# Three different entropies, variational principle and the degree formula.

My inaugural blog post, hurray!! This post is based on a mandatory assignment in a course in complex dynamics held at the university of Oslo 2018. For sources we mainly the book of Walters [1] and handouts. Some of the more basic results will, due to time constraints, be left unproven, but can all be found in [1].

#### – Introduction –

The notion of entropy, first formulated in the context of thermodynamics, is a quantity meant to capture the complexity/chaoticness of the state of a system. It is maybe not surprising that such a broadly defined notion spread like aids throughout different subfields of applied mathematics and information theory. Here, three different definitions of entropy are introduced, two of which turn out to be equivalent when working on compact metric spaces, and all three are related, on compact metric spaces, by the so called Variational Principle.
Lastly, the theorem of Misiurewicz, Przytycki and Gromov, which relates the entropy of a holomorphic map on the $n$-sphere to its degree was proven.

#### – The three entropies –

We give the definitions in chronological order.

#### – Measure theoretic entropy –

Let $(X, \mathcal{B})$ be a measurable space, and $T:X\to X$ a measurable map. The collection of all probability measures will be denoted $M(X)$, and the T-invariant probability measures, that is, measures $\mu \in M(X)$ for which $\mu(A) = \mu(T^{-1}A)$ for all measurable $A$, will be denoted $M(X, T)$. A measurable partition is defined as a collection $\xi \subset \mathcal{B}$ of disjoint measurable sets that cover $X$. We define the join of two measurable partitions as
$\xi \vee \eta = \{ C\cap D ~ | ~ C \in \xi, ~ D\in \eta \}$

The entropy of a measurable partition is defined as the (generalized) sum
$$H_\mu(\xi) = -\sum_{C\in \xi} \mu(C)log(\mu(C))$$
with the conventions that $H_\mu(\xi) = \infty$ if the union of all sets of measure zero in $\xi$ have non-zero measure, and we let $\mu(C)log(\mu(C)) = 0$ if $\mu(C) = 0$.

Now the entropy of T with respect to a T-invariant probability measure $\mu$ and a partition $\xi$ with $H_\mu(\xi) < \infty$ is defined as

$$h_{\mu}(T, \xi) = \inf_{n\in \mathbb{N}} \left\{ \frac{1}{n} H(\bigvee^{n-1}_{k=0}T^{-k}\xi ) \right\} = \lim_{n\to\infty} \frac{1}{n} H_\mu\left(\bigvee^{n-1}_{k=0} T^{-k}\xi\right).$$

It should be noted that some authors define a measurable partition as a collection of measurable sets whose intersection have measure zero and which cover the space almost everywhere. Personally, I prefer the definition of a measurable partition to be independent of choice of measure. It is also customary to remove sets of measure zero in stead (as an alternative to the convention $0log(0) = 0$ we adopt here), but this again requires a choice of measure, which is unfortunate.

The map sending an invariant probability measure $\mu \mapsto h_\mu(T)$ is called the entropy map. It is affine, but in general not continuous.

Definition 1 (Measure Theoretic Entropy)
The measure theoretic entropy of a measurable map $T$ and a measure $\mu\in M(X,T)$ is defined as $$h_\mu(T) = \sup\{ h_\mu(T, \xi) ~|~ H_\mu(\xi) < \infty \}.$$

We also will need the following definition in the proof of the Variational principle below

Definition 2 (Conditional Entropy)
Let $\xi = \{C_1,…., C_n \}, \nu = \{B_1,…., B_m \}$ be two measurable partitions of $X$. We define the conditional entropy of $\xi$ given $\nu$ as
$$H_\mu(\xi | \nu) = – \sum_{i,j} \mu(C_j\cap B_i) \log\left(\frac{\mu(B_i\cap C_j)}{\mu(B_i)}\right).$$

#### – Topological Entropy –

Analogously, we may define a notion of entropy in the category of compact topological spaces, which we will call the topological entropy. It was first given in the article of Adler et. al. [2]. For this we need the following preliminary definitions (here $X$ will denote a compact space).

