$\Big{|}\int\log\|D^{\mathrm{c}}f\|\,d\nu^{\prime}-\chi^{\mathrm{c}}(\nu)\Big{|}<\varepsilon/4.$

Item (a) implies item (1) of the assertion. Item (b) implies that there exists
$\widetilde{K}>0$ such that for every $x\in\Lambda$ and every
$n\in\mathbb{N}$, one has

$\widetilde{K}^{-1}\cdot
e^{n(\chi^{\mathrm{c}}(\nu)-\varepsilon/4)}\leq\|D^{\mathrm{c}}f^{n}(x)\|\leq\widetilde{K}\cdot
e^{n(\chi^{\mathrm{c}}(\nu)+\varepsilon/4)}.$

By Lemma 8.2, every center curve centered at a point $x\in\Lambda$ of inner
radius $r^{\sharp}$ is contained in $\mathcal{W}^{\mathrm{s}}(x,r^{\sharp})$,
proving item (2). It follows from the uniform continuity of the center bundle,
that there exists $\widehat{K}\geq 1$ such that for every $x\in\Lambda$,
$y\in\mathcal{W}^{\mathrm{s}}(x,r^{\sharp})$, and $n\in\mathbb{N}$, one has

$\widehat{K}^{-1}\cdot
e^{n(\chi^{\mathrm{c}}(\nu)-\varepsilon/2)}\leq\|D^{\mathrm{c}}f^{n}(y)\|\leq\widehat{K}\cdot
e^{n(\chi^{\mathrm{c}}(\nu)+\varepsilon/2)}.$

As $f$ is uniformly contracting along the strong stable foliation, there is
$K\geq 1$ such that for every center curve $\gamma$ centered at a point
$x\in\Lambda$ of inner radius $r^{\sharp}$,
$y\in\mathcal{W}^{\mathrm{ss}}\big{(}\gamma,\delta\big{)}$, and
$n\in\mathbb{N}$, one has

$K^{-1}\cdot
e^{n(\chi^{\mathrm{c}}(\nu)-\varepsilon)}\leq\|D^{\mathrm{c}}f^{n}(y)\|\leq
K\cdot e^{n(\chi^{\mathrm{c}}(\nu)+\varepsilon)}.$

This proves (3) and finishes the proof of the lemma. ∎

We now state the main result in this section.

######  Proposition 8.4.

Let $f\in\operatorname{PH}^{1}_{\mathrm{c}=1}(M)$ and $\delta_{0}>0$ be as in
Remark 8.1. Let $\Lambda$ be a basic set of contracting type of $f$ and let
$r^{\sharp}=r^{\sharp}(\Lambda)>0$ and
$a^{\sharp}=a^{\sharp}(\Lambda)\in\mathbb{N}$ be as in Lemma 8.2.

Then there exists $\ell^{\sharp}=\ell^{\sharp}(\Lambda)\in\mathbb{N}$ with the
following properties. For every $\varepsilon>0$, there exists
$b^{\sharp}=b^{\sharp}(\Lambda,\varepsilon)>a^{\sharp}$ such that for every
$n>b^{\sharp}$, there exist an $(n,\delta_{0})$-separated subset
$\mathcal{S}=\mathcal{S}(n)\subset\Lambda$ and an integer
$\ell\in\\{0,\ldots,\ell^{\sharp}-1\\}$ such that

  * (a)

$\operatorname{card}(\mathcal{S})\geq e^{n(h_{\rm
top}(f,\Lambda)-\varepsilon)},$

  * (b)

for every $N\in\mathbb{N}$, there exists an injective map

$\Psi=\Psi^{n,N}\colon\mathcal{S}^{N}\to\Lambda$

having the following properties: given two points

$\mathfrak{X}=(x_{1},\ldots,x_{N}),\quad\mathfrak{X}^{\prime}=(x_{1}^{\prime},\ldots,x_{N}^{\prime})\in\mathcal{S}^{N},$

let $i_{0}$ be the smallest integer such that $x_{i_{0}}\neq
x_{i_{0}}^{\prime}$, then

