# Unifying topological entropy notions for piecewise continuous maps on the
interval

A.E. Calderón111Escuela de Ingeniería, Facultad de Ingeniería y Empresa –
Universidad Católica Silva Henríquez, Santiago, Chile<EMAIL_ADDRESS>and
E. Villar-Sepúlveda222Department of Engineering Mathematics – University of
Bristol, Bristol, England<EMAIL_ADDRESS>
###### Abstract

We generalize the definition of topological entropy given by Adler, Konheim,
and McAndrew (AKM) for piecewise continuous self-maps defined on a compact
interval (pc-maps). For this notion of entropy, we prove that the properties
of the AKM-entropy in the compact-continuous setting get naturally extended,
including that it can be estimated using Bowen’s formula disregarding the
metric used to compute it. Additionally, for piecewise strictly monotonic pc-
maps, we prove that the Misiurewicz-Szlenk formula based on the asymptotic
behavior of the number of monotony pieces for iterations of the map coincides
with our notion of topological entropy. In this way, we unify the notions of
entropy commonly used for piecewise continuous interval maps, proving that
they all coincide under compactness assumptions.

MSC 2022: 37B40, 37E05 (primary).
Keywords: interval map, piecewise continuous, topological entropy.

##  1 Introduction

Rufus Bowen was a pioneer in deepening and giving importance to the concept of
topological entropy, which was introduced in [1] by Adler, Konheim, and
McAndrew (AKM). This concept was originally defined using open covers to study
the ‘complexity’ of continuous self-maps on a compact topological space. A
reformulation of the AKM-entropy based on the dispersion of orbits was given
by Bowen and Dinaburg (BD) [4, 11]. However, that definition requires the
topological space to be metrizable. Fortunately, both definitions yield the
same number when the metric space is compact. Compactness guarantees that the
BD-entropy does not depend on the metric used to calculate it.

On the other hand, when the metric space is not compact, the BD-entropy gets
metric-dependent. It is well-known that if it is computed using two uniformly
equivalent metrics, the results are the same (for more details, see [24, pages
168-176]). Nonetheless, these can change if the metrics used are equivalent,
but not uniformly equivalent (an example of this can be found in [24, page
171]). Another problem with this definition in the non-compact case is that,
in general, the BD-entropy does not seem to measure the complexity of the
map’s dynamics. For example, consider the uniformly continuous map
$T:\mathbb{R}\to\mathbb{R}$ given by $T(x)=2x$, where $\mathbb{R}$ is endowed
with the Euclidean metric. The dynamics of $T$ is really simple: $0$ is the
unique fixed point and, for every $x\neq 0$, $|T^{n}(x)|\to\infty$ as
$n\to\infty$. Nevertheless, for this map, $h_{\text{BD}}(T)\geq\log 2$ (see
[5]), which breaks the premise: “positive entropy implies that the map has a
complicated dynamic behavior”. Unfortunately, this problem is not solved by
defining the entropy of $T$ only considering finite open covers of the non-
compact set (see [15, page 241]).

To address this issue, other definitions of entropy have been considered to
obtain good properties when the space is not compact or the dynamical system
is not continuous. In particular, two notions of topological entropy –based on
BD-entropy– are provided in recent papers [9, 20]. These entropies are
referred to as continuity entropies because they are obtained by restricting
the system to different compact subsets where the map is continuous and
invariant. Then, one has to take the supremum over all these restrictions (see
[20]). These notions are independent of the metric and seem to solve the non-
compactness problem. Such definitions preserve some good properties of the BD-
entropy and serve as a good measure of the complexity of the system’s dynamic
behavior in the non-compact context; nevertheless, the property associated
with topological semiconjugation is generally not valid. Note that if the
system is piecewise continuous, it can always be restricted to the subspace
where the system and its iterates are continuous, which is usually non-empty
and non-compact. Nonetheless, there are cases where complicated dynamics might
not be taken into account by continuity entropies in this context. For
instance, as shown in [6], the asymptotic dynamics of piecewise contracting
interval maps may be supported on minimal Cantor sets, which always intersect
some jump-discontinuities of the system. The good properties of the continuity
entropies strongly depend on the continuity of the map restricted to compact
and invariant subsets, but every piecewise contracting interval map restricted
to an attracting Cantor set –if it exists– is discontinuous, not to mention
that these Cantor sets are not exactly invariant (since these appear
regardless of how the map is defined over discontinuities). These sets satisfy
what is known as pseudo-invariance, which was introduced in [6] and allows
working with a concept of invariance that does not depend on how the map is
defined over its finite set of discontinuity points (a rigorous definition of
pseudo-invariance is provided in Section 2). In short, the problem with the
concepts of continuity entropies is that there are important properties that
are not valid in the discontinuous setting, and –sometimes– they fail to
capture some complicated dynamics.

Other formulations of entropy have been used in a one-dimensional setting. In
[14, 25], symbolic dynamics is used to define the entropy of piecewise
strictly monotonic pc-maps (i.e. the maps that are continuous on a compact
interval and strictly monotonic over a finite number of intervals). This
“symbolic entropy” coincides with the AKM-entropy when the system is
continuous (see [25]) or when the set of pre-images of turning points is dense
on the whole interval (see [14]). For the same class of systems (i.e.
piecewise strictly monotonic pc-maps), we have the following formula due to
Misiurewicz and Szlenk (see [18]): If $T$ is a continuous piecewise strictly
monotonic interval map, then

$h_{\text{AKM}}(T)=\lim_{n\to\infty}\frac{1}{n}\log c_{n},$ (1)

where $c_{n}\geq 1$ is the smallest number of intervals over which $T^{n}$ is
monotonous. However, if $T$ is not continuous, it is natural to consider the
value (1) as the definition of topological entropy (see [2, 3, 17, 12]), which
is also referred to as the singular entropy in the discontinuous context,
differentiating it from the topological entropy defined using Bowen’s formula
(see, for example, [21]), which has also been used to provide an idea of
“dynamic complexity” associated with piecewise continuous dynamical systems
(see [8, 7]). Considering this, the singular entropy of a piecewise strictly
monotonic pc-map can be computed by counting permutations exhibited by
periodic orbits (see [17]) or by initial segments of orbits (see [3]).
However, in [17] it is shown that these approximations via permutations can
fail if $T$ is not piecewise monotonic. Also, note that defining the entropy
of $T$ via symbolic dynamics or formula (1) makes it difficult to obtain good
properties for such a concept. Last, in [10], the concept of topological
entropy for arbitrary self-maps defined on a compact interval is defined using
Bowen’s formula. Whilst it is possible to obtain good properties from this
notion of entropy, the dependence on the metric is explicit.

Our goal in this article is to construct a metric-independent concept of
topological entropy that does not depend on how the map is defined over its
discontinuity points whilst keeping all of the good properties that the AKM-
entropy has in the compact-continuous setting. Particularly, we want this
concept of entropy to capture all complicated dynamics, unlike what happens
with the continuity entropies defined in [9, 20]. We will achieve this by
generalizing the classical topological construction of the AKM-entropy in [1],
ensuring that our definition is independent of the metric that generates the
topology of the space.

Outline of the article: In Section 2, we establish the main results that are
proven in this article. Sections 3 and 4 are dedicated to the construction of
the concept of topological entropy for pc-maps with a finite number of
critical points. Specifically, on the one hand, in Section 3 we introduce and
prove some basic results on preliminary concepts such as collections, open
covers, and minimal cardinality, all adapted to our context. On the other
hand, in Section 4 we define our notion of topological entropy for piecewise
continuous interval maps and prove that it is well-defined. In Section 5, we
prove some classic properties that this notion of entropy satisfies (see
Theorem 2.3). In Section 6, we prove that it is possible to use Bowen’s
formula to compute the topological entropy, concluding that this concept does
not depend on the metric used as long as the space is compact (see Theorem
2.4). In Section 7, we prove that the topological entropy can also be
estimated using Misiurewicz-Szlenk formula (see Theorem 2.6). As a corollary
of this, we prove that every injective pc-map has zero topological entropy.
Furthermore, we present a result on how discontinuities and/or critical points
influence the value of the topological entropy for systems that map each
continuity piece onto the entire interval (see Theorem 2.8). Finally, Section
8 is devoted to showing some examples and applications of the theory.

##  2 Preliminary definitions and main results

Let $X\subset\mathbb{R}$ be a compact interval with a non-empty interior. A
map $f:X\to X$ is said to be a piecewise continuous interval map (in short,
pc-map) if there exists a finite collection of $N\geq 1$ open subsets
$X_{1},X_{2},\ldots,X_{N}\subset X$, called continuity pieces of $f$, such
that

  1. 1.

$X_{i_{1}}\cap X_{i_{2}}=\emptyset\,$ for all $i_{1},i_{2}\in\\{1,\ldots,N\\}$
such that $i_{1}\neq i_{2}\,$;

  2. 2.

$X=\overline{X_{1}}\cup\overline{X_{2}}\cup\ldots\cup\overline{X_{N}}\,$;

  3. 3.

$f|_{X_{i}}$ is a continuous map for each $i\in\\{1,\ldots,N\\}$.

Furthermore, we assume that $f$ has a finite number of critical points (i.e.,
points where $f$ is not locally invertible or where $f$ is discontinuous)
within each continuity piece. The technical reasons behind this assumption are
explained in item (a) of Remark 2.9. Here, the topology considered on $X$ is
the one induced by the usual topology of $\mathbb{R}$. Also, we denote the
topological boundary operator within $X$ as $\partial$. Specifically, if
$A\subset X$, then $\partial A$ represents those points in $X$ for which every
open neighborhood intersects $A$ and $X\setminus A$.

For a pc-map $f:X\to X$ we will consider its continuity pieces to be sorted;
i.e., for all $i,j\in\\{1,\ldots,N\\}$ with $i<j$, we assume that $x<y$ for
every $(x,y)\in X_{i}\times X_{j}$. Let $\Delta_{f}$ be the set of boundaries
of the continuity pieces of $f$; that is,

$\Delta_{f}:=\bigcup_{i=1}^{N}\partial X_{i}.$

For convenience of notation, we assume that $f$ is continuous at the endpoints
of $X$, which implies that the endpoints of $X$ do not belong to $\Delta_{f}$
(see item (b) of Remark 2.9). Note that the continuity pieces $X_{1}$ and
$X_{N}$ are semi-open.

Now, for every $m\geq 0$, we define the set

$\Delta_{f}^{m}=(\Delta_{f})^{m}:=\bigcup_{j=0}^{m-1}f^{-j}(\Delta_{f})\subset
X,$

where, by convention, we consider that $\Delta_{f}^{0}=\emptyset$. As $f$ has
a finite number of critical points, for every $m\geq 0$ the set
$\Delta_{f}^{m}$ is finite and, consequently, a closed set. Furthermore, for
every $k\geq 1$, we define the set of boundaries of the continuity pieces of
the map $f^{k}$, denoted by $\Delta_{f^{k}}$, as the collection induced by
$\Delta_{f}$; that is, $\Delta_{f^{k}}:=\Delta_{f}^{k}$.

We highlight that $\Delta_{f}$ can contain removable or jump discontinuities
of $f$. As usual, the orbit of $x\in X$ (by $f$) is defined as the countable
set $\\{f^{n}(x)\\}_{n\geq 0}$. Note that this notion depends on how $f$ is
defined on $\Delta_{f}$. For this reason, it is convenient to work on the set
$\widetilde{X}_{f}$, which is defined as the collection of points in $X$ whose
orbit does not intersect $\Delta_{f}$; that is,

$\widetilde{X}_{f}:=\bigcap_{n\geq 0}f^{-n}(X\setminus\Delta_{f}).$

Note that $\widetilde{X}_{f}=X$ if and only if $\Delta_{f}=\emptyset$. For the
iterates of $f$, we have the following result:

######  Lemma 2.1.

For every pc-map $f:X\to X$ and every integer $k\geq 1$,
$\widetilde{X}_{f}=\widetilde{X}_{f^{k}}$.

###### Proof.

For every integer $k\geq 1$, it is clear that
$\widetilde{X}_{f}\subset\widetilde{X}_{f^{k}}$. Reciprocally, if
$x\in\widetilde{X}_{f^{k}}$, we have that $x\in
f^{-jk}(X\setminus\Delta_{f^{k}})$ for all $j\geq 0$. This implies that

$f^{jk}(x)\in X\setminus\Delta_{f^{k}}\quad\forall\,j\geq 0.$

Thus, we deduce that $f^{jk+\ell}(x)\notin\Delta_{f}$ for every $j\geq 0$ and
$\ell\in\\{0,\ldots,k-1\\}$. Therefore, $f^{i}(x)\notin\Delta_{f}$ for every
$i\geq 0$. This argument shows that $\widetilde{X}_{f}=\widetilde{X}_{f^{k}}$.
∎

Since each pc-map has a finite number of critical points, it is not difficult
to see that $\widetilde{X}_{f}$ is non-empty and dense in $X$.

Recall that a set $A\subset X$ is $f$-invariant if $f(A)\subset A$. Note that
$\widetilde{X}_{f}$ is an $f$-invariant –usually non-compact– set, and the
restriction $f|_{\widetilde{X}_{f}}:\widetilde{X}_{f}\to\widetilde{X}_{f}$ is
a continuous self-map. As it happens with the concept of orbit, the notion of
invariance depends on the definition of $f$ over $\Delta_{f}$. There are 2
ways to solve this: (1) Require $A\subset X$ to be an $f$-invariant subset
such that $A\cap\widetilde{X}_{f}\neq\emptyset$ (note that, in this case,
$A\cap\widetilde{X}_{f}$ is $f$-invariant as well); (2) Using a new type of
invariance called pseudo-invariance (defined below), which does not depend on
how $f$ is defined on $\Delta_{f}$.

We say that a set $A\subset X$ is $f$-pseudo-invariant if for every $x_{0}\in
A$, the limit of $f(x)$ as $x\to x_{0}$ either from above or below, exists and
belongs to $A$. In particular, note that if $A\subset X$ is such that
$A\cap\Delta_{f}=\emptyset$, then $A$ is $f$-invariant if and only if $A$ is
$f$-pseudo-invariant. Also, if $A$ is an $f$-pseudo-invariant set, then
$f(A\setminus\Delta_{f})\subset A$. Under some conditions, it can be
guaranteed that any $f$-pseudo-invariant set intersects $\widetilde{X}_{f}$:
if $f$ is a pc-map such that for every $c\in\Delta_{f}$, the limit of $f(x)$
as $x\to c$, either from above or below, exists and belongs to
$\widetilde{X}_{f}$, then every $f$-pseudo-invariant subset contained in $X$
intersects $\widetilde{X}_{f}$ (this result follows directly from the
definition). However, it will be convenient to require the corresponding
$f$-(pseudo-)invariant subset intersects $\widetilde{X}_{f}$ in each
statement.