For any open cover $\mathcal{U}$ of $X$ the quantity $N(\mathcal{U})$ is the minimal cardinality of any subcover of $\mathcal{U}$. We define the entropy of the cover $\mathcal{U}$ to be the number $$H_{top}(\mathcal{U}) = log(N(\mathcal{U}))$$

The join of two open covers $\mathcal{U}, \mathcal{W}$ of $X$ is defined as the open cover $$\mathcal{U}\vee \mathcal{W} = \{ U\cap W ~ | ~ U\in \mathcal{U}, ~ W\in \mathcal{W} \} .$$

The topological entropy of a continuous map $\phi: X\to X$, with respect to the open cover $\mathcal{U}$ of $X$ is defined as the number
$$h_{top}(\phi, \mathcal{U}) = \lim_{n\to \infty}\frac{1}{n} H_{top}\left(\bigvee^{n-1}_{i=0} \phi^{-1}\mathcal{U}\right).$$

Definition 3 (Topological Entropy)
The entropy of a continuous map $\phi: X\to X$ on a topological space $X$ is defined as the supremum
$$h_{top}(\phi) = \sup \{h_{top} (\phi, \mathcal{U}) ~| ~ \mathcal{U} \text{ is an open cover of }X \}.$$

Though we will not delve into this, it should be mentioned that there are more definitions of topological entropy in the literature for general topological spaces which all reduce to the above definition if the space is compact.

#### – Metric Entropy –

Lastly we have an entropy which (to the best of my knowledge) was first introduced in [3] by Bowen. It makes explicit use of a metric, hence the name. Some authors call it the topological entropy, since, as we will see, they are actually equivalent on compact metric spaces.
Let $(X, d)$ be a compact metric space, and $T: X\to X$ a continuous map. To define the metric entropy, we need the following preliminary definitions

We define the metric $d_T^n$ on $X$ by
$$d_T^n(x, y )= \max_{0\leq i \leq n-1} d(T^ix, T^iy).$$
The metrics $d$ and $d_T^n$ are  (strongly) equivalent Why?

$B_r^d(x) \subset B_r^{d_T^n}(x)$ since $d \leq d_T^n$, and since $X$ is compact, the $T^i$’s are uniformly continuous, hence there exists a $\delta$ such that $B_\delta^{d_T^n}(x) \subset B_{r}(x)$.

A collection $F \subset X$ of points is said to be $(n, \epsilon)$-separated if $$d^n_T(x,y) \geq \epsilon \qquad \text{for all }x,y \in F, \text{ with } x\neq y.$$ By compactness this set must be finite.
We let $N_d(T, \epsilon, n)$ denote the maximum carnality all $(n, \epsilon)$-separated subsets of $X$. This number is necessarily finite
Why?

If $U$ is a cover of $X$ by balls of $d^n_T$-radius less than $\epsilon/2$, then the cardinality of any $(n, \epsilon)$-separated set, must be smaller than the number of elements in the cover $U$ by a triangle inequality. This relies on the fact that $d_T^n$ and $d$ are equivalent.

Define the quantity $h_d(T, \epsilon)$ to be
$$h_d(T,\epsilon) = \limsup_{n\to \infty} \frac{1}{n} \log(N_d(T, \epsilon, n)).$$

Definition 4 (Metric Entropy)
The metric entropy is defined as the limit
$$h_d(T) = \lim_{\epsilon\to 0} h_d(T,\epsilon).$$

$h_d(T)$ is is well defined by the following lemma

Lemma 1 If $\epsilon_1 < \epsilon_2$ then $N_d(T, \epsilon_2, n) \leq N_d(T, \epsilon_1, n) < \infty$, hence $h_d(T, \epsilon)$ is monotone non-increasing as $\epsilon \to 0$ and bounded below by $0$ (by inspection).