    * (b1)

for every $i\leq i_{0}-1$ and $(i-1)(n+\ell)\leq j<(i-1)(n+\ell)+n$, it holds

$d\big{(}f^{j}\circ\Psi(\mathfrak{X}),f^{j}\circ\Psi(\mathfrak{X}^{\prime})\big{)}<\delta_{0}/5;$

    * (b2)

there exist $(i_{0}-1)(n+\ell)+a^{\sharp}<j_{0}<(i_{0}-1)(n+\ell)+n$ such that

$d\big{(}f^{j_{0}}\circ\Psi(\mathfrak{X}),f^{j_{0}}\circ\Psi(\mathfrak{X}^{\prime})\big{)}>4\delta_{0}/5.$

  * (c)

for every pair of different points $x,y\in\operatorname{Im}(\Psi)$ and $r\leq
r^{\sharp}$, one has

$\mathcal{W}^{\mathrm{s}}(x,r)\cap\mathcal{W}^{\mathrm{s}}(y,r)=\emptyset.$

###### Proof.

We collect two classical facts without proof.

######  Claim 8.5 (Spectral decomposition).

There exist $p^{\sharp}\in\mathbb{N}$ and a decomposition

$\Lambda=\widehat{\Lambda}\cup f(\widehat{\Lambda})\cup\ldots\cup
f^{p^{\sharp}-1}(\widehat{\Lambda})$

such that
$\widehat{\Lambda},f(\widehat{\Lambda}),\ldots,f^{p^{\sharp}-1}(\widehat{\Lambda})$
are pairwise disjoint and $f^{p^{\sharp}}$-invariant compact sets, and for
each $0\leq i<p^{\sharp}$, the map $f^{p^{\sharp}}\colon
f^{i}(\widehat{\Lambda})\to f^{i}(\widehat{\Lambda})$ is topologically mixing.

The following result, that we state without proof (see, for example, [KH95]),
is a consequence of the fact that $f^{p^{\sharp}}\colon
f^{i}(\widehat{\Lambda})\to f^{i}(\widehat{\Lambda})$ has the specification
property for each $i$.

######  Claim 8.6 (Specification property).

There exists $\ell_{1}\in\mathbb{N}$ so that every finite family of orbit
segments which start and end in $\widehat{\Lambda}$ are
$\delta_{0}/10$-shadowed by a true orbit segment in $\Lambda$ with spacing
time $\ell_{1}$. Analogous result holds for orbit segments which start and end
in $f^{i}(\widehat{\Lambda})$, $i\in\\{1,\ldots,p^{\sharp}-1\\}$.

Let $a^{\sharp}=a^{\sharp}(\Lambda)$ as in Lemma 8.2. Let

(8.4) $\ell^{\sharp}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\ell_{1}+p^{\sharp}.$

The following claim provides the set $\mathcal{S}(n)$ and proves property (a)
in the proposition.

######  Claim 8.7 (The set $\mathcal{S}(n)$).

For every $\varepsilon>0$, there exists $b^{\sharp}>a^{\sharp}$ such that for
every $n>b^{\sharp}$, there exists a $(n,\delta_{0})$-separated set
$\mathcal{S}(n)$ which satisfies

  * •

$\operatorname{card}(\mathcal{S}(n))\geq e^{n(h_{\rm
top}(f,\Lambda)-\varepsilon)}$;

  * •

$\mathcal{S}(n)\subset f^{j}(\widehat{\Lambda})$ for some $0\leq
j<p^{\sharp}$;

  * •

$\mathcal{S}(n)$ is contained in $B_{a^{\sharp}}(y_{0},\delta_{0}/4)$ for some
point $y_{0}\in\Lambda.$

###### Proof.

Fix some $(a^{\sharp},\delta_{0}/4)$-separated set of maximal cardinality
$\widetilde{\mathcal{S}}\subset\Lambda$. The fact that
$\widetilde{\mathcal{S}}\subset\Lambda$ is an
$(a^{\sharp},\delta_{0}/4)$-separated set of maximal cardinality, together
with Claim 8.5, implies that

(8.5)
$\Lambda=\bigcup_{i=0}^{p^{\sharp}-1}f^{i}(\widehat{\Lambda})\subset\bigcup_{y\in\widetilde{\mathcal{S}}}B_{a^{\sharp}}(y,\delta_{0}/4).$

By the compactness of $M$, the set $\widetilde{\mathcal{S}}$ is finite.

