}\leq
c$

for all $n$ sufficiently large. In particular, it follows that for every
$P\in\mathcal{P}^{n}$,

$\frac{|P(y)|}{c}=\frac{\operatorname{Leb}(B^{n}_{g}(y,\epsilon))}{c}\leq\epsilon_{n},$

adding both sides of the inequality over $P\in\mathcal{P}^{n}$ we obtain
$C\mathrel{\mathop{:}}=\frac{1}{dc}\leq n\epsilon_{n}$ for all $n$
sufficiently large. This proves (2).

Assume (2), and suppose by contradiction that (1) does not happen. So, for
every $1>k>0$ there exists $n$ very large such that

$\frac{\operatorname{Leb}(B^{n}_{g}(x,\epsilon))}{\operatorname{Leb}(B^{n}_{g}(y,\epsilon))}<k.$

Without loss of generality, we can assume that
$\epsilon_{n}=\operatorname{Leb}(B^{n}_{g}(x,\epsilon))$ and
$\operatorname{Leb}(B^{n}_{g}(y,\epsilon))<\frac{1}{n}$. Then we have that
$n\epsilon_{n}=n\operatorname{Leb}(B^{n}_{g}(x,\epsilon))<k$. In particular,
if $k=C$, which contradicts (2). ∎

We have the main result of this section:

######  Corollary 6.7.

Suppose that $g$ is an idoc, linearly recurrent $d$-IET. Then, for
$a_{\chi}=n^{\chi}$, we have that

(26) $h_{semi,Leb,a_{\chi}}(g)=h_{top,a_{\chi}}=1$

###### Proof.

3.5 and 6.6 imply that
$h_{semi,\operatorname{Leb},a_{\chi}}(g)=h_{top,a_{\chi}}(g)$. 6.5 implies
that the equality is 1. ∎

We note that this does not say anything about the metric slow entropy of IETs.
The class of linearly recurrent IET is uniquely ergodic by [Vee87, Theorem
1.2] then, computing the metric slow entropy of a linearly recurrent IET with
respect to the Lebesgue measure will say whether such IET is variational. We
also want to remark that the conditions in 6.7 are satisfied by a set of
parameters $\lambda$ of Hausdorff dimension $d$. This was noted by D.
Robertson following his proof of [Rob19, Proposition 4] and the proof of
[CCM13, Theorem 1.4].

###  6.3. On 3-IETs

Any vector of positive entries $\lambda=(\lambda_{A},\lambda_{B},\lambda_{C})$
and the symmetric permutation $\pi=\begin{pmatrix}A&B&C\\\ C&B&A\end{pmatrix}$
determines a 3-IET given by Equation 25, which in a simpler form is

$g(x)=\left\\{\begin{array}[]{ll}x+\lambda_{B}+\lambda_{C}&\text{ if
}x\in[0,\lambda_{A}),\\\ x+\lambda_{C}-\lambda_{A}&\text{ if
}x\in[\lambda_{A},\lambda_{A}+\lambda_{B}),\\\
x-\lambda_{A}-\lambda_{B}&\text{ if
}x\in[\lambda_{A}+\lambda_{B},|I|).\end{array}\right.$

It is common to think of 3-IETs over $I=[0,1)$, but we will consider 3-IETs
with $I=[0,1+\xi)$, $\mathcal{A}=\\{A,B,C\\}$, and the symmetric permutation
$\pi=\begin{pmatrix}A&B&C\\\ C&B&A\end{pmatrix}$.

Let $\alpha\in(0,1)$ be an irrational number, and let
$\left\\{\frac{p_{m}}{q_{m}}\right\\}_{m\geq 1}$ be the sequence of best
approximations. Let $|\\!|x|\\!|$ be equal to

$|\\!|x|\\!|\mathrel{\mathop{:}}=\min_{n\in\mathbb{Z}}|x-n|.$

The number $\alpha$ is badly approximable if there exists $C_{\alpha}>0$
depending on $\alpha$, satisfying

$q_{m+1}\leq C_{\alpha}q_{m},$

for every $m\in\mathbb{N}.$ Equivalently, $\alpha$ is badly approximable, if
there exists $D_{\alpha}>0,$ such that

$|\\!|m\alpha|\\!|\geq\frac{D_{\alpha}}{m}$

for every $m\in\mathbb{N}.$

For every real number $\alpha$, we denote $S_{\alpha}$ the set

$S_{\alpha}=\\{\xi\in\mathbb{R}:|\\!|\xi-j\alpha|\\!|\geq\frac{C_{\xi}}{q_{n}}\textit{\;for
every $-q_{n}<j<q_{n}$}\\}.$

We have the following property regarding the Hausdorff dimension of the
numbers.

######  Lemma 6.8.

The set of badly approximable real numbers has Hausdorff dimension one. For
arbitrary $\alpha$, the set $S_{\alpha}$ has Hausdorff dimension one.

###### Proof.

For badly approximable numbers, this was proved in [Sch1, Theorem 3]. Lastly,
[Tse09, Theorem 1] proved that $\dim S_{\alpha}=1$ for every real number
$\alpha$. ∎

######  Lemma 6.9.

Define

$X=\\{(\alpha,\xi)\in(0,1)\times(0,1):\alpha\text{ is badly approximable and
}\xi\in S_{\alpha}\\}.$

Then Hausdorff dimension of $X$, $\dim X=2.$

###### Proof.

We apply the following theorem about the Hausdorff dimension of products
[Fal85, Theorem 5.8 and Excercise 5.2]: Let $A,B$ be Borel subsets of a
Euclidean spaces and let $Y\subset A\times B$ be such that for all $a\in A$,
$\dim\\{b\in B:(a,b)\in Y\\}\geq d.$ Then

$\dim Y\geq\dim A+d.$

Let $A$ be the set of badly approximable numbers, $B=(0,1)$, and replace $X$
into $Y$. The set $\\{b:(a,b)\in X\\}$ is the set $S_{a}.$ From 6.8, $\dim
S_{a}=1.$

Therefore $\dim X\geq\dim A+1.$ Thus $\dim X=2$, because $\dim A=1$ as stated
in 6.8. ∎

The proof of 6.1 follows from 6.10 and 6.11. 6.10 is the core of the argument
and its proof will be postponed until Section 6.5.