Proof
First we show finiteness: Since all $T^i$’s are uniformly continuous for all $i\in \mathbb{N}$, there is, for any $\epsilon >0$, a $\delta >0$ such that $d(T^ix, T^iy) < \epsilon$ whenever $d(x, y ) < ~ \delta$, for all $0 \leq i \leq n-1$. It follows that $N_d(T, \epsilon, n)$ is less than the number of $\delta$ discs needed to cover $X$ (finite since $X$ is compact). For the monotonicity, assume $\epsilon_1 <~ \epsilon_2$. If $F\subset X$ is $(\epsilon_2, n)$-separated, then a forteriori it must by $(\epsilon_1, n)$-separated, which implies that $N_d(T, \epsilon_2, n) \leq N_d(T, \epsilon_1, n)$ .

#### – Alternative Defintions –

We can also treat the topological and measure theoretic entropies as limits over appropriate directed index sets. Let’s see how this plays out in the case of topological entropy, bearing in mind that the same could just as well be done with the measure theoretic entropy.

With $X$ a compact Hausdorff space, the family $$O = \{ \mathcal{U} ~ | ~ \mathcal{U} \text{ is an open cover of } X \}$$ can be made into a directed set with respect to the order relation of refinement, that is $\mathcal{U}_1 \prec \mathcal{U}_2$ if $\mathcal{U}_2$ is a refinement of $\mathcal{U}_1$. It is not hard to check $\prec$ is a partial ordering on the collection of open covers, and for any open cover $\mathcal{U}_1$ and $\mathcal{U}_2$ we have $$\mathcal{U}_1, \mathcal{U}_2 \prec \mathcal{U}_1\vee \mathcal{U}_2.$$

The same can be done for the measure theoretic entropy with the order relation $\xi\prec \eta$ if for all $U\in \eta$ there is some $V\in \xi$ such that $U\subset V$. Again $\xi, \eta \prec \xi\vee \eta$ so in both these cases, $\prec$ determines a directed set.

Now we can define the topological entropy and measure theoretic entropy of a map $\phi$ by

Definition 5 (Topological/Measure theoretic entropy [nets])
The etropies defined above admit a concise definition through nets, given by\begin{align*}
& h_{top}(\phi) = \lim_{\mathcal{U}}h_{top}(\phi, \mathcal{U}) \qquad \text{over all open covers $\mathcal{U}$} \\
& h_\mu(\phi)~ = \lim_{\xi}h_\mu(\phi, \xi) \qquad \text{over all measurable partitions $\xi$}
\end{align*}
Proof
If $\mathcal{U} \prec \mathcal{V}$, we know by Proposition 6 (2) that $H_{top}(\mathcal{U}) \leq H_{top}(\mathcal{V})$ and hence
$$h_{top}(T, \mathcal{U}) \leq h_{top}(T, \mathcal{V}),$$ so $\{h_{ top }( T, \mathcal{U} )\}_{\mathcal{U}}$ is a monotone net and hence must converge in $\mathbb{R}\cup \{ \infty \}$ to it’s supremum.

The situation for $h_\mu$ is completely analogous, using Proposition 3 (3) in place of Proposition 6 (2).

Note that for compact metric spaces, we know that the above defined net of open covers has a cofinal subsequence, that is, a subnet   which is also a sequence, and since all subnets of a convergent net converge to the same limit, we may as well take the limit over this sequence.

To construct the sequence we need the Lebesgue number lemma, which asserts that for any open cover $\mathcal{W}$ of a compact metric space there exists a number $\delta >0$ such that any disc of radius $\delta$ is completely contained in some element in the cover. Let $\mathcal{U}_n = \{ B_{\frac{1}{n}}(x) ~ | ~ x \in X\}$ be the open cover of all balls of radius $\frac{1}{n}$. Now it follows that for any open cover $\mathcal{W}$ of $X$ with a Lebesgue number $\delta$, we have $\mathcal{W} \prec \mathcal{U}_n$ for all $n\in \mathbb{N}$ such that $\frac{1}{n} \leq \delta$, hence the sequence is cofinal.

#### – Basic Properties (Measure Theoretic Entropy)-

Here we list some of the basic properties of the entropy. First and foremost let’s check that the definition given above makes sense.