As $f$ is $\delta_{0}$-entropy expansive, by [B72, Corollary 2.5], for every
$\varepsilon>0,$ there exists $n_{1}\in\mathbb{N}$ such that for every $n\geq
n_{1}$, every $(n,\delta_{0}/4)$-separated set
$\widehat{\mathcal{S}}(n)\subset\Lambda$ with maximal cardinality satisfies

(8.6) $e^{n(h_{\rm
top}(f,\Lambda)-\varepsilon/2)}\leq\operatorname{card}\big{(}\widehat{\mathcal{S}}(n)\big{)}\leq
e^{n(h_{\rm top}(f,\Lambda)+\varepsilon/2)}.$

Hence, it follows straightforwardly from (8.6), that there exists
$n_{2}\in\mathbb{N}$ such that for every $n\geq\max\\{n_{1},n_{2}\\}$, we have

$e^{n(h_{\rm
top}(f,\Lambda)-\varepsilon)}\leq\frac{1}{p^{\sharp}}\frac{\operatorname{card}\big{(}\widehat{\mathcal{S}}(n)\big{)}}{\operatorname{card}\big{(}\widetilde{\mathcal{S}}\big{)}}\leq
e^{n(h_{\rm top}(f,\Lambda)+\varepsilon)}.$

Let

$b^{\sharp}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\max\\{a^{\sharp},n_{1},n_{2}\\}.$

Consider now $n\geq b^{\sharp}$ and any $(n,\delta_{0})$-separated set with
maximal cardinality $\widehat{\mathcal{S}}(n)\subset\Lambda$. Hence, by the
pigeonhole principle applied to the cover in (8.5), there exist $0\leq
j<p^{\sharp}$ and $y_{0}\in\widetilde{\mathcal{S}}$ such that

$\begin{split}\operatorname{card}\big{(}\widehat{\mathcal{S}}(n)\cap
f^{j}(\widehat{\Lambda})\cap
B_{a^{\sharp}}(y_{0},\delta_{0}/4)\big{)}&\geq\frac{1}{p^{\sharp}}\frac{1}{\operatorname{card}\big{(}\widetilde{\mathcal{S}}\big{)}}\operatorname{card}(\widehat{\mathcal{S}}(n))\\\
&\geq e^{n(h_{\rm top}(f,\Lambda)-\varepsilon)}.\end{split}$

Taking $\mathcal{S}(n)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widehat{\mathcal{S}}(n)\cap f^{j}(\widehat{\Lambda})\cap
B_{a^{\sharp}}(y_{0},\delta_{0}/4)$, the claim follows. ∎

Without loss of generality, we can assume that in Claim 8.7 it holds $j=0$ and
hence $\mathcal{S}(n)\subset\widehat{\Lambda}$. The following is a consequence
of Claim 8.5.

######  Claim 8.8.

For every $n>b^{\sharp}$, there exists $j_{1}=j_{1}(n)$ satisfying $0\leq
j_{1}\leq p^{\sharp}-1$ and

$\mathcal{S}(n)\cup f^{n+j_{1}}(\mathcal{S}(n))\subset\widehat{\Lambda}.$

Let us now prove property (b) in the proposition.

######  Claim 8.9.

For every $n>b^{\sharp}$ and $N\in\mathbb{N}$, there exists an injective map
$\Psi\colon(\mathcal{S}(n))^{N}\to\Lambda$ satisfying properties (b1) and
(b2).

###### Proof.

We start with the definition of $\Psi$. Fix $N\in\mathbb{N}$ and
$n>b^{\sharp}$ and let

$\ell=\ell(n)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\ell_{1}+j_{1}(n)<\ell^{\sharp}=\ell_{1}+p^{\sharp}.$

Consider a tuple $(x_{1},\ldots,x_{N})\in\mathcal{S}(n)^{N}$. By construction
and by Claim 8.8, for every $1\leq i\leq N$, the point $x_{i}$ and the “end-
point” $f^{n+j_{1}(n)}(x_{i})$ are in $\widehat{\Lambda}$. Hence, by Claim 8.6
applied to $\widehat{\Lambda}$ and $\delta_{0}/10$ and the orbit segments