######  Proposition 6.10.

Let $\xi\in(0,1)$ and let
$\lambda=(\xi,\lambda_{B},\lambda_{C})\in\mathbb{R}^{3}_{+}$ such that
$\lambda_{B}+\lambda_{C}=1$. Define

$\alpha=\left\\{\begin{array}[]{cc}\lambda_{C}-\xi&\text{ if
}\lambda_{C}>\xi,\\\ 1+\lambda_{C}-\xi&\text{ if
}\lambda_{C}<\xi.\end{array}\right.$

If $\alpha$ is badly approximable, and $\xi\in S_{\alpha}$, then the 3-IET
$f\mathrel{\mathop{:}}=f_{\lambda,\pi}$ determined in this way satisfies
$h_{\operatorname{Leb},a_{\chi}}(f)=1$ with polynomial scale
$a_{\chi}(n)=n^{\chi}.$

######  Lemma 6.11.

Let $\lambda\in\mathbb{R}^{3}_{>0}$, $c>0$ and $\pi$ be a symmetric
permutation. The 3-IETs $g\mathrel{\mathop{:}}=g_{\lambda,\pi}$ and
$h\mathrel{\mathop{:}}=h_{c\lambda,\pi}$ are smoothly conjugated by the map
$x\mapsto cx$. Moreover, any 3-IET
$g\mathrel{\mathop{:}}=g_{(\lambda_{A},\lambda_{B},\lambda_{C}),\pi}$ is
measurably conjugated to its inverse
$g^{-1}=g^{-1}_{(\lambda_{C},\lambda_{B},\lambda_{A}),\pi}$, by the
hyperelliptic involution map $\iota(x)=|I|-x.$

###### Proof.

For $c>0$, let
$f:[0,\sum_{\alpha}\lambda_{\alpha})\to[0,c\sum_{\alpha}\lambda_{\alpha})$ be
the map $x\mapsto cx.$ Let $g$ be the 3-IET determined by the length vector
$\lambda$ and the permutation $\pi$. Let $h$ be the 3-IET determined by the
length vector $c\lambda$ and the permutation $\pi.$ The map $f$ is a
diffeomorphism that sends the Lebesgue measure on
$[0,\sum_{\alpha}\lambda_{\alpha})$ to the Lebesgue measure on
$[0,c\sum_{\alpha}\lambda_{\alpha})$. The IET $g$ maps by translation the
segment $[\sum_{\pi_{t}(\alpha)<i}\lambda_{\alpha},\sum_{\pi_{t}(\alpha)\leq
i}\lambda_{\alpha})$ to the segment
$[\sum_{\pi_{b}(\alpha)<4-i}\lambda_{\alpha},\sum_{\pi_{b}(\alpha)\leq
4-i}\lambda_{\alpha})$. Thus the composition $f\circ g$ maps the segment
$[\sum_{\pi_{t}(\alpha)<i}\lambda_{\alpha},\sum_{\pi_{t}(\alpha)\leq
i}\lambda_{\alpha})$ by stretching and translating to the segment
$[c\sum_{\pi_{b}(\alpha)<4-i}\lambda_{\alpha},c\sum_{\pi_{b}(\alpha)\leq
4-i}\lambda_{\alpha})$.

Similarly, $h$ maps by translation the interval
$[c\sum_{\pi_{t}(\alpha)<i}\lambda_{\alpha},c\sum_{\pi_{t}(\alpha)\leq
i}\lambda_{\alpha})$ to the interval
$[c\sum_{\pi_{b}(\alpha)<4-i}\lambda_{\alpha},c\sum_{\pi_{b}(\alpha)\leq
4-i}\lambda_{\alpha}).$ In conclusion, for all
$x\in[0,\sum_{\alpha}\lambda_{\alpha})$, we have $f\circ g(x)=h\circ f(x).$
Proving that $g$ and $h$ are smoothly conjugated.