Proposition 1 (Existence of the Limit) The limit
$$h_{\mu}(T, \xi) = \lim_{n\to \infty} \frac{1}{n} H_\mu(\bigvee^{n-1}_{k=0} T^{-k}\xi)$$
exists, is finite, and is equal to $\inf_{n\in \mathbb{N}} \{ \frac{1}{n} H(\bigvee^{n-1}_{k=0}T^{-k}\xi ) \}.$
Proof

With $H_m = H_\mu\left(\bigvee^{m-1}_{k=0}T^{-k}\xi \right)$ we will show that the sequence $\{ H_n/n\}$ is decreasing as $n$ increases. This suffices to prove the theorem since the sequence is bounded below by 0. Noting that $H_{m+n} \leq H_m + H_n$, we may, for some fixed $p$, partition the index set into steps of $p$, that is, we may write $n = i+ kp$ where $0\leq i< p$. Now
\begin{align*}
\frac{H_n}{n} & = \frac{H_{i + kp}}{i + kp} \leq \frac{H_{i}}{i+kp} + \frac{H_{kp}}{i+kp} \leq \frac{H_{i}}{kp} + \frac{H_{kp}}{kp}\\
&\leq \frac{H_{i}}{kp} + \frac{k H_{p}}{kp} = \frac{H_{i}}{kp} + \frac{H_{p}}{p}.
\end{align*}
The result now follows by noting that as $n\to \infty$, $k\to \infty$.

Proposition 2

• $h_\mu(id) = 0$
•  $h_\mu(T^k) = kh_\mu(T)$ for every $k\in \mathbb{N}$
Proof
• By definition we know that $H_\mu(\xi\vee\xi) = \sum_{(C, D) \in \xi \times \xi} \mu(C\cap D) \log(\mu(C\cap D))$ but note these summands are zero off of the diagonal of $\xi\times \xi$, hence we get that
$$H_\mu(\xi \vee \xi) = H_\mu(\xi)$$
Consequently
$$h_\mu(id, \xi) = \lim_{n\to \infty} \frac{1}{n} H_\mu(\bigvee_{k=0}^{n-1}id^{-1}\xi) = \lim_{n\to \infty} \frac{1}{n} H_\mu(\bigvee_{k=0}^{n-1}\xi) = \lim_{n\to \infty} \frac{1}{n} H_\mu(\xi) = 0$$ and the claim follows.
• Though it is not generally true that $H(\bigvee^{n-1}_{k=0}T^{rk}\xi)$ equals $H(\bigvee^{r(n-1)}_{k=0}T^{k}\xi)$, this will not mater in the limit case. Let $\xi$ be a measurable partition of $X$ (of finite entropy), then
\begin{align*}
h_\mu(T^k, \bigvee^{k-1}_{i=1}T^i \xi) &= \lim_{n\to \infty} \frac{H(\bigvee^{nk-1}_{i=1} T^{-i} \xi)}{n}\\
& = k \lim_{n\to \infty} \frac{H(\bigvee^{nk-1}_{i=1} T^{-i} \xi)}{kn}\\
& = k h_\mu(T, \xi) \leq k h_\mu(T).\end{align*}
Taking supremums over all partitions, we get$h_\mu(T^k) = \text{sup}_{\xi}\{ h_\mu(T^k, \xi) \} \geq \text{sup}_{\xi}\{ h_\mu(T^k, \bigvee^{k-1}_{i=1}T^i \xi ) \} = \text{sup}_{\xi}\{ k h_\mu(T, \xi)\} = h_\mu(T).$
The reverse inequality follows directly from the inequality
$$H_\mu(\bigvee^{k-1}_{i=1}T^i \xi) \geq H_\mu(\xi).$$

Other noteworthy properties of the measure theoretic entropy are listed in chapter 4.5 in [1]. We state them here without proof. Let $\xi, \eta$ be finite entropy measurable partitions of $X$.