Although invariant and $f$-pseudo-invariant sets generate invariant sets when
we intersect them with $\widetilde{X}_{f}$ (and the intersection is non-
empty), both approaches are interesting because invariance is neither a
necessary nor a sufficient condition for $f$-pseudo-invariance 333For example,
the $f$-pseudo-invariant Cantor sets that can appear in the asymptotic
dynamics of piecewise contracting interval maps are not invariant, in general
(see [6]). On the other hand, any invariant subset containing an isolated
discontinuity $c\in\Delta_{f}$ such that $f(c)=c$ is not $f$-pseudo-
invariant..

######  Definition 2.1.

Let $X$ and $Y$ be two compact intervals and let $f:X\to X$ and $g:Y\to Y$ be
two pc-maps defined on $X$ and $Y$, respectively. We say that $f$ is
topologically semi-conjugate (respectively, conjugate) to $g$ if there exists
a continuous surjection (respectively, homeomorphism) $\varphi:X\to Y$ such
that $\varphi^{-1}(\Delta_{g})=\Delta_{f}$ and $\varphi\circ f=g\circ\varphi$
on $X\setminus\Delta_{f}$.

######  Remark 2.2.

  1. 1.

In Definition 2.1, the condition $\varphi^{-1}(\Delta_{g})=\Delta_{f}$ is not
needed when $\Delta_{f}$ and $\Delta_{g}$ only comprise jump discontinuity
points of $f$ and $g$, respectively. We use it only to be general without
imposing that assumption.

  2. 2.

If $f:X\to X$ and $g:Y\to Y$ are two pc-maps such that $f$ is topologically
semi-conjugate to $g$ by a continuous surjection $\varphi:X\to Y$, then
$\\#\Delta_{f}\geq\\#\Delta_{g}$, $\varphi(\Delta_{f})=\Delta_{g}$ and
$\varphi(\widetilde{X}_{f})=\widetilde{Y}_{g}$.

We recall that the diameter of a set $E\subset X$, denoted by
$\operatorname{diam}(E)$, is defined as the supremum of the distances between
all pairs of points that belong to $E$ (by convention, we say that
$\operatorname{diam}(\emptyset)=0$). Thus, the diameter of a collection
$\mathcal{C}$ of subsets of $X$ is given by

$\operatorname{diam}(\mathcal{C}):=\sup_{C\in\mathcal{C}}\operatorname{diam}(C).$

Note that $\operatorname{diam}(\mathcal{C})\leq\operatorname{diam}(X)<\infty$.

To preserve the usual notation, the topological entropy that we will construct
in Sections 3 and 4 will be denoted by $h_{top}$. Now, we establish the main
results of this paper, which are divided into five theorems.

######  Theorem 2.3.

Let $X$ and $Y$ be two compact intervals and let $f:X\to X$ and $g:Y\to Y$ be
two pc-maps. Then, the following properties hold:

  1. (1)

If $f$ is continuous, then $h_{top}(f)$ coincides with the AKM-entropy of $f$.

  2. (2)

If $A$ is a non-empty, compact and $f$-(pseudo-)invariant subset of $X$ such
that $A\cap\widetilde{X}_{f}\neq\emptyset$, then $h_{top}(f|_{A})\leq
h_{top}(f)$.

  3. (3)

If $f$ is topologically semi-conjugate (respectively, conjugate) to $g$, then
$h_{top}(f)\geq h_{top}(g)$ (respectively, $h_{top}(f)=h_{top}(g)$).

  4. (4)

$h_{top}(f^{k})=k\,h_{top}(f)$ for every integer $k\geq 0$ (by convention,
$f^{0}$ is the identity map).

  5. (5)

If $X=A_{1}\cup\ldots\cup A_{p}$, where $A_{i}$ is a compact
$f$-(pseudo-)invariant subset of $X$ such that
$A_{i}\cap\widetilde{X}_{f}\neq\emptyset$ for each $1\leq i\leq p$, then

$h_{top}(f)=\max_{1\leq i\leq p}\\!\big{\\{}h_{top}(f|_{A_{i}})\big{\\}}.$

  6. (6)

If $h_{top}(f)<\infty$ (respectively, $h_{top}(f)=\infty$) and
$\\{\mathcal{C}_{k}\\}_{k\geq 1}$ is a sequence of open covers of $X$ such
that $\operatorname{diam}(\mathcal{C}_{k})\to 0$ as $k\to\infty$, then
$\lim_{k\to\infty}h_{top}(f,\mathcal{C}_{k})=h_{top}(f)$ (respectively,
$\lim_{k\to\infty}h_{top}(f,\mathcal{C}_{k})=\infty$).

Moreover, the following result will enable us to compute the entropy using the
classic Bowen’s formula while retaining the same good properties of the
compact-continuous setting.

######  Theorem 2.4 ([Bowen’s formula]).

Let $f:X\to X$ be a pc-map. If $A$ is a non-empty, compact and
$f$-(pseudo-)invariant subset of $X$ such that
$A\cap\widetilde{X}_{f}\neq\emptyset$, then

$h_{top}(f|_{A})=\lim_{\epsilon\to 0}\limsup_{n\to\infty}\frac{\log
s_{n}(A;f,\epsilon)}{n}=\lim_{\epsilon\to 0}\limsup_{n\to\infty}\frac{\log
r_{n}(A;f,\epsilon)}{n},$ (2)

where $s_{n}(A;f,\epsilon)$ (respectively, $r_{n}(A;f,\epsilon)$) is the
maximal cardinality of $(n,\epsilon)$-separated sets (respectively, minimal
cardinality of $(n,\epsilon)$-spanning sets) of $A\cap\widetilde{X}_{f}$.
Moreover, for the calculation using Bowen’s formula, the value of
$h_{top}(f|_{A})$ does not depend on the metric that generates the induced
topology on $A\cap\widetilde{X}_{f}$.

######  Remark 2.5.

In (2), the values

$\lim_{\epsilon\to 0}\limsup_{n\to\infty}\frac{\log
s_{n}(A;f,\epsilon)}{n}\quad\text{ and }\quad\lim_{\epsilon\to
0}\limsup_{n\to\infty}\frac{\log r_{n}(A;f,\epsilon)}{n}$

refer to the limits obtained via the classic Bowen’s formula (see [21, page
58]) for the map $f$ restricted to the invariant subset
$A\cap\widetilde{X}_{f}$.

Next, for each $n\geq 1$, we denote by $c_{n}\geq 1$ the smallest number of
intervals on which $f^{n}$ is monotonic and essentially continuous; that is,

$\displaystyle c_{n}:=\min\\!\big{\\{}\\#\mathcal{P}\ :\
\mathcal{P}\in\Phi(f^{n})\big{\\}},$ (3)

where $\Phi(f^{n})$ denotes the collection of partitions $\mathcal{P}$
comprising intervals of $X$ with non-empty interior such that for all
$P\in\mathcal{P}$, the restriction of $f^{n}$ to the interior of $P$ is
monotonic and continuously extendable (i.e., every point within the interior
of $P$ is either a point where $f^{n}$ is continuous or a removable
discontinuity of $f^{n}$). Note that adding removable discontinuities to the
map $f$ increases its number of continuity pieces, but does not modify
$c_{n}$.

The following result relates the entropy with formula (1) adapted to this
context.

######  Theorem 2.6 ([Misiurewicz-Szlenk formula]).

Let $f:X\to X$ be a pc-map. Then

$h_{top}(f)=\lim_{n\to\infty}\frac{1}{n}\log c_{n},$

where $c_{n}$ is defined as in (3).

In addition, we say that a pc-map is injective when it is injective over the
union of its continuity pieces. Thus, as a consequence of Theorem 2.6, we can
state an extension of the following result: “injective continuous maps defined
on a compact interval have zero topological entropy” (see [23, Lemma 8.3.1]).

######  Corollary 2.7.

Let $f:X\to X$ an injective pc-map. Then $h_{top}(f)=0$.

Furthermore, under certain conditions, it is possible to obtain the value of
the topological entropy as a function of the minimum number of monotonic
pieces of $f$:

######  Theorem 2.8.

Assume that $f:X\to X$ is a pc-map such that $\overline{f(X_{i})}=X$ for each
$i\in\\{1,\ldots,N\\}$, then $h_{top}(f)=\log c_{1}$, where $c_{1}$ represents
the number of strictly monotonic pieces of $f$.

######  Remark 2.9.

  * (a)

As any pc-map $f$ has a finite number of critical points, it follows that the
preimages of $\Delta_{f}$ (by $f$) are also finite sets; that is,
$\\#\big{(}f^{-k}(\Delta_{f})\big{)}<\infty$ for every $k\geq 1$. This
condition will be frequently used to ensure, for example, that the covers we
construct are open (see Lemma 3.6), the iteration of a pc-map remains a pc-map
(i.e., for every $k\geq 1$, $f^{k}$ has a finite number of continuity pieces),
and $\widetilde{X}_{f}\neq\emptyset$.

  * (b)

With the assumptions we have made, such as assuming that the endpoints of $X$
are not boundaries of continuity pieces (i.e., they do not belong to
$\Delta_{f}$), some aspects get simplified; for example, the following
equality holds: $N=\\#\Delta_{f}+1$. However, there is no problem with
considering one or both extremes of $X$ as points in $\Delta_{f}$. Doing this
does not generate additional continuity pieces. Nonetheless, this change
implies the modification of $X_{1}$, $X_{N}$, $\widetilde{X}_{f}$, and every
definition that depends on $\Delta_{f}$. In particular, these changes could
modify the statements or some of the procedures used to prove the results
slightly. In any case, the conclusions of this paper are still valid.

##  3 Collections, covers and minimal cardinality

In this section, we state the notation we use to define the entropy of pc-
maps. Throughout this paper, we set a compact interval $X$ and a pc-map
$f:X\to X$. For ease of notation, in all that follows, when there is no place
for confusion, we will use $\Delta:=\Delta_{f}$ and
$\widetilde{X}:=\widetilde{X}_{f}$, together with the concept of
(pseudo-)invariance when talking about $f$-(pseudo-)invariance, omitting the
name of the map.

We denote by $\mathcal{P}(X)$ the power set of $X$ and call
$\mathcal{C}\subset\mathcal{P}(X)$ a collection of subsets of $X$ (or just a
collection of $X$) when $\emptyset\not\in\mathcal{C}$. Furthermore, we say
that the collection $\mathcal{C}$ is finite if $\\#\mathcal{C}<\infty$ (note
that $\emptyset$ is a finite collection since $\\#\emptyset=0<\infty$ and
$\emptyset\notin\emptyset$). In particular, we say that the collection
$\mathcal{C}$ is a cover of a subset $A\subset X$ if
$\mathcal{C}\neq\emptyset$ and $A\subset\bigcup_{C\in\mathcal{C}}C$. Moreover,
we define the $j$-th pre-image of the collection $\mathcal{C}$ as
$f^{-j}(\mathcal{C}):=\\{f^{-j}(C)\
:\;C\in\mathcal{C}\\}\setminus\\{\emptyset\\}$, for every integer $j\geq 0$
(we recall that, by convention, $f^{0}$ is the identity map).

Let $\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{m}$ be collections of
$X$. We define the $\vee$-product of these collections as

$\bigvee_{j=1}^{m}\mathcal{C}_{j}:=\left\\{\bigcap_{j=1}^{m}C_{j}\ \ :\ \quad
C_{j}\in\mathcal{C}_{j}\quad\forall\,j\in\\{1,\ldots,m\\}\right\\}\setminus\\{\emptyset\\},$
(4)

which is a new collection of $X$. Also, if $\mathcal{C}$ and $\mathcal{D}$ are
collections of $X$, we say that $\mathcal{D}$ is finer than $\mathcal{C}$, and
write $\mathcal{C}\preccurlyeq\mathcal{D}$, if for every $D\in\mathcal{D}$
there exists $C\in\mathcal{C}$ such that $D\subset C$. With this, it is clear
that for every pair of collections $\mathcal{C}$ and $\mathcal{D}$,
$\mathcal{C}\preccurlyeq\mathcal{C}\preccurlyeq\mathcal{C}\vee\mathcal{D}\preccurlyeq\emptyset$.

######  Remark 3.1.

Here, we mention some important facts about the previous definitions:

  1. (a)

Suppose that $\mathcal{C}_{1},\ldots,\mathcal{C}_{m}$ are collections of $X$
and $\mathcal{D}_{1},\ldots,\mathcal{D}_{m}$ are pairwise non-empty sub-
collections of the previous collections; that is,
$\emptyset\neq\mathcal{D}_{j}\subset\mathcal{C}_{j}$ for every
$j\in\\{1,\ldots,m\\}$. Then, $\mathcal{D}_{1}\vee\ldots\vee\mathcal{D}_{m}$
is a sub-collection of $\mathcal{C}_{1}\vee\ldots\vee\mathcal{C}_{m}$.

  2. (b)

Any (finite) cover of $X$ is a (finite) cover of any subset $A\subset X$.

  3. (c)

The finite $\vee$-product of covers of a subset $A\subset X$ is a cover of $A$
as well.

Let $\mathcal{C}$ be a collection of $X$. Given $F\subset X$ and $n\geq 1$, we
consider the new collections

$\mathcal{C}\setminus F:=\big{\\{}C\setminus F\ :\
\;C\in\mathcal{C}\big{\\}}\setminus\\{\emptyset\\},\qquad\text{and}\qquad\mathcal{C}^{n}_{f}:=\bigvee_{j=0}^{n-1}f^{-j}\big{(}\mathcal{C}\setminus\Delta\big{)}.$

When there is no place for confusion, we will often use the notation
$\mathcal{C}^{n}:=\mathcal{C}^{n}_{f}$, omitting the name of the map. Note
that for every $n\geq 0$,
$\mathcal{C}^{n}=\mathcal{C}^{n}\setminus\Delta^{n}$. Moreover, if
$\mathcal{C}$ and $\mathcal{D}$ are collections of subsets of $X$ such that
$\mathcal{C}\preccurlyeq\mathcal{D}$, then
$\mathcal{C}^{n}\preccurlyeq\mathcal{D}^{n}$ for every $n\geq 1$.

######  Lemma 3.2.

Let $A$ be a subset of $X$. Then,

  1. 1.

$A\subset f^{-j}(A)$ for all $j\geq 0$ if and only if $A$ is $f$-invariant.
Moreover, if $A$ is an invariant subset of $X$ such that
$A\cap\widetilde{X}\neq\emptyset$ and $\mathcal{C}$ is a cover of $A$, then
$\mathcal{C}^{n}$ is a cover of $A\cap\widetilde{X}$ for every $n\geq 1$.

  2. 2.

If $A$ is a non-empty pseudo-invariant subset of $X$ such that
$A\cap\widetilde{X}\neq\emptyset$, then $A\cap\widetilde{X}\subset
f^{-j}(A\setminus\Delta)$ for all $j\geq 0$. Moreover, if $A$ is a pseudo-
invariant subset of $X$ and $\mathcal{C}$ is a cover of $A$, then
$\mathcal{C}^{n}$ is a cover of $A\cap\widetilde{X}$ for every $n\geq 1$.