Denote $I=[0,\lambda_{A}+\lambda_{B}+\lambda_{C})$, and fix the permutation
$\pi=\begin{pmatrix}A&B&C\\\ C&B&A\end{pmatrix}$. Let $g$ be the 3-IET
determined by the length vector
$\lambda=(\lambda_{A},\lambda_{B},\lambda_{C})$ and the permutation $\pi.$ Let
$g^{-1}$ be the inverse map of $g$. We let the reader verify that this is a
3-IET determined by the length vector
$\lambda^{\prime}=(\lambda^{\prime}_{A},\lambda^{\prime}_{B},\lambda^{\prime}_{C})\equiv(\lambda_{C},\lambda_{B},\lambda_{A})$
and the permutation $\pi.$ Let $\iota:I\to I$ be the map $x\mapsto|I|-x.$ The
map $\iota$ sends the open interval
$(\sum_{\pi_{*}(\alpha)<i}\lambda_{\alpha},\sum_{\pi_{*}(\alpha)\leq
i}\lambda_{\alpha})$ to the open interval
$(\sum_{\pi_{*}(\alpha)<4-i}\lambda^{\prime}_{\alpha},\sum_{\pi_{*}(\alpha)\leq
4-i}\lambda^{\prime}_{\alpha})$ by translation and order reversing, where
$*\in\\{t,b\\}$. The composition $\iota\circ g$ maps the interval
$(\sum_{\pi_{t}(\alpha)<i}\lambda_{\alpha},\sum_{\pi_{t}(\alpha)\leq
i}\lambda_{\alpha})$ to the interval
$(\sum_{\pi_{b}(\alpha)<i}\lambda^{\prime}_{\alpha},\sum_{\pi_{b}(\alpha)\leq
i}\lambda^{\prime}_{\alpha})$ by a translation and order reversing. Also, the
composition $g^{-1}\circ\iota$ maps the open interval
$(\sum_{\pi_{t}(\alpha)<i}\lambda_{\alpha},\sum_{\pi_{t}(\alpha)\leq
i}\lambda_{\alpha})$ to the interval
$(\sum_{\pi_{b}(\alpha)<i}\lambda^{\prime}_{\alpha},\sum_{\pi_{b}(\alpha)\leq
i}\lambda^{\prime}_{\alpha})$ by a translation and order reversing. Thus, we
have the equality $\iota\circ g=g^{-1}\circ\iota$ on the interior of the
intervals $I_{i}$ for $i\in\\{1,2,3\\}$. The equality does not occur at the
left endpoints of the intervals $I_{\alpha}$, for example
$\iota(g(0))=|I|-g(0)=|I|-\lambda_{A}-\lambda_{B}=\lambda_{C}$, but
$g^{-1}(\iota(0))$ is not defined because $\iota(0)=|I|$ and $g^{-1}$ is not
defined at $|I|.$ Thus the maps $g$ and $g^{-1}$ are measurable conjugated,
since the map $\iota$ is a diffeomorphism that preserves the Lebesgue measure,
and $\iota(g(x))=g(\iota(x))$ except at finitely many $x\in I.$ ∎

###  6.4. Suspensions of a 3-IET and a rotation

Let $(T,X,\mu)$ be measure preserving transformation and $f\colon
X\to\mathbb{R}_{>0}$ an $L^{1}(\mu)$ function. Let $X^{f}$ be the quotient

$X^{f}=\left\\{(x,s):x\in X\text{ and }0\leq s\leq f(x)\right\\}/\sim,$

with $(x,f(x))\sim(T(x),0).$ The suspension flow determined by $T$ and $f$ is
the flow $T^{f}_{t}\mathrel{\mathop{:}}=\mathcal{F}_{t}\colon X^{f}\to X^{f}$
defined by $\mathcal{F}_{t}(x,s)=(T^{n}(x),s+t-\sum_{i=0}^{n-1}f(T^{i}x))$,
where $n\geq 0$ is such that $f^{(n)}(x)\leq s+t<f^{(n+1)}(x),$ where
$f^{(n)}(x)$ is the Birkhoff sum

$\displaystyle f^{(n)}(x)\mathrel{\mathop{:}}=\sum_{i=0}^{n-1}f(T^{i}x).$

####  6.4.1. A specific suspension for a 3-IET

Consider a 3-IET with $|I|=1+\xi$, and the conditions
$\lambda_{A}=\xi<\lambda_{B}+\lambda_{C}=1$. The suspension with the constant
function 1 of this 3-IET is presented in Figure 1A. This construction starts
with a rectangle of length $1+\xi$ and height 1, by convenience assume that
the left bottom corner is placed at the origin. The sides of the rectangle are
identified by translation as follows

(27)
$\begin{split}[0,\lambda_{C})\times\\{0\\}&\sim[\lambda_{A}+\lambda_{B},1+\xi)\times\\{1\\}\\\
[\lambda_{C},\lambda_{B}+\lambda_{C})\times\\{0\\}&\sim[\lambda_{A},\lambda_{A}+\lambda_{B})\times\\{1\\}\\\
[1,1+\xi)\times\\{0\\}&\sim[0,\lambda_{A})\times\\{1\\}\\\
\\{0\\}\times[0,1)&\sim\\{1+\xi\\}\times[0,1).\end{split}$

$A$$A$$B$$B$$C$$C$$1$$1$ A

$B$$B$$C_{1}$$C_{2}$$A$$C_{1}$$C_{2}$ B

Figure 1. This is case $\lambda_{C}>\xi$ in Equation 28. In Figure 1A, the
sides of a rectangle $[0,1+\xi)\times[0,1]$ are identified according to
Equation 27. The vertical flow’s first return time to $[0,1+\xi)$ is a 3-IET.
In Figure 1B, suspension of a rotation on $[0,1]$ with
$f=2\mathds{1}_{[0,\xi)}+\mathds{1}_{[\xi,1)}$. This can also be obtained by
cutting over the dotted line in Figure 1A and gluing the small piece over the
segment $[1,1+\xi)\times\\{1\\}.$

$A$$A$$B$$B$$C$$C$11 A

$C$$B_{2}$$B_{1}$$C$$A$$B_{1}$$B_{2}$ B

Figure 2.  This is case $\lambda_{C}<\xi$ in Equation 28. In Figure 2A, the
sides of a rectangle $[0,1+\xi)\times[0,1]$ are identified according to
Equation 27. The vertical flow’s first return time to $[0,1+\xi)$ is a 3-IET.
In Figure 2B, suspension of a rotation on $[0,1]$ with
$f=2\mathds{1}_{[0,\xi)}+\mathds{1}_{[\xi,1)}$. This can also be obtained by
cutting over the dotted line in Figure 2A and gluing the small piece over the
segment $[1,1+\xi)\times\\{1\\}.$