Proposition 3

1.  $h_\mu(T, \xi) \leq H_\mu(\xi)$
2.  $h_\mu(T, \xi \vee \eta) \leq h_\mu(T, \xi) + h_\mu(T, \eta)$
3.  $\xi \prec \eta \qquad \Rightarrow h_\mu(T, \xi) \leq h_\mu(T,\eta)$
4.  if $k \geq 1$, then $h_\mu(T, \xi ) = h_\mu(T, \bigvee^{k-1}_{i=0}T^{-i} \xi)$
5.  (continuity) $| h_\mu(T, \xi) – h_\mu(T, \eta)| \leq d(\xi, \eta)$, where $d$ is the metric on the space of finite partitions defined by
$$d(\xi, \eta) = H_\mu(\xi | \eta) + H_\mu(\eta | \xi).$$
6.  $h_\mu(T, \xi) \leq h_\mu(T, \eta) + H_\mu(\xi | \eta)$
7.  $H_\mu(\xi) \leq log(|\xi|)$ if $\xi$ is finite, where $|\xi|$ denotes the number of elements in $\xi$.

The following theorem is used repeatedly, is easy to state, and has an elegant one-line proof, so there is no good reason not to include it here.

Theorem 1 (Krylov-Bogoliubov)
Let $X$ a compact metric space, $T: X\to X$ a continuous map, and $M(X, T)$ the space T-invariant probability measures on $X$. The space $M(X, T)$ is non-empty.
Proof
The map $\mu \mapsto T_\star \mu$ (the pushforward measure) is continuous with respect to the weak*-topology on $M(X, \mathcal{B})$, (which is compact by Alaoglu), hence by Schauder’s fixed point theorem, we know that it has a fixed point.
Proposition 4
The entropy map is an affine map, ie. it maps convex combinations to convex combinations.
Proof
Here we present a sketch of the proof, which can be found i [1].

It is not too hard to check that by concavity of the function $x\mapsto xlog(x)$ we get the inequality
$$0 \leq H_{p\mu + (1 – p )m}(\xi) – pH_\mu(\xi) – (1-p)H_m(\xi).$$
By log-sum arithmetics, one can also produce the inequality
$$H_{p\mu + (1 – p )m}(\xi) – pH_\mu(\xi) – (1-p)H_m(\xi) \leq log(2)$$
From this we get that

\begin{align*}
\frac{H_{p\mu + (1 – p )m}(\bigvee_{i=0}^{n-1} T^{-1} \xi)}{n} & – \frac{ pH_\mu(\bigvee_{i=0}^{n-1} T^{-1} \xi)}{n} & \\ & – \frac{(1-p)H_m(\bigvee_{i=0}^{n-1} T^{-1} \xi)}{n} \leq \frac{log(2)}{n}
\end{align*}
which, as $n\to 0$ yields
$$h_{p\mu + (1 – p )m}(T, \xi) = ph_\mu(T, \xi)+ (1-p)h_m(T, \xi)$$

Deviating a bit from Walter’s proof, we may take the limit over the net of all measurable partitions (of finite entropy) in the above equality, which produces the equality
$$h_{p\mu + (1 – p )m}(T) = ph_\mu(T)+ (1-p)h_m(T).$$

As the next proposition shows, the above proposition can be extended to arbitrary convex combinations of ergodic probability measures, which are defined as measures $\mu \in M(T,X)$ for which $T^{-1}A = A \Rightarrow \mu(A) \in \{ 0, 1 \}$. It relies on the ergodic decomposition theorem  which I hope to cover in a subsequent post.

Theorem 2 (Jacobs)
If $T: X\to X$ is a continuous map on a compact metrizable space and $\mu \in M(T,X)$ has ergodic decomposition
$$\mu = \int_{EM(T, X)} \nu d\eta(\nu)$$
where $EM(T, X) = \{ \text{ all ergodic probability measures on } X \}$, then
$$h_\mu(T) = \int_{EM(T, X)} h_\nu(T) d\eta(\nu).$$

Consult Theorem 4.11 of [1] for the proof of the following important result,

Proposition 5
Entropy is conjugacy invariant (hence an isomorphism invariant in the category of measure spaces).