###### Proof.

(i) If $A\subset f^{-j}(A)$ for all $j\geq 0$, then taking $j=1$, we see that

$A\subset f^{-1}(A)\quad\Rightarrow\quad f(A)\subset f(f^{-1}(A))\subset A,$

which shows that $A$ is $f$-invariant. Reciprocally, let $j\geq 0$ and assume
that $f(A)\subset A$. Then $f^{-j+1}(A)\subset f^{-j}(f(A))\subset f^{-j}(A)$,
which implies that

$A=f^{0}(A)\subset f^{-1}(A)\subset f^{-2}(A)\subset\ldots\subset f^{-j}(A).$

Now, if $A$ is $f$-invariant and $\mathcal{C}$ is a cover of $A$, then for
every $j\geq 0$,

$A\cap\widetilde{X}\subset\left(\bigcup_{C\in\mathcal{C}}f^{-j}(C)\right)\cap\widetilde{X}\subset\bigcup_{C\in\mathcal{C}}f^{-j}(C\setminus\Delta),$

which proves that $f^{-j}(\mathcal{C}\setminus\Delta)$ is a cover of
$A\cap\widetilde{X}$ for all $j\geq 0$. Thus, we deduce that $\mathcal{C}^{n}$
is a cover of $A\cap\widetilde{X}$ for every $n\geq 1$.
(ii) Let $j\geq 0$. Since $A\subset X$ is pseudo-invariant and $f^{j}$ is a
continuous map on $A\setminus\Delta^{j}$, we deduce that
$A\setminus\Delta^{j}$ is $f^{j}$-invariant and

$A\setminus\Delta^{j}\subset
f^{-j}\big{(}f^{j}\left(A\setminus\Delta^{j}\right)\big{)}\subset
f^{-j}\left(A\setminus\Delta^{j}\right)\subset f^{-j}(A\setminus\Delta).$ (5)

Thus, we deduce that

$A\cap\widetilde{X}=\left(A\setminus\Delta^{j}\right)\cap\widetilde{X}\subset
f^{-j}(A\setminus\Delta)\cap\widetilde{X}\subset f^{-j}(A\setminus\Delta).$
(6)

Finally, note that if $\mathcal{C}$ is a cover of $A$, the last part of the
proof follows from (6), using an argument similar to the one used at the end
of the proof of part (i). ∎

######  Definition 3.1.

Suppose that $\emptyset\neq A\subset X$ and let $\mathcal{C}$ be a cover of
$A$. We define the minimal cardinality of sub-covers of $A$ with respect to
$\mathcal{C}$, denoted by $\aleph_{A}(\mathcal{C})$, as

$\aleph_{A}(\mathcal{C}):=\inf\\!\left\\{m\geq 1\ :\
\exists\,\mathcal{D}\subset\mathcal{C}\text{ with $\\#\mathcal{D}=m$ such that
}A\subset\bigcup_{D\in\mathcal{D}}D\right\\}.$

Note that $1\leq\aleph_{A}(\mathcal{C})\leq\\#\mathcal{C}$, where
$\\#\mathcal{C}$ might be infinity. Some important properties of the minimal
cardinality are established in the following result.

######  Lemma 3.3.

Let $A$ be a subset of $X$ and let $\mathcal{C},\mathcal{D}$ be covers of $A$
such that $\aleph_{A}(\mathcal{C})$ and $\aleph_{A}(\mathcal{D})$ are finite
numbers. Then, the following properties hold:

  1. (a)

If $\mathcal{C}\preccurlyeq\mathcal{D}$, then
$\aleph_{A}(\mathcal{C})\leq\aleph_{A}(\mathcal{D})$.

  2. (b)

If $\mathcal{C}\subset\mathcal{D}$, then
$\aleph_{A}(\mathcal{C})\geq\aleph_{A}(\mathcal{D})$.

  3. (c)

$\aleph_{A}(\mathcal{C}\vee\mathcal{D})\leq\aleph_{A}(\mathcal{C})\cdot\aleph_{A}(\mathcal{D})$.

  4. (d)

If $A$ is $f$-invariant, then
$\aleph_{A}(f^{-j}(\mathcal{C}))\leq\aleph_{A}(\mathcal{C})$ for all $j\geq
0$.

###### Proof.

Define $m:=\aleph_{A}(\mathcal{C})$, and $n:=\aleph_{A}(\mathcal{D})$.
Moreover, let $\mathcal{M}\subset\mathcal{C},\,\mathcal{N}\subset\mathcal{D}$
be finite covers of $A$ such that $\\#\mathcal{M}=m$ and $\\#\mathcal{N}=n$.
These definitions will be used throughout the whole proof.
(a) If $\mathcal{C}\preccurlyeq\mathcal{D}$, then for all $N\in\mathcal{N}$
there exists $M_{N}\in\mathcal{C}$ such that $N\subset M_{N}$. Thus, we deduce
that

$A\subset\bigcup_{N\in\,\mathcal{N}}N\subset\bigcup_{N\in\,\mathcal{N}}M_{N}.$

Therefore, as $n=\\#\mathcal{N}\geq\\#\\{M_{N}\ :\ N\in\mathcal{N}\\}\geq m$,
then we conclude (a).
(b) If $\mathcal{C}\subset\mathcal{D}$, then
$\mathcal{D}\preccurlyeq\mathcal{C}$. Thus, the result follows from part (a).
(c) The collection
$\mathcal{M}\vee\mathcal{N}\subset\mathcal{C}\vee\mathcal{D}$ is a finite
cover of $A$ such that
$\aleph_{A}(\mathcal{C}\vee\mathcal{D})\leq\aleph_{A}(\mathcal{M}\vee\mathcal{N})\leq
mn$, which implies that
$\aleph_{A}(\mathcal{C}\vee\mathcal{D})\leq\aleph_{A}(\mathcal{C})\cdot\aleph_{A}(\mathcal{D})$,
concluding the proof of part (c).
(d) Due to part (i) of Lemma 3.2, since $A$ is $f$-invariant, for $j\geq 0$ we
have that

$A\subset\bigcup_{M\in\,\mathcal{M}}M\qquad\Rightarrow\qquad A\subset
f^{-j}(A)\subset\bigcup_{M\in\,\mathcal{M}}f^{-j}(M),$

which lets us deduce that $f^{-j}(\mathcal{M})$ is a finite cover of $A$ with
no more than $m$ elements. Then,

$\aleph_{A}(f^{-j}(\mathcal{C}))\leq\aleph_{A}(f^{-j}(\mathcal{M}))\leq
m=\aleph_{A}(\mathcal{M})=\aleph_{A}(\mathcal{C}),$

concluding the proof of part (d). ∎

######  Remark 3.4.

Under the same assumptions as in Lemma 3.3, we have that

$A\subset\bigcup_{C\in\mathcal{C}}C\qquad\Leftrightarrow\qquad
A=\bigcup_{C\in\mathcal{C}}(C\cap A).$

Thus, we deduce that $\aleph_{A}(\mathcal{C}\cap A)=\aleph_{A}(\mathcal{C})$,
where

$\mathcal{C}\cap A:=\mathcal{C}\vee\\{A\\}=\big{\\{}C\cap A\ :\
C\in\mathcal{C}\big{\\}}\setminus\\{\emptyset\\}.$

######  Remark 3.5.

Fix $n\geq 1$ and consider the same hypotheses as in Lemma 3.3. Then, due to
part (d) of said lemma, we can always find a sub-collection of
$\mathcal{C}^{n}$ that is a finite cover of $A\cap\widetilde{X}$. Thus,
$\aleph_{A\cap\widetilde{X}}(\mathcal{C}^{n})<\infty$ for every integer $n\geq
1$.

Let $A,B$ be non-empty subsets such that $A\subset B\subset X$ and let
$\mathcal{C}$ be a cover of $A$. We say that $\mathcal{C}$ is a $B$-open cover
of $A$ if every $C\in\mathcal{C}$ is an open set of $B$ (endowed with the
subspace topology). To avoid redundancy, when $A=B$ we will say “$A$-open
cover” when talking about an “$A$-open cover of $A$”. Naturally, a finite
$B$-open cover of $A$ is a $B$-open cover of $A$ with a finite number of
elements.

Assume that $A\subset X$ is a (pseudo-)invariant set such that
$A\cap\widetilde{X}\neq\emptyset$. Let $\mathcal{C}$ be an $A$-open cover.
From Lemma 3.2, we have that $\mathcal{C}^{n}$ is a cover of
$A\cap\widetilde{X}$ for every $n\geq 1$. Now, we also prove that every
element of $\mathcal{C}^{n}$ is $A$-open.

######  Lemma 3.6.

Let $A$ be a (pseudo-)invariant subset of $X$ such that
$A\cap\widetilde{X}\neq\emptyset$ and let $\mathcal{C}$ be an $A$-open cover.
Then, $\mathcal{C}^{n}$ is an $A$-open cover of $A\cap\widetilde{X}$ for every
$n\geq 1$.

###### Proof.

We proceed by induction on $n$. If $n=1$, the result follows. Next, assume
that for $k\geq 1$, $\mathcal{C}^{k}$ is an $A$-open cover of
$A\cap\widetilde{X}$. Note that

$\displaystyle\mathcal{C}^{k}=\left(\bigvee_{j=0}^{k-1}f^{-j}(\mathcal{C})\right)\setminus\Delta^{k}.$

With this, we deduce that

$\displaystyle\mathcal{C}^{k+1}$
$\displaystyle=\mathcal{C}^{k}\vee\big{(}f^{-k}(\mathcal{C})\setminus
f^{-k}(\Delta)\big{)}=\mathcal{C}^{k}\vee\big{(}f^{-k}(\mathcal{C})\setminus\Delta^{k+1}\big{)}.$
(7)

Since $\mathcal{C}^{k}$ is an $A$-open cover of $A\cap\widetilde{X}$, and
$f^{-k}(\Delta)\subset\Delta^{k+1}$, we have that $\mathcal{C}^{k+1}$ is an
$A$-open cover of $A\cap\widetilde{X}$, which ends the proof. ∎

######  Remark 3.7.

Let $\mathcal{C},\mathcal{D}$ be covers of a set $A\subset X$. If
$A\cap\widetilde{X}\neq\emptyset$ and

$\mathcal{C}\cap\widetilde{X}=\mathcal{D}\cap\widetilde{X},$

then the least number of elements required to cover $A\cap\widetilde{X}$ with
either $\mathcal{C}$ or $\mathcal{D}$ is the same. Specifically,

$\aleph_{A\cap\widetilde{X}}(\mathcal{C})=\aleph_{A\cap\widetilde{X}}(\mathcal{D}),$

Note that both values could be infinite.

######  Lemma 3.8.

Let $A$ be a compact (pseudo-)invariant subset of $X$ such that
$A\cap\widetilde{X}\neq\emptyset$, and let $\mathcal{C}$ be an $A$-open cover.
Then, for every $j\geq 0$,

$\aleph_{A\cap\widetilde{X}}\big{(}f^{-j}\big{(}\mathcal{C}\setminus\Delta\big{)}\big{)}\leq\aleph_{A\cap\widetilde{X}}(\mathcal{C})<\infty.$

###### Proof.

Fix $j\geq 0$. As $A$ is compact, then there exists a finite sub-collection of
$\mathcal{C}$ which is an $A$-open cover. Therefore, from Lemma 3.2, we have
that $f^{-j}(\mathcal{C}\setminus\Delta)$ admits a finite sub-cover of
$A\cap\widetilde{X}$ (not necessarily $A$-open). Thus, part (d) of Lemma 3.3
and Remark 3.7 imply what is required, concluding the proof. ∎

######  Remark 3.9.

Fix $n\geq 1$ and consider the same hypotheses as in Lemma 3.8. Then, we can
always find a sub-collection of $\mathcal{C}^{n}$ that is a finite $A$-open
cover of $A\cap\widetilde{X}$ thanks to Lemma 3.6. Thus,
$\aleph_{A\cap\widetilde{X}}(\mathcal{C}^{n})<\infty$ for every integer $n\geq
1$.

##  4 Definition of topological entropy

In this section, we define the concept of topological entropy adapted to pc-
maps. The following result shows that such concept is well-defined.

######  Lemma 4.1.

Let $A$ be a compact (pseudo-)invariant subset of $X$ such that
$A\cap\widetilde{X}\neq\emptyset$ and let $\mathcal{C}$ be an $A$-open cover.
Then, the sequence $\\{a_{n}\\}_{n\geq 1}\subset\mathbb{R}$ given by

$a_{n}:=\log\aleph_{A\cap\widetilde{X}}(\mathcal{C}^{n})\qquad\forall\,n\geq
1,$

is sub-additive; that is $a_{n+k}\leq a_{n}+a_{k}$ for all $n,k\geq 1$.

###### Proof.

Fixing $n,k\geq 1$, from parts (c)–(d) of Lemma 3.3, we have that

$\displaystyle a_{n+k}=\log\aleph_{A\cap\widetilde{X}}(\mathcal{C}^{n+k})$
$\displaystyle=$
$\displaystyle\log\aleph_{A\cap\widetilde{X}}\\!\\!\left(\mathcal{C}^{n}\vee\bigvee_{j=n}^{n+k-1}f^{-j}(\mathcal{C}\setminus\Delta)\right)$
(8) $\displaystyle\leq$
$\displaystyle\log\aleph_{A\cap\widetilde{X}}(\mathcal{C}^{n})+\log\aleph_{A\cap\widetilde{X}}\\!\\!\left(\bigvee_{j=0}^{k-1}f^{-j}\big{(}\mathcal{C}\setminus\Delta\big{)}\\!\right)\\!.$

This implies that $a_{n+k}\leq a_{n}+a_{k}$, concluding the proof. ∎

Note that Lemma 4.1 guarantees that the limit

$\lim_{n\to\infty}\frac{\log\aleph_{A\cap\widetilde{X}}(\mathcal{C}^{n})}{n}$

exists. Now, we proceed to define the concept of topological entropy for pc-
maps with respect to an open cover.

######  Definition 4.1.

Let $A$ be a compact (pseudo-)invariant subset of $X$ such that
$A\cap\widetilde{X}\neq\emptyset$ and let $\mathcal{C}$ be a cover of
$A\cap\widetilde{X}$ satisfying
$\aleph_{A\cap\widetilde{X}}(\mathcal{C})<\infty$. From Remark 3.5, we deduce
that $\aleph_{\widetilde{X}}(\mathcal{C}^{n})<\infty$ for every $n\geq 1$.
Thus, we define the topological entropy of $f|_{A}$ with respect to
$\mathcal{C}$ as

$h_{top}(f|_{A},\mathcal{C}):=\lim_{n\to\infty}\frac{\log\aleph_{A\cap\widetilde{X}}(\mathcal{C}^{n})}{n}\geq
0.$ (9)

In particular, note that $\aleph_{A\cap\widetilde{X}}(\mathcal{C})<\infty$
when $\mathcal{C}$ is an $A$-open cover.