$x_{1},\ldots,f^{n+j_{1}(n)}(x_{1})\quad
x_{2},\ldots,f^{n+j_{2}(n)}(x_{2})\quad\ldots\quad
x_{N},\ldots,f^{n+j_{1}(n)}(x_{N}),$

there exists $x\in\Lambda$ such that for every $1\leq i\leq N$ and $0\leq
j<n$, one has in particular

(8.7) $d\big{(}f^{j+(i-1)(n+\ell)}(x),f^{j}(x_{i})\big{)}<\delta_{0}/10.$

We define a map $\Psi\colon(\mathcal{S}(n))^{N}\to\Lambda$ by letting

$\Psi(x_{1},\ldots,x_{N})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}x.$

Let us now verify properties (b1) and (b2). Given two $N$-tuples

$\mathfrak{X}=(x_{1},\ldots,x_{N}),\quad\mathfrak{X}^{\prime}=(x_{1}^{\prime},\ldots,x_{N}^{\prime})\in(\mathcal{S}(n))^{N},$

let $i_{0}$ be the integer such that $x_{i_{0}}\neq x_{i_{0}}^{\prime}$ and
$x_{i}=x_{i}^{\prime}$ for every $i<i_{0}$. By the definition of $\Psi$ and
(8.7), for every $i\leq i_{0}-1$ and $0\leq j<n$, one has that

$\begin{split}&d\big{(}f^{j+(i-1)(n+\ell)}\circ\Psi(\mathfrak{X}),f^{j+(i-1)(n+\ell)}\circ\Psi(\mathfrak{X}^{\prime})\big{)}\\\
&\leq
d\big{(}f^{j+(i-1)(n+\ell)}\circ\Psi(\mathfrak{X}),f^{j}(x_{i})\big{)}+d\big{(}f^{j}(x_{i}),f^{j+(i-1)(n+\ell)}\circ\Psi(\mathfrak{X}^{\prime})\big{)}\\\
&=d\big{(}f^{j+(i-1)(n+\ell)}\circ\Psi(\mathfrak{X}),f^{j}(x_{i})\big{)}+d\big{(}f^{j}(x_{i}^{\prime}),f^{j+(i-1)(n+\ell)}\circ\Psi(\mathfrak{X}^{\prime})\big{)}\\\
&<\delta_{0}/10+\delta_{0}/10=\delta_{0}/5.\end{split}$

This is property (b1).

To prove (b2), by Claim 8.7, $\mathcal{S}(n)$ is an $(n,\delta_{0})$-separated
set. Hence, as $x_{i_{0}}$ and $x_{i_{0}}^{\prime}$ are distinct points in
$\mathcal{S}(n)$, there exists $0\leq k_{0}<n$ such that

(8.8)
$d\big{(}f^{k_{0}}(x_{i_{0}}),f^{k_{0}}(x_{i_{0}}^{\prime})\big{)}>\delta_{0}.$

Moreover, as Claim 8.7 also asserts that $\mathcal{S}(n)\subset
B_{a^{\sharp}}(y_{0},\delta_{0}/4)$ for some point $y_{0}\in\Lambda$, for
every $k=0,\ldots,a^{\sharp}-1$, we get

$\begin{split}d\big{(}f^{k}(x_{i_{0}}),f^{k}(x_{i_{0}}^{\prime})\big{)}&\leq
d(f^{k}(x_{i_{0}}),f^{k}(y_{0}))+d(f^{k}(y_{0}),f^{k}(x_{i_{0}}^{\prime}))\leq\delta_{0}/4+\delta_{0}/4\\\
&=\delta_{0}/2<\delta_{0}.\end{split}$