#### – Basic Properties (Topological Entropy )-

In this section $X, Y$ are compact metrizable space, the maps in question will always be continuous and $\mathcal{U}$ will denote an open cover of $X$. First we collect some basic results, whose proofs are all more or less straightforward. Recall that the order relation of refinement, which we denoted $\prec$ is used to determine a partial order on the collection of open covers.

Proposition 6

1. $\mathcal{U} \prec \mathcal{U}’$ and $\mathcal{V} \prec \mathcal{V}’$, then $\mathcal{U}\vee \mathcal{V} \prec \mathcal{U}’ \vee \mathcal{V}’$.
2. $\mathcal{U} \prec \mathcal{U}’$, then $N(\mathcal{U})\leq N(\mathcal{U}’)$ and so $H_{top}(\mathcal{U}) \leq H_{top}(\mathcal{U}’)$
3. $N(\mathcal{U} \vee \mathcal{U}’) \leq N(\mathcal{U}) N(\mathcal{U}’)$ and so $H_{top}(\mathcal{U} \vee \mathcal{U}’) \leq H_{top}(\mathcal{U}) + H_{top}(\mathcal{U}’)$
4. For any continuous $F: X\to X$, we have $N(F^{-1}\mathcal{U}) \leq N(\mathcal{U})$, and so $H_{top}(F^{-1}\mathcal{U}) \leq H_{top}(\mathcal{U})$.
5. For an homeomorphism $F: X\to X$ we have $N(F^{-1}\mathcal{U}) = N(\mathcal{U})$, and so $H_{top}(F^{-1}\mathcal{U}) = H_{top}(\mathcal{U})$
Proof
1. Let $C’ \subset \mathcal{U}’\vee \mathcal{V}’$, then $C’ = U’ \cap V’$ for some $U’\in \mathcal{U}’$ and $V’\in \mathcal{V}’,$ hence there are $U\in \mathcal{U}$ and $V \in \mathcal{U}$ such that $U’\subset U$ and $V’ \subset V$, So $C’ \subset C = U\cap V \in \mathcal{U} \vee \mathcal{V}$. The claim follows.
2. If $\mathcal{U} \prec \mathcal{V}$, that is $\mathcal{V}$ refines $\mathcal{U}$, then we can for any subcover $\{V_1, …, V_{N(\mathcal{V})} \}\subset \mathcal{V}$ construct a subcover of $\mathcal{U}$ with (at most) the same cardinality, $\{ U_1, …, U_{N(\mathcal{V})} \} \subset \mathcal{W}$, where $V_i \subset U_i$. It thus follows that $$N(\mathcal{U}) \leq N(\mathcal{V}).$$ Taking log on both sides shows $H_{top}(\mathcal{U}) \leq H_{top}(\mathcal{U})$
3. If $\mathcal{U}’ \subset \mathcal{U}$ and $\mathcal{V}’\subset \mathcal{V}$ are minimal covers of $X$, then $\{ U\cap V | ~ U\in \mathcal{U}’, ~ V \in \mathcal{V}’ \}$ is an open cover in $\mathcal{U}’ \vee \mathcal{V}’$ with (at most) $card(\mathcal{U}’) card(\mathcal{V}’)$ elements. The claim follows.
4. Since $\mathcal{U}$ is a refinement of $F^{-1}(\mathcal{U})$, this follows from (2)
5. This follows by substituting $F$ with $F^{-1}$ in (4).

The first thing we should check is that the definition of the topological entropy given above is well defined,