######  Lemma 4.2.

Let $A$ be a compact (pseudo-)invariant subset of $X$ such that
$A\cap\widetilde{X}\neq\emptyset$.

  1. (a)

If $\mathcal{C}$, $\mathcal{D}$ are $A$-open covers such that
$\mathcal{C}\preccurlyeq\mathcal{D}$, then $h_{top}(f|_{A},\mathcal{C})\leq
h_{top}(f|_{A},\mathcal{D})$.

  2. (b)

Let $B$ be a compact (pseudo-)invariant subset of $X$ such that $A\subset B$,
and let $\mathcal{C}$ be a $B$-open cover. Then,
$h_{top}(f|_{A},\mathcal{C}\cap A)\leq h_{top}(f|_{B},\mathcal{C})$.

###### Proof.

(a) This is a straightforward consequence of part (a) of Lemma 3.3.
(b) If $A\subset B$, then every $B$-open cover $\mathcal{C}$ defines an
$A$-open cover $\mathcal{C}\cap A$. Therefore, it follows that
$\aleph_{A\cap\widetilde{X}}\big{(}(\mathcal{C}\cap
A)^{n}\big{)}\leq\aleph_{B\cap\widetilde{X}}(\mathcal{C}^{n})$ for all $n\geq
1$, which concludes the proof. ∎

Before proceeding to give a formal definition of entropy note that, by Remark
3.4, if $\mathcal{C}$ is an $X$-open cover of $A$, then

$\aleph_{A\cap\widetilde{X}}(\mathcal{C}^{n})=\aleph_{A\cap\widetilde{X}}\big{(}(\mathcal{C}\cap
A)^{n}\big{)}\qquad\forall\,n\geq 1.$

######  Definition 4.2.

Let $A$ be a compact (pseudo-)invariant subset of $X$ such that
$A\cap\widetilde{X}\neq\emptyset$. We define the topological entropy of
$f|_{A}$ as

$\displaystyle h_{top}(f|_{A}):=\sup\\!\big{\\{}h_{top}(f|_{A},\mathcal{C})\
:\ \mathcal{C}\text{ is an $A$-open cover}\big{\\}}\geq 0.$

Note that, due to the compactness of $A$, one can take the supremum over all
finite $A$-open covers in the previous definition. This follows from part (a)
of Lemma 4.2.

######  Remark 4.3.

While $f$ is defined over the entire interval $X$, the definition of
$h_{top}(f|_{A})$ does not depend on how $f$ is defined over $A\cap\Delta$.
That is, if $f$ and $g$ are pc-maps such that $\Delta_{f}=\Delta_{g}$ and
$f=g$ on $X\setminus\Delta_{f}$, then $h_{top}(f|_{A})=h_{top}(g|_{A})$ for
every compact (pseudo-)invariant subset $A\subset X$.

##  5 Properties of the topological entropy

In this section, we prove each part of Theorem 2.3. First, note that its part
(1) follows from the definition of entropy (it suffices to take
$\Delta=\emptyset$).

######  Lemma 5.1 ([Part (2) of Theorem 2.3]).

If $A\subset X$ is a compact and (pseudo-) invariant subset of $X$ such that
$A\cap\widetilde{X}\neq\emptyset$, then $h_{top}(f|_{A})\leq h_{top}(f)$.

###### Proof.

Let $\mathcal{C}$ be an $X$-open cover. Then, by part (b) of Lemma 4.2 we know
that $h_{top}(f|_{A},\mathcal{C}\cap A)\leq h_{top}(f,\mathcal{C})$, which
implies that $h_{top}(f|_{A})\leq h_{top}(f)$. ∎

######  Lemma 5.2 ([Part (3) of Theorem 2.3]).

Let $f:X\to X$ and $g:Y\to Y$ be pc-maps defined on compact intervals $X$ and
$Y$, respectively. If $f$ is topologically semi-conjugate (respectively,
conjugate) to $g$, then $h_{top}(f)\geq h_{top}(g)$ (respectively,
$h_{top}(f)=h_{top}(g)$).

###### Proof.

Let $n\geq 1$ be an integer and assume that $f$ is topologically semi-
conjugate to $g$. Also, let $\varphi:X\to Y$ be a continuous surjection such
that $\Delta_{f}=\varphi^{-1}(\Delta_{g})$ and $\varphi\circ f=g\circ\varphi$
on $X\setminus\Delta_{f}$. If $\mathcal{B}$ is a $Y$-open cover, we have that
$\varphi^{-1}(\mathcal{B})$ is an $X$-open cover. Moreover, for every
collection $B_{0},\ldots,B_{n-1}\in\mathcal{B}$, we have that

$\bigcap_{j=0}^{n-1}f^{-j}\\!\left(\varphi^{-1}(B_{j})\setminus\Delta_{f}\right)\subset\varphi^{-1}\\!\\!\left(\bigcap_{j=0}^{n-1}g^{-j}(B_{j}\setminus\Delta_{g})\\!\right).$

This implies that
$\varphi^{-1}(\mathcal{B}^{n}_{g})\preccurlyeq\left(\varphi^{-1}(\mathcal{B})\right)^{n}_{f}$,
where both are $X$-open covers of $\widetilde{X}$ having a sub-collection that
is a finite $X$-open cover of $\widetilde{X}$. Then, by part (a) of Lemma 3.3
we deduce that

$\aleph_{\widetilde{X}}\\!\left(\varphi^{-1}(\mathcal{B}^{n}_{g})\right)\leq\aleph_{\widetilde{X}}\\!\left(\left(\varphi^{-1}(\mathcal{B})\right)^{n}_{f}\right)\\!.$
(10)

Next, as $\varphi$ is a continuous surjection, we have that a $Y$-open sub-
collection of $\mathcal{B}_{g}^{n}$ covers $\widetilde{Y}:=\widetilde{Y}_{g}$
if and only if its preimage with respect to $\varphi$ is an $X$-open cover of
$\widetilde{X}$. Thus,

$\aleph_{\widetilde{Y}}(\mathcal{B}^{n}_{g})=\aleph_{\widetilde{X}}\\!\left(\varphi^{-1}(\mathcal{B}^{n}_{g})\right).$

Next, since $n\geq 1$ is arbitrary, by (10) we have that

$\aleph_{\widetilde{Y}}(\mathcal{B}^{n}_{g})\leq\aleph_{\widetilde{X}}\\!\left(\left(\varphi^{-1}(\mathcal{B})\right)^{n}_{f}\right)\quad\forall\,n\geq
1,$

which implies $h_{top}(g,\mathcal{B})\leq
h_{top}\\!\left(f,\varphi^{-1}(\mathcal{B})\right)$. Finally, as $\mathcal{B}$
is an arbitrary $Y$-open cover, we conclude that

$\displaystyle
h_{top}(g)\leq\sup\\!\left\\{h_{top}\\!\left(f,\varphi^{-1}(\mathcal{B})\right)\
:\ \text{$\mathcal{B}$ is a $Y$-open cover}\right\\}\leq h_{top}(f).$

Now, if in addition $f$ and $g$ are topologically conjugate, we also have that
$g$ is topologically semi-conjugate to $f$. Therefore, the inequality
$h_{top}(f)\leq h_{top}(g)$ also holds. ∎

######  Lemma 5.3 ([Part (4) of Theorem 2.3]).

$h_{top}(f^{k})=k\,h_{top}(f)$ for every integer $k\geq 0$.

###### Proof.

First, if $k=0$, the result follows as $h_{top}(\operatorname{id})=0$. Now,
fix $k\geq 1$. Let $\mathcal{C}$ be an $X$-open cover. Therefore, for all
$n\geq 1$,

$\displaystyle\mathcal{C}_{f^{k}}^{n}$
$\displaystyle\preccurlyeq\mathcal{C}_{f^{k}}^{n}\vee\left[\bigvee_{\begin{subarray}{c}\scriptscriptstyle
1\leq j\leq nk-1\\\ \scriptscriptstyle
k\,\nmid\,j\end{subarray}}\\!f^{-j}(\mathcal{C}\setminus\Delta_{f})\right]=\mathcal{C}_{f}^{nk}$

where $k\nmid j$ means that $k$ does not divide $j$. Now, from part (a) of
Lemma 3.3 we deduce that

$\displaystyle\frac{1}{k}\lim_{n\to\infty}\frac{\log\aleph_{\widetilde{X}}\\!\big{(}\mathcal{C}_{f^{k}}^{n}\big{)}}{n}\leq\lim_{n\to\infty}\frac{\log\aleph_{\widetilde{X}}\\!\\!\,\big{(}\mathcal{C}_{f}^{nk}\big{)}}{nk}$
$\displaystyle=$ $\displaystyle h_{top}(f,\mathcal{C})\leq h_{top}(f).$

Thus, we conclude that

$\frac{h_{top}(f^{k},\mathcal{C})}{k}\leq h_{top}(f).$

Thus, as $\mathcal{C}$ is an arbitrary $X$-open cover, then
$h_{top}(f^{k})\leq k\,h_{top}(f)$.

Reciprocally, if $k\geq 1$ and $\mathcal{C}$ is an $X$-open cover, from Lemma
3.6 and Remark 3.5 we deduce that

$\mathcal{B}:=\bigvee_{i=0}^{k-1}f^{-i}(\mathcal{C}\setminus\Delta_{f})=\left(\bigvee_{i=0}^{k-1}f^{-i}(\mathcal{C})\right)\setminus\Delta^{k}_{f}$

is an $X$-open cover of $\widetilde{X}$ satisfying
$\aleph_{\widetilde{X}}(\mathcal{B})<\infty$. Note that the only points in $X$
that $\mathcal{B}$ does not cover are those given by $\Delta_{f}^{k}$. To
construct an $X$-open cover, it suffices to add $\mathcal{B}$ a finite number
of open intervals:

$\mathcal{B}(\epsilon):=\mathcal{B}\cup\big{\\{}(y-\epsilon,y+\epsilon)\cap X\
:\ y\in\Delta^{k}_{f}\big{\\}}\qquad\forall\,\epsilon>0.$

From the compactness of $X$,
$\aleph_{\widetilde{X}}\big{(}\mathcal{B}(\epsilon)\big{)}<\infty$ for every
$\epsilon\geq 0$ (where $\mathcal{B}(0):=\mathcal{B}$). Also, note that
$\mathcal{B}(\epsilon)\preccurlyeq\mathcal{B}(\eta)$ whenever
$\epsilon\geq\eta$. Then, from Lemma 2.1 and part (a) of Lemma 4.2, we have

$\displaystyle
h_{top}(f^{k})\geq\sup_{\epsilon>0}\big{\\{}h_{top}\big{(}f^{k},\mathcal{B}(\epsilon)\big{)}\big{\\}}$
$\displaystyle=h_{top}(f^{k},\mathcal{B})=\lim_{n\to\infty}\frac{\log\aleph_{\widetilde{X}}\big{(}\mathcal{B}^{n}_{f^{k}}\big{)}}{n}$
$\displaystyle\qquad=\lim_{n\to\infty}\frac{1}{n}\log\aleph_{\widetilde{X}}\\!\\!\left(\bigvee\limits_{j=0}^{n-1}f^{-jk}\big{(}\mathcal{B}\setminus\Delta_{f^{k}}\big{)}\right)$
$\displaystyle\qquad=\lim_{n\to\infty}\frac{1}{n}\log\aleph_{\widetilde{X}}\\!\\!\left(\left[\bigvee\limits_{j=0}^{n-1}f^{-jk}(\mathcal{B})\right]\setminus\big{(}\Delta_{f^{k}}\big{)}^{\\!n}_{\\!}\right)\\!\\!.$
(11)

Now, it is not difficult to see that:

$\bigvee_{j=0}^{n-1}f^{-jk}(\mathcal{B})=\left(\bigvee_{\ell=0}^{nk-1}f^{-\ell}(\mathcal{C})\right)\setminus\Delta_{f}^{nk}\qquad\text{and}\qquad\big{(}\Delta_{f^{k}}\big{)}^{\\!n}_{\\!}=\Delta_{f}^{nk}.$
(12)

Thus, from (11) and (12), we deduce that

$h_{top}(f^{k})\geq
k\cdot\lim_{n\to\infty}\frac{1}{nk}\log\aleph_{\widetilde{X}}\\!\\!\left(\left[\bigvee_{\ell=0}^{nk-1}f^{-\ell}(\mathcal{C})\right]\setminus\Delta_{f}^{nk}\right)=k\cdot
h_{top}(f),$

concluding the proof. ∎

To prove the following main result, it will be necessary to make use of the
following elementary lemma.

######  Lemma 5.4.

Let $p\geq 1$. Suppose that for every $i\in\\{1,\ldots,p\\}$,
$\\{a_{i,n}\\}_{n\geq 1}$ is a sequence of real numbers greater than or equal
to 1 such that

$\lim_{n\to\infty}\frac{\log a_{i,n}}{n}$

exists and is equal to $a_{i}\in\mathbb{R}$. Then,

$\lim_{n\to\infty}\frac{1}{n}\log\\!\left(\sum_{i=1}^{p}a_{i,n}\right)=\max_{1\leq
i\leq p}\\{a_{i}\\}.$

###### Proof.

The proof is based on induction and a basic lemma that is found just before
Theorem 4 in [1]. ∎

######  Lemma 5.5 ([Part (5) of Theorem 2.3]).

Let $p\geq 1$. Suppose that $X=A_{1}\cup\ldots\cup A_{p}$, where $A_{i}$ is a
compact (pseudo-)invariant subset of $X$ such that
$A_{i}\cap\widetilde{X}\neq\emptyset$ for every $1\leq i\leq p$. Then,

$h_{top}(f)=\max_{1\leq i\leq p}\\!\big{\\{}h_{top}(f|_{A_{i}})\big{\\}}.$

###### Proof.

From Lemma 5.1 we deduce that

$\max_{1\leq i\leq p}\\!\big{\\{}h_{top}(f|_{A_{i}})\big{\\}}\leq h_{top}(f).$

Now, if $\mathcal{C}$ is an $X$-open cover, we have that

$\log\aleph_{\widetilde{X}}(\mathcal{C}^{n})\leq\log\\!\left(\sum_{i=1}^{p}\aleph_{A_{i}\cap\widetilde{X}}\big{(}(\mathcal{C}\cap
A_{i})^{n}\big{)}\\!\right)\qquad\forall\,n\geq 1.$

Therefore, from Lemma 5.4 we deduce that

$h_{top}(f,\mathcal{C})\leq\max_{1\leq i\leq
p}\\!\big{\\{}h_{top}(f|_{A_{i}},\mathcal{C}\cap A_{i})\big{\\}}.$

Thus, it is enough to take supremum over all the $X$-open covers to obtain the
desired result. ∎

To finish the proof of Theorem 2.3, we will use the following lemma from
topology:

######  Lemma 5.6 ([Lebesgue’s number, [19]]).

Let $A\subset X$ be a compact set, and let $\mathcal{C}$ be an arbitrary
$X$-open cover of $A$. Then, there exists $\delta>0$ (Lebesgue’s number) such
that for every $x\in A$, there exists $C\in\mathcal{C}$ such that
$B(x;\delta)\subset C$.