Hence, by (8.8), we get $n>k_{0}\geq a^{\sharp}$. Then, we get

$\begin{split}&d\big{(}f^{(i_{0}-1)(n+\ell)+k_{0}}\circ\Psi(\mathfrak{X}),f^{(i_{0}-1)(n+\ell)+k_{0}}\circ\Psi(\mathfrak{X}^{\prime})\big{)}\\\
&\geq
d\big{(}f^{k_{0}}(x_{i_{0}}),f^{k_{0}}({x}_{i_{0}}^{\prime})\big{)}-d\big{(}f^{(i_{0}-1)(n+\ell)+k_{0}}\circ\Psi(\mathfrak{X}),f^{k_{0}}(x_{i_{0}})\big{)}\\\
&\phantom{\geq}-d\big{(}f^{(i_{0}-1)(n+\ell)+k_{0}}\circ\Psi(\mathfrak{X}^{\prime}),f^{k_{0}}(x_{i_{0}}^{\prime})\big{)}\\\
\text{\small(by \eqref{eq:separated-at-k0} and
\eqref{consspec})}\quad&>\delta_{0}-\delta_{0}/10-\delta_{0}/10=4\delta_{0}/5.\end{split}$

This proves property (b2). ∎

It remains to prove property (c). Let $r\leq r^{\sharp}$ and recall that
$r^{\sharp}\in(0,\delta_{0}/4)$. For every two different points
$x,y\in\operatorname{Im}(\Psi)$, by property (b2), there exists
$j_{0}>a^{\sharp}$ such that $d(f^{j_{0}}(x),f^{j_{0}}(y))>4\delta_{0}/5$. By
contradiction, suppose that there is
$z\in\mathcal{W}^{\mathrm{s}}(x,r)\cap\mathcal{W}^{\mathrm{s}}(y,r)$. Then, by
(8.2) for all $j\geq a^{\sharp}$ one has

$d(f^{j}(x),f^{j}(y))\leq d(f^{j}(x),f^{j}(z))+d(f^{j}(z),f^{j}(y))\leq
2r<\delta_{0}/2,$

which contradicts with $d(f^{j_{0}}(x),f^{j_{0}}(y))>4\delta_{0}/5$. This
finishes the proof of Proposition 8.4. ∎

##  9\. A fractal set with large entropy and forward center exponent zero

The heart of this section is the construction of a limsup set with _forward_
zero center Lyapunov exponent and large entropy, whose existence and
properties are stated in Theorem 9.1. This theorem contains combinatorial and
fractal properties (item (1)), proved in Section 9.4, and ergodic ingredients
(items (2) and (3)), shown in Sections 10.3 and 11, respectively. Theorem 9.1
is our main step towards the proof of Theorem 3.

We first introduce some terminology. In analogy to the classical definition of
Bowen balls (recall (8.1)), given $n\in\mathbb{N}$, $\varepsilon>0$, and a
center curve $\gamma$, let

$B^{\mathrm{cu}}_{n}(\gamma,\varepsilon)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\bigcap_{i=0}^{n-1}f^{-i}\Big{(}\overline{\mathcal{W}^{\mathrm{uu}}(f^{i}(\gamma),\varepsilon)}\Big{)}.$

Given $\delta>0$, sequences of natural numbers $(t_{k})_{k}$ and
$(T_{k})_{k}$, and a sequence $(\mathcal{E}_{k})_{k}$ of families of center
curves, let

(9.1)
$L=L\big{(}\delta,(t_{k})_{k},(T_{k})_{k},(\mathcal{E}_{k})_{k}\big{)}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\bigcap_{j\in\mathbb{N}}L_{j},\quad\text{ where }\quad
L_{j}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\bigcup_{\gamma_{j}\in\mathcal{E}_{j}}B^{\mathrm{cu}}_{t_{j-1}+T_{j}}\big{(}\gamma_{j},\delta/2^{j-1}\big{)}.$

We will see in the proof of the following theorem that $(L_{k})_{k}$ is a
decreasing nested sequence of compact sets and therefore $L$ is compact
nonempty.

######  Theorem 9.1.

Let $f\in\operatorname{BM}^{1}(M)$. For every $\delta>0$ sufficiently small,
there are increasing sequences $(t_{k})_{k}$ and $(T_{k})_{k}$, and a sequence
$(\mathcal{E}_{k})_{k}$ of families of center curves such that the set
$L=L\big{(}\delta,(t_{k})_{k},(T_{k})_{k},(\mathcal{E}_{k})_{k}\big{)}$ in
(9.1) has the following properties:

  1. (1)

$L$ is a compact nonempty set that consists of pairwise disjoint non-
degenerate compact center curves $\gamma_{\infty}$; each $\gamma_{\infty}$ is
$(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\xi,2\delta))$-complete for some
$\xi\in\mathcal{E}_{1}$,

  2. (2)

$h_{\rm top}(f,L)=\mathfrak{h}(f)$, where $\mathfrak{h}(f)$ is given in (2.2),

  3. (3)

for every $x\in L$,
$\lim_{n\to+\infty}\frac{1}{n}\log\|D^{\mathrm{c}}f^{n}(x)\|=0$ and this
convergence is uniform on $L$.