####  6.4.2. Suspension flow of an irrational rotation and proof of 6.1

Towards the computation of the slow entropy of the 3-IET mentioned in Section
6.4.1, we will compute the slow entropy of the vertical flow in the
construction involving the suspension of the 3-IET in Figure 1B. An equivalent
construction is the suspension of a rotation $T:[0,1)\to[0,1),x\mapsto
x+\alpha\mod 1$. The suspension function is
$f=2\mathds{1}_{[0,\xi)}+\mathds{1}_{[\xi,1)}.$ Computations for the first
return time of the vertical flow to the segment $[0,1)\times\\{0\\}$ show that

(28) $\alpha=\begin{dcases}\lambda_{C}-\xi&\text{if }\lambda_{C}>\xi,\\\
1+\lambda_{C}-\xi&\text{if }\lambda_{C}<\xi.\end{dcases}$

Although we are suspending a rotation, the 3-IET is still present in the first
return map of the vertical flow to the segment
$I=[0,1)\times\\{0\\}\cup[0,\xi)\cup\\{1\\}\subset[0,1)^{f}.$ Figure 1A and
Figure 1B are proof by picture of the case $\lambda_{C}>\lambda_{A}=\xi$.
Because the rotation angle $\alpha$ in Figure 1B is equal to the length of the
segment labeled $C_{1}$, then $\alpha$ is equal to
$\lambda_{C}-\lambda_{A}=\lambda_{C}-\xi$. Figure 2A and Figure 2B are proof
by picture of the case $\lambda_{C}<\lambda_{A}=\xi$. The rotation angle
$\alpha$ of the suspension in Figure 2B is equal to the sum of the lengths of
the segments labeled $C$ and $B_{1}$. The length of the latter is equal to
$1-\lambda_{A}=1-\xi$, then $\alpha=1+\lambda_{C}-\xi.$

Starting with $\alpha,\xi\in[0,1]$, the corresponding 3-IET is given by the
length vector:

(29)
$F_{0}(\alpha,\xi)\mathrel{\mathop{:}}=(\lambda_{A},\lambda_{B},\lambda_{C})=\begin{dcases}(\xi,1-\alpha-\xi,\alpha+\xi)&\text{if
}\alpha+\xi<1,\\\ (\xi,2-\alpha-\xi,\alpha+\xi-1)&\text{if
}\alpha+\xi>1.\end{dcases}$

The proof of Equation 29 follows from similar computations and pictures as a
verification of Equation 28.

###### Proof of 6.1.

Let $\Delta$ be the set $\\{(x,y,z):x+y+z=1,\,x,y,z>0\\}.$ The subset $X$ in
6.9 is of Hausdorff dimension 2. Let $F:[0,1]\times[0,1]\to\Delta$ be the
function defined by $F(\alpha,\xi)=\frac{1}{1+\xi}F_{0}(\alpha,\xi).$ Since
the function $F_{0}$ is linear and of rank 2 in connected components, the
Hausdorff dimension of the set $F_{0}(X)$ is 2. Additionally, the factor of
$1/(1+\xi)$ preserves the Hausdorff dimension since it is the normalization
factor, the $l_{1}$-norm of any point in the image of $F_{0}$ is $1/(1+\xi).$
This proves that the set $A\mathrel{\mathop{:}}=F(X)$ is of Hausdorff
dimension 2.

Note that by 6.10, the metric slow entropy of the 3-IET determined by the
vector $F_{0}(\alpha,\xi)$ is 1, by 6.11, the 3-IET determined by
$F_{0}(\alpha,\xi)$ and the 3-IET determined by
$F(\alpha,\xi)=\frac{1}{1+\xi}F_{0}(\alpha,\xi)$ are measurably conjugated.
Then, every 3-IET determined by a vector in $F(X)$ must have metric slow
entropy 1 with $a_{\chi}=n^{\chi}.$ ∎

###  6.5. Computation of slow entropy and proof of 6.10

We aim to compute the growth rate of the number of Hamming balls of time $R$
of the suspension flow $T^{f}_{t}$, because as we mentioned in the previous
section, this suspension is equivalent to the suspension of a 3-IET with
constant roof function 1. Therefore, these Hamming balls’ growth rate is the
same as the Hamming balls’ of the suspended 3-IET.

Let $\alpha$ and $\xi$ as before and
$f=d_{1}\mathds{1}_{[0,\xi)}+d_{2}\mathds{1}_{[\xi,1)}$. The conditions on
$\alpha$ and $\xi$ will be given later. Given a subset $A\subset\mathbb{T}$,
we use $A^{f}$ to denote the set $\\{(y,t)\in\mathbb{T}^{f}:y\in A\\}$. We
also let $\lambda^{f}$ denote the normalized Lebesgue measure restricted to
$\mathbb{T}^{f}$. Let
$M=\max\\{d_{1},d_{2},\frac{1}{d_{1}},\frac{1}{d_{2}}\\}$ and
$e=|d_{1}-d_{2}|.$ The special situation in which we prove 6.10 is by setting
$d_{1}=2$ and $d_{2}=1.$ Given $R>0$ very large and $\epsilon>0$, fix a
generating partition $\mathcal{P}$ such that all atoms in the partition are
squares with length between $1/2k$ to $1/k$ for some large value $k$ to be
specified later. We want to show that $S_{T^{f},\mathcal{P}}(\epsilon,R)\geq
CR$ holds for some constant $C$, hence

$\displaystyle\limsup_{R\rightarrow\infty}\frac{S_{T^{f},\mathcal{P}}(\epsilon,R)}{R^{\chi}}=\infty$

for any $\chi<1$. This gives us a lower bound for the metric slow entropy,
i.e. $h_{\mu,a_{\chi}}(T^{f})\geq 1.$

From now on, we will assume that $\alpha$ is badly approximable; otherwise, we
will specify.