With this, we are ready to prove the last part of Theorem 2.3.

######  Lemma 5.7 ([Part (6) of Theorem 2.3]).

Suppose that $\\{\mathcal{C}_{k}\\}_{k\geq 1}$ is a sequence of $X$-open
covers such that $\operatorname{diam}(\mathcal{C}_{k})\to 0$ as $k\to\infty$.
Therefore,

  1. (a)

If $h_{top}(f)<\infty$, then $\lim_{k\to\infty}h_{top}(f,\mathcal{C}_{k})$
exists and is equal to $h_{top}(f)$.

  2. (b)

If $h_{top}(f)=\infty$, then
$\lim_{k\to\infty}h_{top}(f,\mathcal{C}_{k})=\infty$.

###### Proof.

(a) Assume that $h_{top}(f)<\infty$. Given $\epsilon>0$, choose an $X$-open
cover $\mathcal{C}$ such that

$h_{top}(f,\mathcal{C})>h_{top}(f)-\epsilon.$ (13)

Let $\delta$ be a Lebesgue number for $\mathcal{C}$. Now, choose $k^{*}\geq 1$
such that $\operatorname{diam}(\mathcal{C}_{k})<\delta$ for every $k\geq
k^{*}$. Thus, for any $D\in\mathcal{C}_{k}$, with $k\geq k^{*}$, there exists
$C\in\mathcal{C}$ such that $D\subset C$, which implies that
$\mathcal{C}\preccurlyeq\mathcal{C}_{k}$ for every $k\geq k^{*}$. From (13)
and part (a) of Lemma 4.2 we deduce that

$h_{top}(f)\geq h_{top}(f,\mathcal{C}_{k})>h_{top}(f)-\epsilon.$

As $\epsilon$ was arbitrary, this concludes the proof of part (a).
(b) If $h_{top}(f)=\infty$, let $\rho>0$ and choose an $X$-open cover
$\mathcal{C}$ such that $h_{top}(f,\mathcal{C})>\rho$. Following the same
approach as in the proof of part (a), it can be shown that
$\lim_{k\to\infty}h_{top}(f,\mathcal{C}_{k})=\infty$. ∎

##  6 Using Bowen’s formula to compute the topological entropy

Let us recall that P. Walters provided an example in which the value of the
topological entropy calculated using Bowen’s formula depends on the metric
used. In particular, the value of the entropy can be different for metrics
that are even equivalent (see [24, page 171]). Walters’ example results from
the non-compactness of the space where the system is defined.

This section is devoted to proving Theorem 2.4, which establishes equality (2)
and that our concept of topological entropy does not depend on the choice of
the metric (as long as it generates the induced topology on the respective
invariant space where the system is defined). In short, we relate our concept
of topological entropy with Bowen’s formula for the map restricted to
$\widetilde{X}$.

We begin this section by stating the so-called dynamical distance or Bowen
metric on $X$. For all $n\geq 1$, we consider the metric on $X$ defined by the
map $\rho_{n}:X\times X\to[0,\infty)$ such that

$\rho_{n}(x,y):=\max_{0\leq j\leq
n-1}\big{\\{}\operatorname{dist}\big{(}f^{j}(x),f^{j}(y)\big{)}\big{\\}}\qquad\forall\,(x,y)\in
X\times X,$

where “$\operatorname{dist}$” is the Euclidean metric. We highlight that it is
possible to consider any other metrics equivalent to $\operatorname{dist}$, as
it will be proved later (see the comment following the proof of Theorem 6.6).

We denote the open ball in $(X,\rho_{n})$ (respectively,
$(X,\operatorname{dist})$) of center $z\in X$ and radius $\eta>0$ by
$B^{\rho_{n}}(z,\eta)$ (respectively, $B(z,\eta)$). The set
$B^{\rho_{n}}(z,\eta)$ is commonly referred to as a dynamical ball. As usual,
note that $B^{\rho_{n}}(z,\eta)$ depends on how $f$ is defined over $\Delta$.
To relate the concepts in this Section with our definition of entropy, we
consider a modified ball as follows.

######  Definition 6.1.

Consider $n\geq 1$, $\epsilon>0$, and let $A$ be a (pseudo-)invariant subset
of $X$ such that $A\cap\widetilde{X}\neq\emptyset$. We define the modified
dynamical ball of center $z\in X$ and radius $\eta>0$ as

$Q_{A}^{n}(z;\eta):=B^{\rho_{n}}(z;\eta)\cap\bigcap_{j=0}^{n-1}f^{-j}(A\setminus\Delta).$
(14)

Observe that the compactness of $A$ is not required for this definition nor
most of the results below. Next, we show that the set defined by (14) is an
open set of $A$.

######  Lemma 6.1.

Consider $n\geq 1$, $\epsilon>0$, and a (pseudo-)invariant subset $A\subset X$
such that $A\cap\widetilde{X}\neq\emptyset$. Then, for every $x\in X$,
$Q_{A}^{n}(x;\epsilon)$ is a relative open set of $A$.

###### Proof.

Fix $n\geq 1$, $\epsilon>0$ and $x\in X$. Then, $y\in Q_{A}^{n}(x;\epsilon)$
if and only if

$f^{j}(y)\in
A\setminus\Delta\quad\land\quad\operatorname{dist}\big{(}f^{j}(x),f^{j}(y)\big{)}<\epsilon\qquad\forall\,j\in\\{0,\ldots,n-1\\},$

which happens if and only if
$f^{j}(y)\in\big{(}B\big{(}f^{j}(x);\epsilon\big{)}\cap
A\big{)}\setminus\Delta$ for all $j\in\\{0,\ldots,n-1\\}$. Thus, we deduce

$Q_{A}^{n}(x;\epsilon)=\bigcap_{j=0}^{n-1}f^{-j}\big{(}\big{(}B(f^{j}(x);\epsilon)\cap
A\big{)}\setminus\Delta\big{)}.$ (15)

Now, observe that,

$\displaystyle\bigcap_{j=0}^{n-1}f^{-j}(A\setminus\Delta)=\bigcap_{j=0}^{n-1}\big{(}f^{-j}(A)\setminus
f^{-j}(\Delta)\big{)}\subset A\setminus\Delta^{n}.$

Moreover, by using this together with (5), we deduce

$\bigcap_{j=0}^{n-1}f^{-j}(A\setminus\Delta)=A\setminus\Delta^{n}.$ (16)

Therefore, from (15) and (16) we have

$Q_{A}^{n}(x;\epsilon)=\left[A\cap\bigcap_{j=0}^{n-1}f^{-j}\big{(}B(f^{j}(x);\epsilon)\big{)}\right]\setminus\Delta^{n}.$
(17)

Last, as $\Delta^{n}$ is a finite set, we conclude that
$Q_{A}^{n}(x;\epsilon)$ is a relative open set of $A$. ∎

Now, notice that

$\displaystyle\operatorname{dist}(x,y)\leq\rho_{n}(x,y)\leq\sum_{j=0}^{n-1}\operatorname{dist}\big{(}f^{j}(x),f^{j}(y)\big{)}\qquad\forall\,x,y\in
X,$ (18)

which implies that any Cauchy sequence with respect to $\rho_{n}$ is also a
Cauchy sequence with respect to $\operatorname{dist}$. Furthermore, if
$\\{f^{j}(x_{i})\\}_{i\geq 0}$ is a Cauchy sequence with respect to
$\operatorname{dist}$ for all $j\in\\{0,\ldots,n-1\\}$, then the sequence
$\\{x_{i}\\}_{i\geq 0}$ is a Cauchy sequence with respect to $\rho_{n}$. Now,
we prove that the subspace $\big{(}A\cap\widetilde{X},\rho_{n}\big{)}$ is
totally bounded, which is equivalent to the following result.

######  Lemma 6.2.

Let $A\subset X$ be a (pseudo-)invariant subset such that
$A\cap\widetilde{X}\neq\emptyset$. Then, for all $n\geq 1$, every sequence in
$\big{(}A\cap\widetilde{X},\rho_{n}\big{)}$ has a Cauchy subsequence.

###### Proof.

Consider a sequence $\\{x_{k}\\}_{k}\subset A\cap\widetilde{X}$, where the
subscripts $k$ are non-negative integers. Clearly,
$\\{x_{k}\\}_{k}\subset\overline{A}\subset X$, which implies that there exists
a $\operatorname{dist}$-convergent subsequence
$\\{x_{k}^{0}\\}_{k}\subset\\{x_{k}\\}_{k}$ within $\overline{A}$. Since
$x_{k}^{0}\in A\cap\widetilde{X}$ for every $k\geq 0$, we conclude that
$\\{x_{k}^{0}\\}_{k}$ forms a $\operatorname{dist}$-Cauchy sequence.
Similarly, as $\big{\\{}f(x^{0}_{k})\big{\\}}_{k}\subset A\cap\widetilde{X}$,
there exists a $\operatorname{dist}$-Cauchy subsequence
$\\{x_{k}^{1}\\}_{k}\subset\\{x^{0}_{k}\\}_{k}$ such that
$\big{\\{}f(x^{1}_{k})\big{\\}}_{k}$ is a $\operatorname{dist}$-Cauchy
subsequence of $\big{\\{}f(x^{0}_{k})\big{\\}}_{k}$. By repeating this process
iteratively, we obtain a $\operatorname{dist}$-subsequence
$\\{x_{k}^{j}\\}_{k}$ of $\\{x_{k}\\}_{k}$ such that
$\big{\\{}f^{j}(x_{k}^{j})\big{\\}}_{k}\subset A\cap\widetilde{X}$ forms a
Cauchy sequence with respect to $\operatorname{dist}$ for all
$j\in\\{0,\ldots,n-1\\}$. Therefore, by (18), we have that
$\\{x_{k}^{j}\\}_{k}\subset A\cap\widetilde{X}$ is a Cauchy sub-sequence of
$\\{x_{k}\\}_{k}$ with respect to $\rho_{n}$. ∎

Next, we recall the classical concepts used to define Bowen’s formula. Let
$A\subset X$ be a (pseudo-)invariant subset such that
$A\cap\widetilde{X}\neq\emptyset$. The set $E\subset A\cap\widetilde{X}$ is
said to be an $(n,\epsilon)$-separated set of $A\cap\widetilde{X}$ if for all
$x,y\in E$ with $x\neq y$, there exists $j\in\\{0,\ldots,n-1\\}$ such that
$\operatorname{dist}\big{(}f^{j}(x),f^{j}(y)\big{)}\geq\epsilon$. We denote
the maximal cardinality of $(n,\epsilon)$-separated sets of
$A\cap\widetilde{X}$ by $s_{n}(A;f,\epsilon)$, where

$s_{n}(A;f,\epsilon)=\max\\!\big{\\{}\\#E\ :\ \text{$E$ is an
$(n,\epsilon)$-separated set of $A\cap\widetilde{X}$}\big{\\}}.$ (19)

Now, we prove that this quantity is well-defined.

######  Lemma 6.3.

Let $A$ be a (pseudo-)invariant subset of $X$ such that
$A\cap\widetilde{X}\neq\emptyset$. Then, for every $n\geq 1$ and $\epsilon>0$,
$s_{n}(A;f,\epsilon)<\infty$.

###### Proof.

Let $n\geq 1$ and $\epsilon>0$. We begin the proof assuming, by contradiction,
that $s_{n}(f,\epsilon)=\infty$. This implies the existence of an
$(n,\epsilon)$-separated set $E\subset A\cap\widetilde{X}$ such that
$\\#E=\infty$. Thus, there exists an infinite sequence $\\{x_{k}\\}_{k\geq
0}\subset E$ where each element is different from one another (i.e.,
$x_{i}\neq x_{j}$ for $i\neq j$). Since $\\{x_{k}\\}_{k\geq 0}\subset
A\cap\widetilde{X}$, we can use Lemma 6.2 to conclude that there exists a
$\rho_{n}$-Cauchy subsequence $\\{x_{k_{j}}\\}_{j\geq 0}\subset E$.
Consequently, there exists an integer $m\geq 1$ such that

$\rho_{n}(x_{k_{i}},x_{k_{j}})<\epsilon\qquad\forall\,i,j\geq m.$

However, this contradicts the fact that $E$ is an $(n,\epsilon)$-separated set
of $A\cap\widetilde{X}$. Therefore, we conclude that
$s_{n}(A;f,\epsilon)<\infty$. ∎

In a dual way, let $A\subset X$ be a (pseudo-)invariant subset such that
$A\cap\widetilde{X}\neq\emptyset$, and let $E\subset A\cap\widetilde{X}$. The
set $E$ is called $(n,\epsilon)$-spanning set of $A\cap\widetilde{X}$ if for
all $x\in A\cap\widetilde{X}$, there exists $y\in E$ such that
$\operatorname{dist}\big{(}f^{j}(x),f^{j}(y)\big{)}<\epsilon$ for all
$j\in\\{0,\ldots,n-1\\}$, which is equivalent to

$A\cap\widetilde{X}\subset\bigcup_{y\in E}Q_{A}^{n}(y;\epsilon).$

Now, we prove that the collection of finite spanning sets is non-empty.

######  Lemma 6.4.

Let $A\subset X$ be a (pseudo-)invariant subset such that
$A\cap\widetilde{X}\neq\emptyset$. Then, for every $n\geq 1$ and $\epsilon>0$,
there exists a finite set $E\subset A\cap\widetilde{X}$ that is an
$(n,\epsilon)$-spanning set of $A\cap\widetilde{X}$.

###### Proof.

Let $n\geq 1$ and $\epsilon>0$. Since $\Delta^{n}$ is finite, then there exist
open intervals $I_{1},\ldots,I_{m}\subset X$ such that $I_{j}\cap
A\cap\widetilde{X}\neq\emptyset$ for all $j\in\\{1,\ldots,m\\}$, and

$A\cap\widetilde{X}=(A\cap\widetilde{X})\setminus\Delta^{n}=\bigcup_{j=1}^{m}I_{j}\cap
A\cap\widetilde{X}.$

It is clear that, if $x,y\in I_{i}\cap A\cap\widetilde{X}$ for some
$i\in\\{1,\ldots,m\\}$, then $f^{k}(x)$ and $f^{k}(y)$ belong to the same
continuity piece for each $k\in\\{0,\ldots,n-1\\}$. Now, note that for every
$j\in\\{1,\ldots,m\\}$, the collection $\big{\\{}B(x,\epsilon)\ :\ x\in
I_{j}\cap A\cap\widetilde{X}\big{\\}}$ is an open cover of
$\overline{I_{j}\cap A\cap\widetilde{X}}\subset X$. Therefore, there exist
$x_{0}^{j},x_{1}^{j},\ldots,x_{p_{j}}^{j}\in I_{j}\cap A\cap\widetilde{X}$
such that

$I_{j}\cap A\cap\widetilde{X}\subset\overline{I_{j}\cap
A\cap\widetilde{X}}\subset B(x_{0}^{j},\epsilon)\cup\ldots\cup
B(x_{p_{j}}^{j},\epsilon).$

Thus, we have that the finite set
$E:=\bigcup_{j=1}^{m}\\{x_{0}^{j},x_{1}^{j},\ldots,x_{p_{j}}^{j}\\}$ is an
$(n,\epsilon)$-spanning set of $A\cap\widetilde{X}$, concluding the proof. ∎

Observe that, under the same assumptions as in Lemma 6.4, it makes sense to
define the minimal cardinality of $(n,\epsilon)$-spanning sets of
$A\cap\widetilde{X}$, by

$r_{n}(A;f,\epsilon):=\min\\{\\#E\ :\ \text{$E$ is an $(n,\epsilon)$-spanning
set of $A\cap\widetilde{X}$}\\}<\infty.$

Now, we prove some properties of comparison and monotonicity of
$r_{n}(A;f,\epsilon)$ and $s_{n}(A;f,\epsilon)$.