Proposition 7 (Existence of a Limit)
If $\mathcal{U}$ is any open cover of $X$, the limit
$$h_{top}(\phi, \mathcal{U}) = \lim_{n\to \infty}\frac{1}{n} H_{top}(\bigvee^{n-1}_{i=0} \phi^{-1}\mathcal{U})$$
exists and is finite.
Proof
\begin{align*}
H_{top}(\bigvee^{n + m -1}_{i=0} \phi^{-i}\mathcal{U})
& \leq H_{top}(\bigvee^{ m -1}_{i=0} \phi^{-i}\mathcal{U} ) + H_{top}(\phi^{-m}(\bigvee^{n -1}_{i=0} \phi^{-i}\mathcal{U})) \\
& \leq H_{top}(\bigvee^{ m -1}_{i=0} \phi^{-i}\mathcal{U} ) + H_{top}(\bigvee^{n – 1}_{i=0} \phi^{-i}\mathcal{U})
\end{align*}
where the first and second inequalitites follow from Proposition 6 (3) and (4) respectively.
Writing $H_m$ for $H_{top}(\mathcal{U} \vee … \vee \phi^{-m +1}\mathcal{U})$ we see that
$$H_{m+n} \leq H_m + H_n$$ so we may repeat the proof of Proposition 1.

We have the following collection of basic properties relating to the topological entropy.

Proposition 8

1. Entropy is a topological invariant.
2. $h(\phi^k) = kh(\phi)$ for all $k\in \mathbb{N}$.
Proof
1. Assume $\psi: X\to Y$ is a homeomorphism of compact spaces, $\phi:X\to X$ is continuous, and $\mathcal{O}$ an open cover of $X$, then
\begin{align*}
h_{top}(\psi\phi\psi^{-1}, \psi \mathcal{O}) & = \lim_{n\to \infty} H_{top}(\psi\mathcal{O} \vee … \vee \psi\phi^{-n+1}\psi^{-1}\psi\mathcal{O})\\
&= \lim_{n\to \infty} H_{top}(\psi( \mathcal{O} \vee \phi^{-1}\mathcal{O} \vee… \vee \phi^{-n+1}\mathcal{O} ) ) \\
& = \lim_{n\to \infty} H_{top}(\mathcal{O} \vee \phi^{-1}\mathcal{O} \vee… \vee \phi^{-n+1}\mathcal{O} ) ~ \text{ (Prop. 6.(5)) }\\
& = h_{top}(\phi, \mathcal{O})
\end{align*}
since $\psi$ is a homeomorphism it induces an isomorphism of measure algebras, so taking supremums over all cover covers $\mathcal{O}$ concludes the proof.
2. we have
\begin{align*}
h_{top}(\phi^k) & \geq h_{top}(\phi^k, \mathcal{O} \vee \phi^{-1}\mathcal{O} \vee… \vee \phi^{-k+1}\mathcal{O})\\
&= k \lim_{n\to \infty} \frac{1}{nk} H_{top}(\mathcal{O} \vee … \vee \phi^{-nk+1}\mathcal{O})\\
&= k h_{top}(\phi, \mathcal{O})
\end{align*}
so we get $h_{top}(\phi^k)\geq kh_{top}(\phi)$. Conversely, since $\bigvee^{n-1}_{i=1} \phi^{-ki } \mathcal{U} \prec \bigvee^{kn -1}_{i=1} \phi^{-i} \mathcal{U}$ Proposition 6 (2) yields
\begin{align*}
h_{top}(\phi) & = \lim_{n\to \infty} \frac{H_{top}(\bigvee^{kn -1}_{i=1} \phi^{-i} \mathcal{U})}{kn} \\
& \geq \lim_{n\to \infty} \frac{H_{top }(\bigvee^{n-1}_{i=0} \phi^{-ki}\mathcal{U})}{nk} = \frac{h_{top}(\phi^k, \mathcal{U})}{k}.
\end{align*}
so the reverse inequality also holds.

#### – Basic Properties (Metric Entropy)-

In this section $(X,d)$ is a compact metric space.

Some properties of Metric entropy

• If $T$ is an isometry, then $h_d(T) = 0$
• If two metrics $d$ and $d’$ on $X$ are equivalent, then $$h_d(T) = h_{d’}(T).$$
• If $(X, \phi)$ is a compact dynamical system, such that $\{\phi^n\}$ is equicontinuous, $h_{top}(\phi) = 0$

Two metrics $d$ and $d’$ are said to be equivalent (denoted $d \sim d’$) if they induce the same topology. On a compact metric space $X$, this is equivalent to the existence of two fixed real numbers $\alpha, \beta > 0$ such that $\alpha d(x, y) \leq d'(x,y) \leq \beta d(x,y)$ for all $x, y \in X$. On a general metric space, $\alpha$ and $\beta$ are allowed to depend $x$ or $y$.