To prove 6.10, we combine several lemmas which describe recurrence properties
for the base circle rotation $R_{\alpha}$. In fact, this will allow us to
compute the slow entropy of some other special flows with piecewise constant
roof functions (see 6.16). We therefore state a few lemmas (6.12 \- 6.15),
which we use to prove 6.10, delaying their proof until later in this section.

The following proposition uses similar ideas in [FLL07, Lemma 4], and is a
Ratner-type property for the special flows we consider.

######  Proposition 6.12.

Suppose $\alpha\in(0,1)$ is badly approximable, and $\xi\in S_{\alpha}$. There
exists $\kappa,c>0$ and a finite set $V\subset{\mathbb{R}\setminus\\{0\\}}$
such that for all large enough $m$, and $x,y\in\mathbb{T}$ with

$\frac{C_{\xi}}{2q_{m+1}}\leq|\\!|x-y|\\!|<\frac{C_{\xi}}{2q_{m}},$

for some time $N$ where

$q_{m}\leq N\leq cq_{m+1}+2q_{m},$

the following set

$\Lambda=\\{n\in[0,N]\cap\mathbb{Z}:f^{(n)}(x)-f^{(n)}(y)\in V\\}$

has cardinality larger than $\kappa N$.

6.12 shows that in some definite proportion, the Birkhoff sum $f^{(n)}$ of
nearby points will differ. Moreover, the next lemma shows that the splitting
phenomenon occurring in the Birkhoff sums implies the splitting time of two
orbits in special flow space by a definite proportion.

######  Lemma 6.13.

Let $\tilde{x}=(x,s)$, $\tilde{y}=(y,s^{\prime})$ be two elements in the same
atom of partition. If

$\frac{C_{\xi}}{2q_{m+1}}\leq|\\!|x-y|\\!|<\frac{C_{\xi}}{2q_{m}}$

for some $m$, there exists some time $T$ and constant $D$ where $T\leq Dq_{m}$
and $\kappa^{\prime}>0$ such that the following set

$\tilde{\Lambda}=\\{t<T:T_{t}^{f}{\tilde{x}}\text{\;and\;}T_{t}^{f}{\tilde{y}}\text{\;are
not in the same atom }\\}$

has measure $|\tilde{\Lambda}|$ larger than $\kappa^{\prime}T$.

Let $\tilde{x}=(x,s)$ be an element in $\mathbb{T}^{f}$. Consider a Hamming
ball $B_{T^{f},\mathcal{P}}^{R}(\tilde{x},\epsilon)\eqqcolon
B^{R}(\tilde{x},\epsilon)$ centered at $\tilde{x}$, then for any
$\tilde{y}=(y,s^{\prime})\in B^{R}(\tilde{x},\epsilon)$ we want to show there
exists $n_{1}$ and $n_{2}$ such that
$|\\!|x+n_{1}\alpha-(y+n_{2}\alpha)|\\!|<H/R$ for some constant $H$.

######  Lemma 6.14.

There exists a constant $H,$ such that for any $\tilde{y}=(y,s^{\prime})\in
B^{R}(\tilde{x},\epsilon)$, there are $t_{0},s_{0},s_{0}^{\prime}\in(0,R)$,
$n_{1}=n_{1}(t_{0},x,s)\in\mathbb{Z}$ and
$n_{2}=n_{2}(t_{0},y,s^{\prime})\in\mathbb{Z}$ satisfying
$|\\!|x+n_{1}\alpha-(y+n_{2}\alpha)|\\!|<H/R$, where
$T_{t_{0}}^{f}\tilde{x}=(x+n_{1}\alpha,s_{0})$ and
$T_{t_{0}}^{f}\tilde{y}=(y+n_{2}\alpha,s_{0}^{\prime})$ are in the same atom
of $\mathcal{P}$.

From 6.14, for every $\tilde{y}=(y,s^{\prime})\in B^{R}(\tilde{x},\epsilon)$,
we obtain $|\\!|x-y-(n_{2}-n_{1})\alpha|\\!|\leq H/R.$ Therefore, $\tilde{y}$
is in

$\displaystyle\bigcup_{j=-|n_{2}-n_{1}|}^{|n_{2}-n_{1}|}\left[x+j\alpha-\frac{H}{R},x+j\alpha+\frac{H}{R}\right]^{f}.$

To get a lower bound of $S_{T^{f},\mathcal{P}}(\epsilon,R)$, we will compute a
uniform upper bound for the Lebesgue measure of $B^{R}(\tilde{x},\epsilon)$,
it suffices to find an upper bound for $|n_{2}-n_{1}|$ for all $\tilde{x}$ and
$\tilde{y}\in B^{R}(\tilde{x},\epsilon)$. 6.15 below, summarizes this idea.

######  Lemma 6.15.

There exists a constant $G>0$ depending only on $\alpha$ and $\xi$ such that
$|n_{2}-n_{1}|\leq G$, where $n_{1}$ and $n_{2}$ are the integers in 6.14.

Let $\mathcal{D}\subset[0,1]$ be the set consisting of irrational numbers
$\alpha$ such that there exists $C_{1}>0$ depending on $\alpha$, satisfying

$q_{m+1}\leq C_{1}q_{m}\log^{2}q_{m}$

for every $m.$ In particular $\mathcal{D}$ contains badly approximable
irrational numbers.