######  Lemma 6.5.

Consider $n\geq 1$, $\epsilon>0$ and let $A\subset X$ be a (pseudo-)invariant
subset such that $A\cap\widetilde{X}\neq\emptyset$. Then, the following
statements hold:

  1. (i)

If $0<\epsilon^{\prime}<\epsilon$, then $r_{n}(A;f,\epsilon)\leq
r_{n}(A;f,\epsilon^{\prime})$ and $s_{n}(A;f,\epsilon)\leq
s_{n}(A;f,\epsilon^{\prime})$.

  2. (ii)

$r_{n}(A;f,\epsilon)\leq s_{n}(A;f,\epsilon)\leq r_{n}(A;f,\epsilon/2)$.

###### Proof.

(i) Assume that $0<\epsilon^{\prime}<\epsilon$. Let $E^{\prime}\subset
A\cap\widetilde{X}$ be an $(n,\epsilon^{\prime})$-spanning set of
$A\cap\widetilde{X}$ such that $\\#E^{\prime}=r_{n}(A;f,\epsilon^{\prime})$.
We have that

$A\cap\widetilde{X}\subset\bigcup_{x\in
E^{\prime}}Q_{A}^{n}(x;\epsilon^{\prime})\subset\bigcup_{x\in
E^{\prime}}Q_{A}^{n}(x;\epsilon).$

Thus, we deduce that $E^{\prime}$ is an $(n,\epsilon)$-spanning set and,
therefore, there exists an $(n,\epsilon)$-spanning set $E\subset E^{\prime}$
such that $\\#E=r_{n}(A;f,\epsilon)$. We conclude that
$r_{n}(A;f,\epsilon)\leq r_{n}(A;f,\epsilon^{\prime})$.

Now, suppose that $E\subset A\cap\widetilde{X}$ is an $(n,\epsilon)$-separated
set of $A\cap\widetilde{X}$ such that $\\#E=s_{n}(A;f,\epsilon)$. If $x,y\in
E$ are different points, then there exists $j\in\\{0,\ldots,n-1\\}$ such that

$\operatorname{dist}\big{(}f^{j}(x),f^{j}(y)\big{)}\geq\epsilon>\epsilon^{\prime}.$

Thus, we deduce that $E$ is an $(n,\epsilon^{\prime})$-separated set of
$A\cap\widetilde{X}$ and, therefore, there exists $E^{\prime}\subset
A\cap\widetilde{X}$, an $(n,\epsilon^{\prime})$-separated set of
$A\cap\widetilde{X}$ such that $E\subset E^{\prime}$ and
$\\#E^{\prime}=s_{n}(A;f,\epsilon^{\prime})$. We conclude that
$s_{n}(A;f,\epsilon)\leq s_{n}(A;f,\epsilon^{\prime})$.
(ii) Let $E\subset A\cap\widetilde{X}$ be an $(n,\epsilon)$-separated set of
$A\cap\widetilde{X}$ such that $\\#E=s_{n}(A;f,\epsilon)$. Then, for all
$y\in(A\cap\widetilde{X})\setminus E$ we have that $E\cup\\{y\\}$ is not an
$(n,\epsilon)$-separated set of $A\cap\widetilde{X}$. Therefore, there exists
$x_{y}\in E$ such that

$\operatorname{dist}\big{(}f^{j}(x_{y}),f^{j}(y)\big{)}<\epsilon\qquad\forall\,j\in\\{0,\ldots,n-1\\}.$

We deduce that $y\in Q_{A}^{n}(x_{y};\epsilon)$. Since
$y\in(A\cap\widetilde{X})\setminus E$ is arbitrary, we deduce that

$A\cap\widetilde{X}\subset\bigcup_{x\in E}Q_{A}^{n}(x;\epsilon),$

which implies that $E$ is an $(n,\epsilon)$-spanning set of
$A\cap\widetilde{X}$. Thus, we have that

$r_{n}(A;f,\epsilon)\leq\\#E=s_{n}(A;f,\epsilon).$

To prove the second inequality, consider $E\subset A\cap\widetilde{X}$, an
$(n,\epsilon)$-separated set of $A\cap\widetilde{X}$, and $F\subset
A\cap\widetilde{X}$, an $(n,\epsilon/2)$-spanning set of $A\cap\widetilde{X}$.
Then, we have

$E\subset A\cap\widetilde{X}\subset\bigcup_{y\in F}Q_{A}^{n}(y;\epsilon/2).$

Now, define the map $\phi:E\to F$ as

$\phi(x):=\min\\!\big{\\{}y\in F\ :\ x\in
Q_{A}^{n}(y;\epsilon/2)\big{\\}}\qquad\forall\,x\in E.$

We claim that $\phi$ is an injective map. Indeed, suppose that $x,z\in E$
satisfy $\phi(x)=y=\phi(z)$. Then,

$\operatorname{dist}\big{(}f^{j}(x),f^{j}(z)\big{)}\leq\operatorname{dist}\big{(}f^{j}(x),f^{j}(y)\big{)}+\operatorname{dist}\big{(}f^{j}(y),f^{j}(z)\big{)}<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon,$

for all $j\in\\{0,\ldots,n-1\\}$. As $E$ is an $(n,\epsilon)$-separated set of
$A\cap\widetilde{X}$, we deduce that $x=z$, which proves the claim. Thus, we
have that $\\#E\leq\\#F$ and, as $E$ and $F$ are arbitrary, we conclude that
$s_{n}(A;f,\epsilon)\leq r_{n}(A;f,\epsilon/2)$. ∎

Here, note that under the same assumptions as in Lemma 6.5, its part (i) lets
us conclude that

$\epsilon\mapsto\limsup_{n\to\infty}\frac{\log
s_{n}(A;f,\epsilon)}{n}\quad\text{ and
}\quad\epsilon\mapsto\limsup_{n\to\infty}\frac{\log r_{n}(A;f,\epsilon)}{n}$

are non-increasing maps (as long as the involved limits are finite). Thus, we
conclude that

$s(A;f):=\lim_{\epsilon\to 0}\limsup_{n\to\infty}\frac{\log
s_{n}(A;f,\epsilon)}{n}\quad\text{ and }\quad r(A;f):=\lim_{\epsilon\to
0}\limsup_{n\to\infty}\frac{\log r_{n}(A;f,\epsilon)}{n}$ (20)

always exist, considering that they could be infinite. Next, we prove that the
topological entropy can be obtained via Bowen’s formula under the assumption
of compactness. The following result is a reformulation of Bowen’s formula
established in Theorem 2.4.

######  Theorem 6.6.

Let $A\subset X$ be a compact (pseudo-)invariant subset such that
$A\cap\widetilde{X}\neq\emptyset$, then $h_{top}(f|_{A})=s(A;f)=r(A;f)$.

###### Proof.

Using part (ii) of Lemma 6.5 and a squeeze argument, we conclude that both
limits in (20) are equal. Thus, it is sufficient to show that $s(A;f)\leq
h_{top}(f|_{A})\leq r(A;f)$ to conclude the desired result.

Consider $n\geq 1$, $\epsilon>0$, and let $E\subset A\cap\widetilde{X}$ be an
$(n,\epsilon)$-separated set of $A\cap\widetilde{X}$ such that
$\\#E=s_{n}(A;f,\epsilon)$. Consider $\mathcal{C}$, an $A$-open cover such
that $\operatorname{diam}(\mathcal{C})<\epsilon$. Since $A$ is compact, we can
assume that $\mathcal{C}$ is finite. Note that if $x,y\in A\cap\widetilde{X}$
belong to the same element of $\mathcal{C}^{n}$, then

$\operatorname{dist}\big{(}f^{j}(x),f^{j}(y)\big{)}<\max_{D\in\,\mathcal{C}}\\{\operatorname{diam}(D)\\}<\epsilon\qquad\forall\,j\in\\{0,\ldots,n-1\\}.$

In particular, $\\#(E\cap C)\leq 1$ for all $C\in\mathcal{C}^{n}$. Thus, we
conclude that

$s_{n}(A;f,\epsilon)=\\#E\leq\aleph_{A\cap\widetilde{X}}(\mathcal{C}^{n})\qquad\forall\,n\geq
1.$

Therefore,

$\limsup_{n\to\infty}\frac{\log
s_{n}(A;f,\epsilon)}{n}\leq\lim_{n\to\infty}\frac{\log\aleph_{A\cap\widetilde{X}}(\mathcal{C}^{n})}{n}=h_{top}(f|_{A},\mathcal{C}\cap
A)\leq h_{top}(f|_{A}),$

which lets us conclude that $s(A;f)\leq h_{top}(f|_{A})$, when taking the
limit as $\epsilon\to 0$.

To prove the second inequality, let $\mathcal{C}$ be an $X$-open cover of $A$
and let $\epsilon>0$ be the Lebesgue’s number associated with $\mathcal{C}$.
Again, since $A$ is compact, we can assume that $\mathcal{C}$ is finite. Also,
let $E\subset A\cap\widetilde{X}$ be an $(n,\epsilon)$-spanning set of
$A\cap\widetilde{X}$ such that $\\#E=r_{n}(A;f,\epsilon)$. Then, for all $x\in
A\cap\widetilde{X}$ and $j\in\\{0,\ldots,n-1\\}$, there exists
$C_{x,j}\in\mathcal{C}$ such that

$\big{[}B\big{(}f^{j}(x),\epsilon\big{)}\cap
A\big{]}\setminus\Delta\subset\big{[}C_{x,j}\cap A\big{]}\setminus\Delta.$

Then, by using (15), we conclude that

$Q_{A}^{n}(x;\epsilon)=\bigcap_{j=0}^{n-1}f^{-j}\big{(}[B(f^{j}(x);\epsilon)\cap
A]\setminus\Delta\big{)}\subset\bigcap_{j=0}^{n-1}f^{-j}([C_{x,j}\cap
A]\setminus\Delta).$

Since $E$ is an $(n,\epsilon)$-spanning set of $A\cap\widetilde{X}$, we deduce
that the collection

$\mathcal{T}:=\left\\{\bigcap_{j=0}^{n-1}f^{-j}([C_{x,j}\cap
A]\setminus\Delta)\in(\mathcal{C}\cap A)^{n}\ :\ x\in E\right\\}$

is a finite $A$-open cover of $A\cap\widetilde{X}$. Thus, from part (b) of
Lemma 3.3, as $\mathcal{T}\subset(\mathcal{C}\cap A)^{n}$ we have that

$\aleph_{A\cap\widetilde{X}}(\mathcal{C}^{n})\leq\\#E=r_{n}(A;f,\epsilon)\qquad\forall\,n\geq
1.$

Therefore,

$h_{top}(f|_{A};\mathcal{C}\cap
A)=\lim_{n\to\infty}\frac{\log\aleph_{A\cap\widetilde{X}}(\mathcal{C}^{n})}{n}\leq\limsup_{n\to\infty}\frac{\log
r_{n}(A;f,\epsilon)}{n}.$

Now, taking the limit as $\epsilon\to 0$, we have that
$h_{top}(f|_{A};\mathcal{C}\cap A)\leq r(A;f)$. Since $\mathcal{C}$ is an
arbitrary $X$-open cover of $A$, we conclude $h_{top}(f|_{A})\leq r(A;f)$.
Thus, $h_{top}(f|_{A})=s(A;f)=r(A;f)$. ∎

Since $h_{top}(f)$ is a purely topological concept, the equality in Theorem
6.6 shows that there is no dependence on the metric used to compute our
entropy. In fact, Note that all the results in this section are valid for any
choice of metric, as long as it is equivalent to $\operatorname{dist}$.

##  7 Proof of Theorems 2.6 and 2.8

This section is devoted to proving Theorems 2.6 and 2.8, together with
Corollary 2.7 (see Corollary 7.11). Without loss of generality, we assume that
$f$ is piecewise strictly monotonic throughout this section; that is, the
restriction of $f$ to each of its continuity pieces is injective (note that,
as $f$ has a finite number of critical points, it is always possible to adapt
the continuity pieces of $f$ so that the map is piecewise strictly monotonic).
Furthermore, to enhance the clarity of the arguments, we assume that the
points of $\Delta$ fall into two categories: jump discontinuities or critical
continuity points of $f$ (that is, we do not allow for removable
discontinuities). The proof of Theorem 2.6 is divided into 2 parts:
Subsections 7.1 and 7.2; whilst the proof of Theorem 2.8 is given in
Subsection 7.3.

We begin by defining a special cover of $\widetilde{X}$, which will be key for
the proofs we provide in this section. We define
$\mathcal{D}:=\\{X_{1},\ldots,X_{N}\\}$, the natural $X$-open cover of
$\widetilde{X}$, which fulfills $\mathcal{D}\cap\Delta=\\{\emptyset\\}$ and,
therefore,

$\mathcal{D}^{n}=\bigvee_{j=0}^{n-1}f^{-j}(\mathcal{D})\qquad\forall\,n\geq
1.$

In addition, for an arbitrary collection $\mathcal{C}$ of $X$, we define the
topological boundary of $\mathcal{C}$, denoted by $\partial\mathcal{C}$, as
the topological boundary (with respect to $X$) of the union of its elements.
In particular, observe that $\partial\mathcal{D}=\Delta$.

###  7.1 Preimages of critical points, monotonicity intervals, and entropy

We state a straightforward result using that, for every
$k\in\\{0,1,\ldots,n\\}$, $f^{k}$ is a continuous and strictly monotonic
function on the connected components of $X\setminus\Delta^{n}$. In particular,
when $k=n$, these components coincide with the continuity pieces of $f^{n}$.
This idea serves as a straightforward proof of the following result.

######  Lemma 7.1.

Let $n\geq 1$. Then, for every open interval $U\subset X$ and connected
component $V$ of $X\setminus\Delta^{n}$, the set $f^{-k}(U)\cap V$ is either
$\emptyset$ or an open interval for each $k\in\\{0,1,\ldots,n\\}$.