Note that Theorem 9.1 does not yet provide any estimate of the entropy of
$\mathcal{L}(0)$, as $L$ only contains points whose _forward_ center exponent
is zero. The estimate of the entropy of $\mathcal{L}(0)$ will be completed in
Section 12.

The following result is an immediate consequence of the constructions in the
proof of Theorem 9.1, see Section 10.4 for brief arguments. Compare [T21] for
further references on the concept of unstable entropy and related results.

######  Scholium 9.2.

Assume the hypotheses of Theorem 9.1. Given $\delta>0$ sufficiently small, let
$L=L\big{(}\delta,(t_{k})_{k},(T_{k})_{k},(\mathcal{E}_{k})_{k}\big{)}$ as
provided by Theorem 9.1. Then for every $\xi\in\mathcal{E}_{1}$ and $x\in\xi$,

$h_{\rm top}\big{(}f,\mathcal{W}^{\mathrm{uu}}(x,2\delta)\cap
L\big{)}=\mathfrak{h}(f).$

This section is devoted to proving item (1) in Theorem 9.1. For that, in
Section 9.1, we choose the sequences $(t_{k})_{k}$ and $(T_{k})_{k}$ and the
sequence of center curves $(\mathcal{E}_{k})_{k}$ having controlled behavior,
see Proposition 9.3. Its proof is given in Section 9.2. In Section 9.3 we
investigate separation properties of the curves in $(\mathcal{E}_{k})_{k}$. In
Section 9.4, we complete the proof of item (1) in Theorem 9.1. Throughout this
section, we let $f\in\operatorname{BM}^{1}(M)$ with a stable blender-horseshoe
$\mathfrak{B}^{-}$.

###  9.1. Families of center curves with prescribed behavior

In this section, we construct families of curves $(\mathcal{E}_{k})_{k}$. The
concatenation method in Proposition 7.1 plays a key role.

Let $\delta_{0}>0$ be as in Remark 8.1. Recall the choice of the quantifier
$r^{\mathrm{u}}_{2}$ in Remark 6.11.

######  Proposition 9.3 (Families of center curves).

Let $f\in\operatorname{BM}^{1}(M)$. For every

(9.2)
$\delta\in\Big{(}0,\min\big{\\{}\frac{1}{20}\delta_{0},\frac{1}{2}r^{\mathrm{u}}_{2}\big{\\}}\Big{)},$

there are sequences $(t_{k})_{k}$ and $(T_{k})_{k}$ of natural numbers,
sequences of families of center curves $(\mathcal{D}_{k})_{k}$ and
$(\mathcal{E}_{k})_{k}$, and bijections

$\phi_{k}\colon\mathcal{E}_{k}\times\mathcal{D}_{k+1}\to\mathcal{E}_{k+1}$

such that for every $\gamma_{k}\in\mathcal{E}_{k}$,
$\xi_{k+1}\in\mathcal{D}_{k+1}$, and
$\gamma_{k+1}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\phi_{k}(\gamma_{k},\xi_{k+1})$,

  1. (1)

$\gamma_{k+1}$ is
$(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\gamma_{k},\delta/2^{k}))$-complete,

  2. (2)

$\mathcal{W}^{\mathrm{uu}}(f^{i}(\gamma_{k+1}),\delta/2^{k})\subset\mathcal{W}^{\mathrm{uu}}(f^{i}(\gamma_{k}),\delta/2^{k-1})$
for $i=0,\ldots,t_{k-1}+T_{k}$,

  3. (3)

$f^{t_{k}}(\gamma_{k+1})\subset\mathcal{W}^{\mathrm{ss}}(\xi_{k+1},\delta/2^{k})$,

  4. (4)

$f^{t_{k-1}}(\gamma_{k+1})$ is
$\big{(}\mathrm{c},\mathcal{W}^{\mathrm{uu}}(f^{t_{k-1}}(\gamma_{k}),\delta/2^{k})\big{)}$-complete.