######  Proposition 6.16.

If $\alpha\in\mathcal{D}$, $\xi\in\mathbb{T}$ and we consider the roof
function $f=d_{1}\mathds{1}_{[0,\xi)}+d_{2}\mathds{1}_{[\xi,1)}$, the metric
slow entropy of the special flow system is at most 1 for scale
$a_{\chi}(t)=t^{\chi}$.

###### Proof of 6.10.

From 6.14, for $R$ large enough and $\tilde{y}\in B^{R}(x,\epsilon)$, there
are $n_{1},n_{2}$ natural numbers, such that

$\tilde{y}\in\displaystyle\bigcup_{j=-|n_{2}-n_{1}|}^{|n_{2}-n_{1}|}\left[x+j\alpha-\frac{H}{R},x+j\alpha+\frac{H}{R}\right]^{f}.$

From 6.15, there is an upper bound $G$ for $|n_{2}-n_{1}|$ independent of
$\tilde{y}.$ Thus $\operatorname{Leb}(B^{R}(x,\epsilon))\leq\frac{4GH}{R}.$ In
other words, for any large $R>0$, the measure of each Hamming ball is bounded
above by $C^{\prime}/R$ for some constant $C^{\prime}$. Thus, there exists a
constant $C>0$ independent of $R$ and $\epsilon$ such that
$S_{T^{f},\mathcal{P}}(\epsilon,R)\geq CR$. This shows the slow entropy $\chi$
for this special flow is at least 1 using time scale $a_{\chi}(t)=t^{\chi}$.

Since $\alpha\in\mathcal{D}$ ($\alpha$ is badly approximable), applying 6.16
the upper bound of the metric slow entropy is 1. ∎

The remaining of this paper is dedicated to prove 6.12,
6.13, 6.14, 6.15, and 6.16.

###### Proof of 6.12.

By the definition of $f^{(n)}$, and assuming that $0<x<y<1$, it follows:

$f^{(n)}(x)-f^{(n)}(y)=e\sum_{j=0}^{n-1}\mathds{1}_{(x,y]}(\xi-j\alpha)-e\sum_{j=0}^{n-1}\mathds{1}_{(x,y]}(-j\alpha).$

By assumption on $\xi$, the interval of $(x,y]$ can be crossed by the orbit of
$\xi$ (and the orbit of $0$) for at most
$\frac{cq_{m+1}+2q_{m}}{q_{m}}+1\leq[cC_{\alpha}]+4\eqqcolon U$ times. Hence,
choose

$V=\\{ne:-U\leq n\leq U\text{ and }n\neq 0\\}.$

Taking $\kappa=\frac{1}{2cC_{\alpha}+4}$, assume for all $0\leq n\leq q_{m},$
$f^{(n)}(x)-f^{(n)}(y)\in V$, then we are done. Suppose $n_{0}$ is the minimum
number such that

$f^{(n_{0})}(x)-f^{(n_{0})}(y)=0.$

There exists $I_{0}$, where $I_{0}$ is the minimum time such that
$\bigcup\limits_{j=0}^{I_{0}}R_{\alpha}^{j}(x,y]$ covers $0$ or $\xi$. Because
$\alpha$ is badly approximable, let $c$ be the constant such that $I_{0}\leq
cq_{m+1}.$ Here $c$ is a constant depending only on $\alpha$ and $\xi.$ Assume
without loss of generality, $\xi\in(x+I_{0}\alpha,y+I_{0}\alpha].$ Assume at
time $J_{0},$ $0\in(x+(I_{0}+J_{0})\alpha,y+(I_{0}+J_{0})\alpha].$ This means

$|\\!|\xi-(x+I_{0}\alpha)|\\!|\leq\frac{C_{\xi}}{2q_{m}}$

and

$|\\!|x+(I_{0}+J_{0})\alpha|\\!|\leq\frac{C_{\xi}}{2q_{m}}.$

By triangle inequality, we obtain
$|\\!|\xi+J_{0}\alpha|\\!|\leq\frac{C_{\xi}}{q_{m}}.$ By definition of $\xi$,
this means $J_{0}\geq q_{m}.$ Therefore, when $n\in[I_{0}+1,I_{0}+J_{0}]$,

$f^{(n)}(x)-f^{(n)}(y)\in V.$

Hence, choose $N$ to be the time $n_{0}+I_{0}+q_{m}$, we complete the proof. ∎

###### Proof of 6.13.

From 6.12, we can find such constants $\kappa$, and $N$, and the interval
$\Lambda=[a,b]\cap\mathbb{N}$. For any time
$t_{0}\in[f^{(a)}(x)-s,f^{(b)}(x)-s]$,
$T_{t_{0}}^{f}{\tilde{x}}=(x+n_{1}\alpha,s_{0})$,
$T_{t_{0}}^{f}{\tilde{y}}=(y+n_{2}\alpha,s_{0}^{\prime})$. Then

$t_{0}=f^{(n_{2})}(y)+s_{0}^{\prime}-s^{\prime}=f^{(n_{1})}(x)+s_{0}-s.$

If $T_{t_{0}}^{f}(\tilde{x})$ and $T_{t_{0}}^{f}(\tilde{y})$ are in the same
atom, we can obtain $|s_{0}-s_{0}^{\prime}|<1/k$. Since $\tilde{x}$ and
$\tilde{y}$ are also in a same atom, we have

$|f^{(n_{2})}(y)-f^{(n_{1})}(x)|\leq 2/k.$

But from 6.12, there exists $p\in V$ such that

$f^{(n_{1})}(x)=f^{(n_{1})}(y)+p.$

Thus, by combining the above two equations, we deduce:

$|f^{(n_{2})}(y)-f^{(n_{1})}(y)-p|\leq 2/k.$

Since $V$ is a fixed finite set, we can assume $2/k\ll\min{\\{|q|:q\in V}\\}$,
so $n_{2}\neq n_{1}$. Because $T_{t_{0}}^{f}(\tilde{x})$ and
$T_{t_{0}}^{f}(\tilde{y})$ are in the same atom and $f\geq\frac{1}{M}$, we
have the following relations:

(30)
$\frac{D_{\alpha}}{|n_{2}-n_{1}|}\leq|\\!|(n_{2}-n_{1})\alpha|\\!|<\frac{2}{k},$

and

(31) $|n_{2}-n_{1}|\leq(\max{\\{|q|:q\in V\\}}+1)M.$

The above two equations are contradictory since we can choose $k$ to be
arbitrarily large. Therefore, for any such time
$t_{0}\in[f^{(a)}(x)-s,f^{(b)}(x)-s]$, $T_{t_{0}}^{f}(\tilde{x})$ and
$T_{t_{0}}^{f}(\tilde{y})$ are not in the same atom. Choosing
$T=f^{(b)}(x)-s$, and $\kappa^{\prime}=\kappa/100M$. We know that

$T\leq(N+1)M\leq(cC_{\alpha}+3)q_{m}M.$

Hence, choose $D=(cC_{\alpha}+3)M$, we complete the proof. ∎

###### Proof of 6.14.

By our choice of generating partitions, the set of times for which
$T_{t}^{f}\tilde{x}$ and $T_{t}^{f}\tilde{y}$ belong to the (closure of) the
same partition element is a union of closed intervals. We may therefore write
$(0,R)$ as a disjoint union of intervals
$\displaystyle\bigcup_{j=1}^{J}I_{j}$, where for each $I_{j}$,
$T_{t}^{f}(\tilde{x})$ and $T_{t}^{f}(\tilde{y})$ are either in the same atom
of $\mathcal{P}$, or they are in different atoms of $\mathcal{P}$ and the
intervals $I_{i}$ are maximal among such choices. In particular, either for
all odd values or all even values of $i$, the interior of $I_{i}$ consists of
matching times for $\tilde{x}$ and $\tilde{y}$.
Case 1. If $|J|=1$ or 2, $T_{t}^{f}(\tilde{x})$ and $T_{t}^{f}(\tilde{y})$
stays in the same atom all time in an interval of length at least
$(1-\epsilon)R$. Let $x$ and $y$ be the corresponding first coordinates when
they stay in the same atom for the first time. We know that

(32)
$f^{(n)}(x)-f^{(n)}(y)=e\sum_{j=0}^{n-1}\mathds{1}_{(x,y]}(\xi-j\alpha)-e\sum_{j=0}^{n-1}\mathds{1}_{(x,y]}(-j\alpha).$

Since $f$ is bounded between $d_{1}$ and $d_{2}$, $R/2M<N<2MR$ where $N$ is an
upper bound of the number of terms in Equation 32. Let $d=|\\!|x-y|\\!|$, as
in the proof of 6.12, there exists a constant $N(d)$ where
$N(d)\leq\frac{K}{d}$ for some constant $K$, s.t.
$\mathbb{T}\subset\displaystyle\bigcup_{j=1}^{N(d)}(x+j\alpha,y+j\alpha]$. By
assumption, $(x,y]$ cannot cross discontinuity points before $N/2$ times of
rotation when the orbit of $\tilde{x}$ and $\tilde{y}$ stay in the same atom.
Hence, $N(d)\geq R/4M$. Therefore, there exists a constant $H_{1}$ s.t. $d\leq
H_{1}/R$.
Case 2. If $|J|>2$, we regroup those intervals in the following way.

Let $t_{0}$ be the first time when $T_{t_{0}}(\tilde{x})$ and
$T_{t_{0}}(\tilde{y})$ stay in the same atom. Denote
$\tilde{I}_{0}=\\{t:t<t_{0}\\}$, note $\tilde{I}_{0}$ could be an empty set.
Then let $T_{t_{0}}^{f}(\tilde{x})=(x_{0},s_{0})$,
$T_{t_{0}}^{f}(\tilde{y})=(y_{0},s_{0}^{\prime})$. Since they are in the same
atom, there exists an integer $m$ s.t.

$\frac{C_{\xi}}{2q_{m+1}}\leq|\\!|x_{0}-y_{0}|\\!|<\frac{C_{\xi}}{2q_{m}}\leq
1/k.$

Applying 6.13, we can find the corresponding time $T$. By definition,
$t_{0}\in I_{j_{0}}$ for some $j_{0}$. Let $j_{1}>j_{0}$ be the minimal
natural number (if exists) such that the following interval
$\displaystyle\bigcup_{l=j_{0}}^{j_{1}-1}I_{l}$, has a length larger than $T$,
and $I_{j_{1}}$ is the interval when $T_{t}(\tilde{x})$ and $T_{t}(\tilde{y})$
are in the same atom. If such $j_{1}$ exists, denote

$\tilde{I}_{1}:=\displaystyle\bigcup_{l=j_{0}}^{j_{1}-1}I_{l}.$

Otherwise, denote $\tilde{I}_{1}=[t_{0},R)$. When
$\displaystyle\cup_{j=0}^{1}\tilde{I}_{j}\neq(0,R)$, we know at the starting
time $t_{1}$ of $I_{j_{1}}$, $T_{t_{1}}(\tilde{x})$ and $T_{t_{1}}(\tilde{y})$
are in the same atom, we can then repeat the above procedure to find
$\tilde{I}_{2}$. Inductively, we can get