Proof
• If $T$ is an isometry, then $d = d_n^T$, so $N_d(T, n, \epsilon)$ is constant as $n$ varies, and has been shown previously to be finite, hence $\limsup_{n\to \infty} \frac{1}{n} N_d(T, n, \epsilon) = 0$.
• With $h_d(T,\epsilon) = \limsup_{n\to \infty} \frac{1}{n} \log(N_d(T, n, \epsilon))$, we noted in the definition that $h_d(T, \epsilon)$ is monotone non-increasing in $\epsilon$. If $d\sim d’$, then there are positive real numbers $0 < m \leq M < \infty$ , such that
$$m d(x, y) \leq d'(x,y ) \leq M d(x, y) \qquad \text{for all } x, y \in X.$$
This immediately yields
$m d_T^n(x, y) \leq d_T^{n’}(x,y) \leq M d_T^n(x, y)$, so $N_d(T, m\epsilon, n) \leq N_{d’}(T, n, \epsilon) \leq N_d (T, M \epsilon, n)$
and
$$h_{d’}(T, m\epsilon) \leq h_{d}(T, \epsilon) \leq h_{d’}(T, M\epsilon).$$
Letting $\epsilon \to 0$, since $h_{d’}$ converges, we see that the we get equality, and conclude that $h_d(T) = h_{d’}(T)$.
• If $\{ f^n\}$ is equicontinuous, then there exists a $\delta > 0$ such that $d^n_f(x, y) < \epsilon$ for all $n$, if $d(x,y) < \delta$. It follows that for a fixed $\epsilon$ and any $n$, $N_d(f, \epsilon, n)$ is less than the number of $\frac{\delta}{2}$ balls needed to cover $X$. Hence $\limsup_{n\to \infty} \frac{log(N_d(f, \epsilon, n))}{n} \to 0$ for all $\epsilon$.

#### – Relations between the definitions –

In the literature the topological and metric entropies are used interchangeably, the reason for which is the following theorem.

Theorem 3
If $X$ is a compact metric space, and $f: X\to X$ is any continuous map, then
$$h_{top}(f) = h_d(f)$$
Proof
We will show $N_d(T, n, \epsilon) \leq N(\mathcal{U}^n) \leq N_d(T, n, \frac{\delta}{2})$, where $\delta$ is a Lebesgue number of the open cover $\mathcal{U}$, $\mathcal{U}^n := \bigvee^{n-1}_{i=0}T^{-i}\mathcal{U}$, and $\epsilon$ is the supremum of the diameters of $\mathcal{U}$ which we assume, without loss of generality, to be finite.

Step 1: First let’s produce the following inequality
$$N_d(T, n, \epsilon) \leq N(\mathcal{U}^n).$$
Letting $V\in \mathcal{U}^n$, that is $V = C_0 \cap T^{-1}C_1 \cap … \cap T^{-n+1}C_{n-1}$ for some $C_i \in \mathcal{U}$. Note that we have $diam(T^{i}V) < \epsilon$ since

#### – Sources –

[1] Walters, Peter. An introduction to ergodic theory. Vol. 79. Springer Science & Business Media, 2000.
[2] Adler, Roy L., Alan G. Konheim, and M. Harry McAndrew. “Topological entropy.” Transactions of the American Mathematical Society 114.2 (1965): 309-319.
[3] Bowen, Rufus. “Entropy for group endomorphisms and homogeneous spaces.” Transactions of the American Mathematical Society 153 (1971): 401-414.
[4] Katok, Anatole, and Boris Hasselblatt. Introduction to the modern theory of dynamical systems. Vol. 54. Cambridge university press, 1997.
[5] Milnor, John Willard. Dynamics in one complex variable. Vol. 160. Princeton: Princeton University Press, 2006.

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