######  Remark 7.2.

$\mathcal{D}^{n}$ is a collection of open pairwise disjoint intervals for
every $n\geq 1$. In what follows, we prove that
$\partial(\mathcal{D}^{n})=\Delta^{n}$, which will allow us to deduce that
$\mathcal{D}^{n}$ coincides with the collection of all connected components of
$X\setminus\Delta^{n}$.

######  Lemma 7.3.

For every $n\geq 0$,
$f^{-n}(\Delta)\subset\partial\big{(}f^{-n}(\mathcal{D})\big{)}\subset\Delta^{n+1}$.

###### Proof.

Let $n\geq 0$ and $y\in f^{-n}(\Delta)$. This implies that
$f^{n}(y)\in\Delta$. Also, let $U$ be an arbitrary neighbourhood of $y$. Then,
it is clear that

$\displaystyle
U\cap\left(X\setminus\bigcup_{i=1}^{N}f^{-n}(X_{i})\right)=U\cap
f^{-n}(\Delta)\neq\emptyset.$

Furthermore, from the density of $X\setminus f^{-n}(\Delta)$ in $X$, we deduce
that

$\displaystyle\emptyset\neq U\cap\big{(}X\setminus
f^{-n}(\Delta)\big{)}=U\cap\left(\bigcup_{i=1}^{N}f^{-n}(X_{i})\right)\\!.$

Therefore, we have that $y\in\partial\big{(}f^{-n}(\mathcal{D})\big{)}$, which
implies $f^{-n}(\Delta)\subset\partial\big{(}f^{-n}(\mathcal{D})\big{)}$.

For the second inclusion, let $y\notin\Delta^{n+1}$. Therefore, $y$ belongs to
a connected component of $X\setminus\Delta^{n+1}$, which is an open interval.
Let $\epsilon>0$ be such that $V:=(y-\epsilon,y+\epsilon)$ is completely
contained in such connected component of $X\setminus\Delta^{n+1}$. Therefore,

$\displaystyle
V\cap\left(X\setminus\bigcup_{i=1}^{N}f^{-n}(X_{i})\right)=V\cap
f^{-n}(\Delta)=\emptyset,$

which proves that $y\notin\partial\big{(}f^{-n}(\mathcal{D})\big{)}$. ∎

A consequence of this is the following result:

######  Lemma 7.4.

For every $n\geq 1$, $\partial(\mathcal{D}^{n})=\Delta^{n}$.

###### Proof.

We proceed by induction on $n$. First, if $n=1$, the result follows. Next, let
us assume that $\partial\big{(}\mathcal{D}^{k}\big{)}=\Delta^{k}$ for $k\geq
1$. Thus, by Lemmas 7.1 and 7.3, in addition to the induction hypothesis, we
have that $\mathcal{D}^{k+1}=\mathcal{D}^{k}\vee f^{-k}(\mathcal{D})$.
Therefore,

$\displaystyle\partial\big{(}\mathcal{D}^{k+1}\big{)}=\partial\big{(}\mathcal{D}^{k}\big{)}\cup\partial\big{(}f^{-k}(\mathcal{D})\big{)}=\Delta^{k+1},$

which concludes the proof. ∎

######  Lemma 7.5.

For every $n\geq 1$,
$\aleph_{\widetilde{X}}(\mathcal{D}^{n})=\\#(\mathcal{D}^{n})$.

###### Proof.

This is a straightforward consequence of the fact that $\mathcal{D}^{n}$ is a
pairwise disjoint collection of intervals, for every $n\geq 1$. ∎

We recall that, for every $n\geq 1$, $c_{n}\geq 1$ is the smallest number of
intervals on which $f^{n}$ is monotonic and essentially continuous (see (3)).
Since $f$ is piecewise strictly monotonic and the map does not allow for
removable discontinuities, it follows that $c_{n}$ coincides with the
cardinality of the collection of all connected components of
$X\setminus\Delta^{n}$; that is, $c_{n}=\\#(\mathcal{D}^{n})$. With this, we
prove the first part of Theorem 2.6.

######  Lemma 7.6.

The following inequality holds:

$h_{top}(f)\geq\lim_{n\to\infty}\frac{1}{n}\log c_{n}.$

###### Proof.

For each $\epsilon>0$, denote by $\mathcal{C}_{\epsilon}$ the open cover of
$X$ given by $\mathcal{D}\cup\mathcal{D}_{\epsilon}$, where

$\displaystyle\mathcal{D}_{\epsilon}:=\bigcup_{d\in\Delta}\big{\\{}(d-\epsilon,d+\epsilon)\cap
X\big{\\}}.$

Naturally, it is convenient to denote $\mathcal{C}^{n}_{0}:=\mathcal{D}^{n}$
for every positive integer $n\geq 1$.

Next, if $\mathbb{N}$ denotes the set of positive integers starting at $1$,
let $a:[0,\infty)\times\mathbb{N}\to[0,\infty)$ be the function given by

$a(\epsilon,n):=\frac{\log\aleph_{\widetilde{X}}(\mathcal{C}_{\epsilon}^{n})}{n}\qquad\forall\,(\epsilon,n)\in[0,\infty)\times\mathbb{N}.$

Now, note that for every $\xi>\eta\geq 0$, we have that
$\mathcal{C}_{\xi}\preccurlyeq\mathcal{C}_{\eta}\preccurlyeq\mathcal{C}_{0}=\mathcal{D}$,
which implies that $a(\xi,n)\leq a(\eta,n)\leq a(0,n)$ for all $n\geq 1$. In
particular, this lets us ensure that
$\lim\limits_{n\to\infty}a(\xi,n)\leq\lim\limits_{n\to\infty}a(\eta,n)$. Next,
we define $g:[0,\infty)\to[0,\infty)$ by

$\displaystyle
g(\epsilon):=\lim_{n\to\infty}a(\epsilon,n)\qquad\forall\,\epsilon\geq 0.$

This function is clearly well-defined. Also, $g$ is a decreasing map and,
therefore, $\sup\limits_{\epsilon\geq 0}\\{g(\epsilon)\\}=g(0)$. Thus, we have
that

$\displaystyle h_{top}(f)\geq\sup_{\epsilon\geq
0}\\!\big{\\{}h_{top}(f,\mathcal{C}_{\epsilon})\big{\\}}=g(0)=\lim_{n\to\infty}\frac{\log\aleph_{\widetilde{X}}(\mathcal{D}^{n})}{n}.$
(21)

Finally, from Lemma 7.5, we have that
$c_{n}=\\#(\mathcal{D}^{n})=\aleph_{\widetilde{X}}(\mathcal{D}^{n})$ for every
$n\geq 1$. Therefore,

$\lim_{n\to\infty}\frac{\log\aleph_{\widetilde{X}}(\mathcal{D}^{n})}{n}=\lim_{n\to\infty}\frac{\log
c_{n}}{n}.$ (22)

Thus, from (21) and (22), we conclude the result. ∎

###  7.2 Entropy conditioned to a cover and conditional entropy

The ideas and definitions in this sub-section are adapted versions of what is
defined in [16, 18] to our context.

Let $\mathcal{B},\mathcal{C}$ be two covers of $\widetilde{X}$ such that
$\\#\mathcal{B}<\infty$ and $\aleph_{\widetilde{X}}(\mathcal{C})<\infty$.
Then, we can define

$\aleph_{\widetilde{X}}(\mathcal{C}|\mathcal{B}):=\max_{B\in\mathcal{B}}\aleph_{B\cap\widetilde{X}}(\mathcal{C}),$

where, by definition, we have that $\aleph_{\emptyset}(\mathcal{C})=1$. In
particular, note that
$\aleph_{\widetilde{X}}(\mathcal{C}|\\{X\\})=\aleph_{\widetilde{X}}(\mathcal{C})$.
Also, if $\mathcal{C}$ is an $X$-open cover, then
$\aleph_{\widetilde{X}}(\mathcal{C}|\mathcal{B})$ is well-defined for every
finite cover $\mathcal{B}$ of $\widetilde{X}$. From Remark 3.5, we deduce that
$\aleph_{\widetilde{X}}(\mathcal{C}^{n})<\infty$ for every $n\geq 1$.
Therefore, we can also define

$\displaystyle h_{top}(f;\mathcal{C}|\mathcal{B})$
$\displaystyle:=\lim_{n\to\infty}\frac{\log\aleph_{\widetilde{X}}(\mathcal{C}^{n}|\mathcal{B}^{n})}{n}.$
(23)

Finally, the topological entropy of $f$ conditioned to $\mathcal{B}$ is given
by:

$\displaystyle h_{top}(f|\mathcal{B})$
$\displaystyle:=\sup\big{\\{}h_{top}(f;\mathcal{C}|\mathcal{B})\ :\
\text{$\mathcal{C}$ is an $X$-open cover}\big{\\}}.$ (24)

By an argument similar to the one given right after Definition 4.2, as $X$ is
compact, it is possible to take this supremum over all finite $X$-open covers.
The following result will be used to prove a key inequality (see (26)), which
will be used to conclude Theorem 2.8.

######  Lemma 7.7.

$\aleph_{\widetilde{X}}((\mathcal{B}\vee\mathcal{C})|\mathcal{E})\leq\aleph_{\widetilde{X}}(\mathcal{B}|\mathcal{E})\cdot\aleph_{\widetilde{X}}(\mathcal{C}|(\mathcal{B}\vee\mathcal{E}))$
for every $\mathcal{B},\mathcal{C},\mathcal{E}$ covers of $\widetilde{X}$ such
that $\aleph_{\widetilde{X}}(\mathcal{C})<\infty$ and
$\max\\{\\#\mathcal{B},\\#\mathcal{E}\\}<\infty$.

###### Proof.

Let $E\in\mathcal{E}$. Then, there exists a sub-collection
$\mathcal{M}\subset\mathcal{B}$ such that

$E\subset\bigcup_{M\in\mathcal{M}}M\qquad\text{and}\qquad\\#\mathcal{M}\leq\aleph_{\widetilde{X}}(\mathcal{B}|\mathcal{E}).$

Now, note that for every $M\in\mathcal{M}$, there exists a finite collection
$\mathcal{C}_{M}\subset\mathcal{C}$ such that

$E\cap
M\subset\bigcup_{C\in\mathcal{C}_{M}}C\qquad\text{and}\qquad\\#\mathcal{C}_{M}\leq\aleph_{\widetilde{X}}(\mathcal{C}|(\mathcal{B}\vee\mathcal{E})).$

Now, observe that $\mathcal{F}:=\\{M\cap C\ :\
M\in\mathcal{M},\;C\in\mathcal{C}_{M}\\}$ is a finite subfamily of
$\mathcal{B}\vee\mathcal{C}$ satisfying
$\aleph_{E}(\mathcal{B}\vee\mathcal{C})\leq\\#\mathcal{F}\leq\aleph_{\widetilde{X}}(\mathcal{B}|\mathcal{E})\cdot\aleph_{\widetilde{X}}(\mathcal{C}|(\mathcal{B}\vee\mathcal{E}))$,
which implies the result. ∎

######  Remark 7.8.

Taking $\mathcal{E}=\\{X\\}$, from part (a) of Lemma 3.3 and Lemma 7.7, we
deduce that

$\aleph_{\widetilde{X}}(\mathcal{C}^{n})\leq\aleph_{\widetilde{X}}(\mathcal{B}^{n}\vee\mathcal{C}^{n})\leq\aleph_{\widetilde{X}}(\mathcal{B}^{n})\cdot\aleph_{\widetilde{X}}(\mathcal{C}^{n}|\mathcal{B}^{n}),$
(25)

for every $n\geq 1$, and $\mathcal{B},\mathcal{C}$ covers of $\widetilde{X}$
such that $\\#\mathcal{B}<\infty$ and
$\aleph_{\widetilde{X}}(\mathcal{C})<\infty$. Moreover, taking the logarithm
and dividing by $n$ in (25), as well as taking the supremum over all $X$-open
covers $\mathcal{C}$, it follows that

$h_{top}(f)\leq h_{top}(f,\mathcal{B})+h_{top}(f|\mathcal{B}).$ (26)

To prove the following result, we recall that $\mathcal{D}$ is the _n_ atural
$X$-open cover given by $\mathcal{D}=\\{X_{1},\ldots,X_{N}\\}$.

######  Lemma 7.9.

$h_{top}(f,\mathcal{C}|\mathcal{D})=0$ for every finite $X$-open cover
$\mathcal{C}$.

###### Proof.

Fix $n\geq 1$ and $D\in\mathcal{D}^{n}$. For $k\in\\{1,\ldots,n\\}$, and
$C\in\mathcal{C}$, we define

$D\cap f^{-k}\big{(}I(C)\big{)}=(f|_{D})^{-k}\big{(}I(C)\big{)},$ (27)

where $I(C):=\big{(}\inf C,\sup C\big{)}$. From Lemma 7.1, the set defined by
(27) is either empty or an open interval contained in a connected component of
$X\setminus\Delta^{n}$. Then, one of the following four cases occurs:

  * •

$\inf f^{-k}\big{(}I(C)\big{)},\sup f^{-k}\big{(}I(C)\big{)}\in D$;

  * •

$\inf f^{-k}\big{(}I(C)\big{)}<\inf D$ and $\sup f^{-k}\big{(}I(C)\big{)}\in
D$;

  * •

$\inf f^{-k}\big{(}I(C)\big{)}\in D$ and $\sup D<\sup
f^{-k}\big{(}I(C)\big{)}$;

  * •

$\inf f^{-k}\big{(}I(C)\big{)},\sup f^{-k}\big{(}I(C)\big{)}\notin D$.

In this way, $D$ contains at most two end-points of
$I\big{(}f^{-k}\big{(}I(C)\big{)}\big{)}$. Therefore, we deduce that $D$ has
at most $2n(\\#\mathcal{C})$ different end-points of the open intervals in

$\mathcal{E}_{n}:=\big{\\{}I\big{(}f^{-k}\big{(}I(C)\big{)}\big{)}\ :\
C\in\mathcal{C},\,\;0\leq k\leq n-1\big{\\}},\qquad\text{where}\quad
I\big{(}f^{0}\big{(}I(C)\big{)}\big{)}=I(C).$

Since the elements of $\mathcal{C}^{n}$ correspond to intersections of pre-
images of elements in $\mathcal{C}$, we can use the endpoints of intervals in
$\mathcal{E}_{n}$ to bound the number of elements in $\mathcal{C}^{n}$ that
intersect $D$. In particular, as there are 4 possibilities in which an
interval can intersect $D$, and each element in $\mathcal{C}^{n}$ is contained
in an interval whose endpoints can be chosen from the (at most)
$2n(\\#\mathcal{C})$ elements belonging to $D$; that is,

$\\#\big{\\{}E\in\mathcal{C}^{n}\ :\ E\cap D\neq\emptyset\big{\\}}\leq
4(2n\\#\mathcal{C})^{2}.$ (28)

Thus, from (28) we deduce

$\displaystyle h_{top}(f;\mathcal{C}|\mathcal{D})$
$\displaystyle=\lim_{n\to\infty}\frac{\log\aleph_{\widetilde{X}}(\mathcal{C}^{n}|\mathcal{D}^{n})}{n}\leq\lim_{n\to\infty}\frac{\log\big{(}16n^{2}(\\#\mathcal{C})^{2}\big{)}}{n}=0.$

∎

With this, to close this sub-section we prove a reformulation of the
Misiurewicz-Szlenk formula established in Theorem 2.6.