Moreover,

  1. (5)

$\gamma_{k}$ is $(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\xi,\delta))$-complete
for some $\xi\in\mathcal{D}_{1}$.

Before proving this proposition, let us first introduce some terminology.

######  Definition 9.4.

Given $\gamma_{k+1}\in\phi_{k}(\\{\gamma_{k}\\}\times\mathcal{D}_{k+1})$, we
call $\gamma_{k}$ a _predecessor_ of $\gamma_{k+1}$ and $\gamma_{k+1}$ a
_successor_ of $\gamma_{k}$. Given $\ell>k$, we call
$\gamma_{k}\in\mathcal{E}_{k}$ a _$(\ell-k)$ -predecessor_ of
$\gamma_{\ell}\in\mathcal{E}_{\ell}$ if there exist curves
$\\{\gamma_{i}\in\mathcal{E}_{i}\colon k+1\leq i\leq\ell-1\\}$ such that
$\gamma_{i}$ is a predecessor of $\gamma_{i+1}$ for every $k\leq i\leq\ell-1$.
In this case, $\gamma_{\ell}$ is a _$(\ell-k)$ -successor_ of $\gamma_{k}$.

###  9.2. Proof of Proposition 9.3

In Section 9.2.1, we fix some quantifiers. In Section 9.2.2, we define the
families $\mathcal{D}_{k}$. The proof is completed in Section 9.2.3.

####  9.2.1. Choice of quantifiers

Let $\delta>0$ be as in (9.2). Fix a sequence $(\varepsilon_{k})_{k}$
monotonically decreasing to zero.

Recall that, by Theorem 1, one has

$\lim_{\beta\nearrow 0}\sup\big{\\{}h_{\mu}(f)\colon\mu\in\mathcal{M}_{\rm
erg}(f),\chi^{\mathrm{c}}(\mu)=\beta\big{\\}}=\mathfrak{h}(f).$

Thus, there exist ergodic measures $\nu\in\mathcal{M}_{\rm erg}(f)$ such that
$\chi^{\mathrm{c}}(\nu)$ is negative and arbitrarily close to zero and
$h_{\nu}(f)$ is arbitrarily close to $\mathfrak{h}(f)$. Hence, together with
Lemma 8.3, this implies that there exist a sequence $(\Lambda_{k})_{k}$ of
basic sets of contracting type and sequences of real numbers $\chi_{k}<0$,
$r_{k}\in(0,r^{\sharp}(\Lambda_{k}))$, $\varepsilon_{k}>0$, and $K_{k}\geq 1$
such that

  1. (i)

$h_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}h_{\rm
top}(f,\Lambda_{k})\geq\mathfrak{h}(f)-\varepsilon_{k}$,

  2. (ii)

$\chi_{k}$, $\varepsilon_{k}/\chi_{k}$, and $r_{k}$ are strictly monotonic and
tend to zero as $k\to\infty$,

  3. (iii)

every center curve centered at $x\in\Lambda_{k}$ of inner radius less than
$r_{k}$ is contained in the stable manifold of $x$,

  4. (iv)

for every center curve $\gamma$ centered at $x\in\Lambda_{k}$ of inner radius
less than $r_{k}$, for every
$y\in\mathcal{W}^{\mathrm{ss}}\big{(}\gamma,\delta\big{)}$ and
$n\in\mathbb{N}$, one has

$K_{k}^{-1}\cdot
e^{-n(\chi_{k}-\varepsilon_{k})}\leq\|D^{\mathrm{c}}f^{n}(y)\|\leq K_{k}\cdot
e^{n(\chi_{k}+\varepsilon_{k})}.$

Let

$\begin{split}&a_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}a^{\sharp}(\Lambda_{k})\quad\text{ as in Lemma \ref{l.local-stable-
manifold}},\\\ &\ell^{\sharp}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\ell^{\sharp}(\Lambda_{k})\text{ and
}b^{\sharp}_{k}=b^{\sharp}(\Lambda_{k},\varepsilon_{k})\quad\text{ as in
Proposition \ref{p.separation-in-symbol},}\\\
&\ell^{\flat}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\ell^{\flat}(\delta,\varepsilon_{k})\quad\text{ the distortion-
control time in Lemma \ref{lemsomecount}}.\end{split}$