$(0,R)=\displaystyle\bigcup_{j=0}^{L}\tilde{I}_{j}$

where $L+1$ is the number of such intervals. By assumption, $L\geq 1.$

If $L=1$, for the first time when the orbit of $\tilde{x}$ and $\tilde{y}$ are
in the same atom, let $m$ be the number s.t. the distance of the first
coordinate is between $\frac{C_{\xi}}{2q_{m+1}}$ and $\frac{C_{\xi}}{2q_{m}}$,
then we can obtain $R<T$. Otherwise, from 6.13, there will be a constant
$\kappa^{\prime}$ which is independent of $R$ and $\epsilon$, s.t. the total
time when those two orbits are not in the same atom is larger than
$\kappa^{\prime}R$. Since we can take $\epsilon<\kappa^{\prime}$ because
$\kappa^{\prime}$ is a fixed constant, this is a contradiction. Hence,

$\frac{1}{Dq_{m}}\leq\frac{1}{T}\leq\frac{1}{R}.$

Therefore, the distance of first coordinate if less than $\frac{H_{2}}{R}$ for
some constant $H_{2}.$

If $L>1$, the sum of the length of $\tilde{I}_{j}$ where $j=1,\dots,L-1$ is
less than $\epsilon R/\kappa^{\prime}$. Therefore, the last interval is of
length larger than $(1-\epsilon/\kappa^{\prime})R$. Using a similar argument,
there exists a constant $H_{3}$ satisfying the distance of the first
coordinate is less than $H_{3}/R$. Finally, taking
$H=\max\\{H_{1},H_{2},H_{3}\\}$, we finish the proof. ∎

###### Proof of 6.15.

From the definition of $n_{1}$ and $n_{2}$, we know

$|f^{(n_{1})}(x)-f^{(n_{2})}(y)|\leq|f^{(n_{1})}(x)-f^{(n_{2})}(y)+s_{0}-s_{0}^{\prime}|+|s_{0}-s_{0}^{\prime}|\leq
G_{1}$

for some constant $G_{1}.$ Since $n_{1}\leq 2MR$, we know there exists
constant $G_{2}$ s.t.

$|f^{(-n_{1})}(x+n_{1}\alpha)-f^{(-n_{2})}(y+n_{2}\alpha)|\leq G_{2},$

this is because the interval $[x+n_{1}\alpha,y+n_{2}\alpha)$ only cross
discontinuity points finitely many times. Using triangle inequality, cocycle
identity, and $f\geq\frac{1}{M}$, we get

(33) $\displaystyle G_{1}$ $\displaystyle\geq|f^{(n_{1})}(x)-f^{(n_{2})}(y)|$
$\displaystyle=|f^{(-n_{1})}(x+n_{1}\alpha)-f^{(-n_{2})}(y+n_{2}\alpha)+f^{(n_{2}-n_{1})}(y+n_{2}\alpha)|$
$\displaystyle\geq|f^{(n_{2}-n_{1})}(y+n_{2}\alpha)|-G_{2}$
$\displaystyle\geq\frac{|n_{2}-n_{1}|}{M}-G_{2}$

Thus we can obtain $|n_{2}-n_{1}|\leq(G_{2}+G_{1})M\eqqcolon G$. ∎

###### Proof of 6.16.

Choose $\delta=\frac{\eta}{2}$. Let $Z_{n}$ be the subset

$\displaystyle\bigcup_{j=-q_{n}}^{q_{n}}\left\\{x\in\mathbb{T}:x+j\alpha\in(-2q_{n}^{-1-\delta},2q_{n}^{-1-\delta})\cup(-2q_{n}^{-1-\delta}+\xi,\xi+2q_{n}^{-1-\delta})\right\\}.$

Since $\sum_{n=1}^{\infty}\frac{1}{q_{n}^{\delta}}<\infty,$ when $N$ is large
enough,
$\lambda\left(\displaystyle\cup_{n=N}^{\infty}Z_{n}\right)<\epsilon/4.$

Denote $\tilde{X}=\left(\displaystyle\cup_{n=N}^{\infty}Z_{n}\right)^{c},$
then $\lambda^{f}\left(\tilde{X}^{f}\right)>1-\epsilon/2$. Define

$\Omega_{\kappa}=\\{\tilde{y}\in\mathbb{T}^{f}:d(\tilde{y},\partial\mathcal{P})<\kappa\\}.$

There exists $0<\tilde{\epsilon}<\frac{\epsilon}{100Mk}$, such that
$\lambda^{f}(\Omega_{\tilde{e}})<\epsilon/4.$ There exists some integer $l$
such that $q_{l}\leq R<q_{l+1},$ choose $R$ to be a large value such that
$l\geq 2N$ and $q_{l}^{-1}<\tilde{e}/100M.$ Take
$A=\tilde{X}\cap\Omega_{\tilde{e}}.$ Note that for $\tilde{x}=(x,s)\in A$ the
Hamming ball $B^{R}(\tilde{x},\epsilon)$ centered at $\tilde{x}$ containing

$\left\\{(y,t)\in\mathbb{T}^{f}:y\in\left(-\frac{1}{q_{l+1}^{1+\delta}}+x,x+\frac{1}{q_{l+1}^{1+\delta}}\right),t\in(s-\tilde{e},s+\tilde{e})\right\\}.$

Hence, $S_{T^{f},\mathcal{P}}(\epsilon,R)\leq CR^{1+\delta}\log^{2+2\delta}R$
for some constant $C$ independent of $R$. It follows that
$\delta_{T^{f},\mathcal{P},\chi}^{S}(\epsilon)=0$ when $\chi=1+\eta.$ ∎

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