######  Theorem 7.10.

The following equality holds:

$h_{top}(f)=h_{top}(f,\mathcal{D})=\lim_{n\to\infty}\frac{\log c_{n}}{n}.$

###### Proof.

From Lemma 7.9 and the observation given right after (24), it follows that
$h_{top}(f|\mathcal{D})=0$. Furthermore, from Remark 7.8, we deduce that

$h_{top}(f)\leq
h_{top}(f,\mathcal{D})+h_{top}(f|\mathcal{D})=h_{top}(f,\mathcal{D}).$ (29)

Now, from (22) and Lemma 7.6, we conclude that

$\lim_{n\to\infty}\frac{\log c_{n}}{n}\leq
h_{top}(f)\leq\lim_{n\to\infty}\frac{\log\aleph_{\widetilde{X}}(\mathcal{D}^{n})}{n}=\lim_{n\to\infty}\frac{\log
c_{n}}{n},$

which concludes the proof. ∎

Furthermore, we can give a straightforward proof of Corollary 2.7 by using
Theorem 7.10. Recall that –in our context– a pc-map is injective when it is
injective over the union of its continuity pieces.

######  Corollary 7.11.

Let $f:X\to X$ be an injective pc-map. Then, $h_{top}(f)=0$.

###### Proof.

First, as $f$ is injective, then $f$ has no critical continuity points (see
the beginning of Section 7 to recall the context).

Now we prove that, as $f$ is injective, then $c_{n}\leq N+(n-1)(N-1)$ for
every $n\geq 1$. To do this, we proceed by induction on $n$. First, if $n=1$,
the result follows. Now, assume that for some $k\geq 1$, we have

$\displaystyle c_{k}\leq N+(k-1)(N-1).$

This implies that $f^{k}$ has at most $N+(k-1)(N-1)$ continuity pieces. Now,
note that the discontinuities of $f^{k+1}$ comprise two kinds of points. On
the one hand, the extreme points of continuity pieces of $f^{k}$ and, on the
other hand, the values of $x$ such that $f^{k}(x)\in\Delta$. However, as $f$
is injective, then $f^{k}(X\setminus\Delta^{k})$ can intersect $\Delta$ at
most $N-1$ times. Therefore,

$\displaystyle c_{k+1}\leq c_{k}+N-1\leq N+k(N-1),$

which concludes the first part of the proof. Now, thanks to Theorem 2.6, we
have that

$\displaystyle h_{top}(f)=\lim_{n\to\infty}\frac{1}{n}\,\log
c_{n}\leq\lim_{n\to\infty}\frac{1}{n}\,\log(N+(n-1)(N-1))=0,$

which lets us conclude the result. ∎

###  7.3 Entropy of pc-maps with surjective branches

We conclude Section 7 by proving the last main result of this article. Let us
recall that, throughout this section, we have globally assumed that $f$ is
strictly monotonic on each of its continuity pieces. For this part, we assume
that “$f^{-j}(\Delta)\cap\Delta=\emptyset$ for all $j\geq 1$”. Although this
condition depends on how $f$ is defined over $\Delta$, it is only an auxiliary
assumption used to accurately count the number of preimages of $\Delta$.
Nevertheless, as the definition of the map over $\Delta$ does not affect its
topological entropy, then this condition can be easily induced by redefining
the map over said set (see Remark 4.3).

######  Lemma 7.12.

If $\overline{f(X_{i})}=X$ for each $i\in\\{1,\ldots,N\\}$ and
$f^{-j}(\Delta)\cap\Delta=\emptyset$ for every $j\geq 1$, then
$\\#(\Delta^{n})=N^{n}-1$ for every $n\geq 0$.

###### Proof.

We proceed by induction on $n$. Taking $n=0$ we have that
$\\#(\Delta^{0})=\\#\emptyset=N^{0}-1=0$. Next, we assume that
$\\#(\Delta^{k})=N^{k}-1$ for some $k\geq 0$. Note that

$f^{-k}(\Delta)\cap\Delta^{k}=\bigcup_{j=0}^{k-1}f^{-j}(\Delta)\cap
f^{-k}(\Delta)=\bigcup_{j=0}^{k-1}f^{-j}\big{(}\Delta\cap
f^{-k+j}(\Delta)\big{)}=\emptyset,$

since, by hypothesis, the pre-images of $\Delta$ do not intersect $\Delta$. It
follows that

$\\#(\Delta^{k+1})=\\#(f^{-k}(\Delta))+\\#(\Delta^{k}).$ (30)

On the other hand, using the fact that $f$ is surjective on each branch, we
deduce

$\\#(f^{-k}(\Delta))=N\cdot\\#f^{-(k-1)}(\Delta)=N^{k}\cdot\\#\Delta=N^{k}(N-1).$
(31)

Finally, from the induction hypothesis and equalities (30) and (31), we
conclude that

$\\#(\Delta^{k+1})=N^{k}(N-1)+N^{k}-1=N^{k+1}-1,$

which ends the proof. ∎

######  Lemma 7.13.

For every $n\geq 1$,
$\big{|}\aleph_{\widetilde{X}}(\mathcal{D}^{n})-\\#(\Delta^{n})\big{|}\leq 1$.

###### Proof.

Fix $n\geq 1$ and assume that $X=[a,b]$, where $a<b$. From Lemma 7.4, we have
four cases. In one case, $a,b\notin\Delta^{n}$, which implies that
$\\#(\mathcal{D}^{n})=\\#(\Delta^{n})+1$, whilst the other cases arise from
the possible alternatives to the former condition. In all of them, the
inequality holds. ∎

Finally, we are in a position to provide a proof for the last main result of
this article. For this, we reformulate Theorem 2.8 recalling that $N=c_{1}$
due to what we assumed at the beginning of this section.

######  Theorem 7.14.

If $\overline{f(X_{i})}=X$ for each $i\in\\{1,\ldots,N\\}$, then
$h_{top}(f)=\log N$.

###### Proof.

Without loss of generality, we assume that
$f^{-j}(\Delta)\cap\Delta=\emptyset$ for all $j\geq 1$. From Theorem 7.10, and
Lemmas 7.12 and 7.13, it follows that

$\lim_{n\to\infty}\frac{\log\big{(}N^{n}-2\big{)}}{n}\leq
h_{top}(f)=\lim_{n\to\infty}\frac{\log\aleph_{\widetilde{X}}(\mathcal{D}^{n})}{n}\leq\lim_{n\to\infty}\frac{\log(N^{n})}{n},$

which concludes the proof. ∎

##  8 Some examples and applications

We conclude this article by showing some examples and applications of the
theory we have presented.

######  Example 8.1.

Consider pc-maps with two continuity pieces, whose representations are defined
by the graphs shown in Figure 1. As can be observed, pc-maps with two strictly
monotonic continuity pieces such as the (discontinuous) doubling map, tent
map, and even one-dimensional Lorenz maps that are surjective on each branch
have a topological entropy equal to $\log 2$ (see Theorem 2.8). We highlight
that the pieces of $X$ do not need to have the same length and the map does
not need to be only expanding or contracting in those intervals. Moreover,
these examples can be generalized to maps with any number of continuity pieces
as long as their number of critical points is finite (see Example 8.2).

Figure 1: Examples of pc-maps with two strictly monotonic continuity pieces
with a topological entropy of $\log 2$.

######  Example 8.2.

Let $N\geq 2$. Using Theorem 2.8, it can be shown that the pc-map
$T_{N}:[0,1]\to[0,1]$ given by

$T_{N}(x)=\left\\{\\!\\!\begin{array}[]{ll}Nx,&\text{ if
$x\in[0,1/N)$,}\\\\[2.15277pt] Nx-j,&\text{ if $x\in\big{[}j/N,(j+1)/N\big{)}$
and $1\leq j\leq N-2$,}\\\\[2.15277pt] Nx-N+1,&\text{ if
$x\in\big{[}(N-1)/N,1\big{]}$,}\end{array}\right.$

has a topological entropy equal to $\log N$. In particular, $T_{2}$
corresponds to the discontinuous doubling map defined on $[0,1]$.

######  Example 8.3.

In Example 8.1, we highlighted the class of maps with more than one continuity
piece that are surjective in each of them. Due to Lemma 5.5, we have that some
maps that are not surjective in all of their continuity pieces can also have
positive entropy. In Figure 2, we show a map with four strictly monotonic
continuity pieces with two pseudo-invariant sets: $X$, and the set comprising
the last two continuity pieces. Furthermore, as the tent map in the last two
branches is surjective in each of them, we can ensure that the topological
entropy of this map will be greater than or equal to $\log 2$.

Figure 2: Graph of a pc-map with four continuity pieces with a positive
entropy greater than or equal to $\log 2$.

######  Example 8.4 ([Entropy of interval exchange transformations]).

The generalized interval exchange transformations (GIET) defined in [13] are a
type of bijective pc-map with zero topological entropy (see Corollary 2.7).
From this fact, together with Lemma 5.2, we deduce that any GIET cannot be
topologically conjugate to a map like those shown in Examples 8.1 – 8.3.

######  Example 8.5.

[Entropy of injective piecewise contracting interval maps] In [12], the
authors show that an injective piecewise contracting interval map, which is
increasing on each of its continuity pieces, has singular entropy equal to
zero; that is,

$\lim_{n\to\infty}\frac{1}{n}\log c_{n}=0.$

Therefore, from Corollary 2.7, we infer that the topological entropy of these
maps equals zero even when not all their branches are increasing, as long as
these maps are injective.
Independently, the authors in [7] prove –using arguments related to atoms and
symbolic complexity– that injective piecewise contracting interval maps
restricted to $\widetilde{X}$ have topological entropy equal to zero when
computed via Bowen’s formula. Moreover, from Theorem 2.4 and Corollary 2.7, we
infer that their topological entropy equals zero and it is independent of the
metric used to compute it.

Closing remark: Naturally, the next step in this study is the variational
principle, which is out of the scope of this article. B. Pires addressed a
delicate point along this line in [26], where he proves the existence of
regular Borel probability measures that are invariant for pc-maps without
connections, which are the maps $f$ such that the one-sided limits of $f(x)$
as $x$ approaches a point in $\Delta$ belong to $\widetilde{X}$ (see rigorous
definition in [26]). Thus, the challenge of establishing a variational
principle in this context will primarily rely on adapting the classical proof
using our notion of topological entropy for pc-maps, assuming the “no
connections” hypothesis.

Acknowledgments: A.E.C. was supported by ANID Fondecyt Research Initiation
N∘11230064, ANID Fondecyt Regular N∘1230569 and MathAmsud 22-MATH-10. E.V-S.
was funded by ANID, Beca Chile Doctorado en el extranjero, number 72210071.

## References

  * [1] Adler, R.L., Konheim, A.G. & McAndrew, M.H. Topological entropy. Transactions of the American Mathematical Society 114(2) (1965) 309-319.
  * [2] Alsedà, L. & Misiurewicz, M. Semiconjugacy to a map of a constant slope. arXiv preprint arXiv:1410.1718 (2014).
  * [3] Bandt, C., Keller, G. & Pompe, B. Entropy of interval maps via permutations. Nonlinearity, 15(5) (2002) 1595.
  * [4] Bowen, R. Topological entropy and Axiom A, Global Analysis. Proc. Sympos. Pure Math., American Mathematical Society, 14 (1970) 23–42.
  * [5] Bowen, R. Entropy for group endomorphisms and homogeneous spaces. Transactions of the American Mathematical Society 153 (1971) 401–414.
  * [6] Calderón, A.E., Catsigeras, E. and Guiraud, P. A spectral decomposition of the attractor of piecewise-contracting maps of the interval. Ergodic Theory and Dynamical Systems 41(7) (2021) 1940-1960.
  * [7] Calderón, A.E. and Villar-Sepúlveda E. Piecewise contracting maps on the interval: Hausdorff dimension, entropy, and attractors. Canadian Mathematical Bulletin (2023) 1-10.
  * [8] Catsigeras, E., Guiraud, P., Meyroneinc, A. and Ugalde, E. On the asymptotic properties of piecewise contracting maps. Dynamical Systems 31 (2016) 107–135.
  * [9] Cánovas, J. S. and Rodríguez, J. M. Topological entropy of maps on the real line. Topology and its Applications 153(5-6) (2005) 735-746.
  * [10] Čiklová, M. Dynamical systems generated by functions with connected $G_{\delta}$ graphs. Real Analysis Exchange 30(2) (2004) 617-638.
  * [11] Dinaburg, E.I. A correlation between topological entropy and metric entropy. In Doklady Akademii Nauk, Russian Academy of Sciences, 190(1) (1970) 19-22
  * [12] Gaivão, J.P. and Nogueira, A. Dynamics of piecewise increasing contractions. Bulletin of the London Mathematical Society, 54(2) (2022) 482–500.
  * [13] Ghazouani, S., and Khanin, K. The symplectic structure for renormalization of circle diffeomorphisms with breaks. Communications in Contemporary Mathematics, 24(01) (2022) 175-200.
  * [14] Hofbauer, F. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Israel Journal of Mathematics, 34 (1979) 213-237.
  * [15] Hofer, J. E. Topological entropy for noncompact spaces. The Michigan Mathematical Journal 21(3) (1975) 235-242.
  * [16] Misiurewicz, M. Topological conditional entropy. Studia Mathematica, 55(2) (1976) 175-200.
  * [17] Misiurewicz, M. Permutations and topological entropy for interval maps. Nonlinearity, 16(3) (2003) 971.
  * [18] Misiurewicz, M. and Szlenk, W. Entropy of piecewise monotone mappings. Studia Math 67 (1980) 45–63.
  * [19] Munkres, J. Topology. Pearson Education Second Edition (2000).
  * [20] Rucká, L., Block, L. & Keesling, J. A generalized definition of topological entropy. Topology Proceedings, Auburn University (2018).
  * [21] Ruette, S. Chaos on the Interval. American Mathematical Society 67 (2017).
  * [22] Veech, W. A. Interval exchange transformations. Journal d’Analyse Mathématique, 33(1) (1978) 222-272.
  * [23] Vries, J. Topological dynamical systems: an introduction to the dynamics of continuous mappings. Walter De Gruyter, 59 (2014).
  * [24] Walters, P. An introduction to ergodic theory. Springer Science & Business Media 79 (2000).
  * [25] Young, L. S. On the prevalence of horseshoes. Transactions of the American Mathematical Society, 263(1) (1981) 75-88.
  * [26] Pires, Benito. Invariant measures for piecewise continuous maps. C. R. Math. Acad. Sci. Paris, 354(7) (2016) 717-722.
  * [27] Viana, M., Oliveira, K, Foundations of ergodic theory Cambridge Studies in Advanced Mathematics