Recall the constants $N(\cdot)$ and $R(\cdot)$ in Lemma 6.6, and
$r^{\sharp}(\cdot)$ in Lemma 8.2. Up to shrinking $r_{k}$, we can assume that
for every $k$

(9.3) $r_{k}\leq r^{\sharp}(\Lambda_{k}),\quad
2r_{k}<R(N(\delta/2^{k-1})),\quad\text{ and
}\quad\sum_{k}r_{k}<\delta_{0}/20.$

Recall the constant $\eta^{\mathrm{s}}(\cdot)$ in Remark 6.7. For every
$k\in\mathbb{N}$, consider the corresponding numbers as in (7.2),

(9.4) $\begin{split}&m_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}m(2r_{k+1},\delta/2^{k})=2N(\delta/2^{k})+M_{\text{\rm
blend}}\big{(}\eta^{\mathrm{s}}(2r_{k+1},\delta/2^{k})\big{)},\\\
&T^{\sharp}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}T^{\sharp}(2r_{k},K_{k},\varepsilon_{k},\delta/2^{k})=\left\lceil\frac{1}{\chi_{k}+\varepsilon_{k}}\big{(}\log
R(N(\delta/2^{k}))-\log K_{k}-\log{(2r_{k})}\big{)}\right\rceil+1.\end{split}$

Let

(9.5) $C_{\rm max}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}2\cdot\sup_{x\in M}\big{|}\log\|D^{\mathrm{c}}f(x)\|\big{|}.$

We choose a sequence of natural numbers $(n_{k})_{k}$ such that

(9.6)
$\displaystyle\max\big{\\{}\ell^{\flat}_{k},b^{\sharp}_{k},T^{\sharp}_{k}\big{\\}}<n_{k},$
(9.7) $\displaystyle\displaystyle\frac{m_{k}}{n_{k}}C_{\rm
max}<\varepsilon_{k},$ (9.8)
$\displaystyle\displaystyle\frac{n_{k}}{n_{k}+\ell_{k}^{\sharp}+m_{k}}\geq
1-\varepsilon_{k},$ (9.9) $\displaystyle\displaystyle\frac{\log
K_{k+1}}{n_{k}}<\varepsilon_{k},$ (9.10)
$\displaystyle\displaystyle\frac{1}{n_{k}}\big{(}\log
K_{k+1}+(m_{k+1}+\ell^{\flat}_{k+1})\cdot C_{\rm max}\big{)}<\varepsilon_{k}.$

Inequalities (9.7) and (9.8) are used to control the entropy in Section 10.2.
Inequalities (9.9) and (9.10) are used to control the exponents in Section 11.

Let $(N_{k})_{k}$ be a sufficiently fast growing sequence of natural numbers
(to be specified in (9.14)–(9.16)). By (9.6), we can apply Proposition 8.4 to
$\Lambda_{k}$ and $\varepsilon_{k}$ and obtains an integer
$\ell_{k}\in\\{0,\ldots,\ell_{k}^{\sharp}\\}$, an
$(n_{k},\delta_{0})$-separated set

(9.11) $\mathcal{S}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathcal{S}(n_{k})\subset\Lambda_{k},\quad\text{ with
}\quad\operatorname{card}(\mathcal{S}_{k})\geq
e^{n_{k}(h_{k}-\varepsilon_{k})},$

and an injective map

(9.12) $\Psi_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Psi^{n_{k},N_{k}}\colon\mathcal{S}_{k}^{N_{k}}\to\Lambda_{k}.$

######  Notation 9.5.

For every $k\in\mathbb{N}$, let

(9.13) $\begin{split}t_{0}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}0\\\ t_{k}&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}t_{k-1}+(T_{k}+m_{k})=\sum_{i=1}^{k}\big{(}T_{i}+m_{i}\big{)},\quad\text{
where }\quad T_{i}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}N_{i}(n_{i}+\ell_{i}).\end{split}$

Choosing $(N_{k})_{k}$ in the way that it is growing fast enough, not
affecting the choices in (9.6)–(9.10), we can assume that

(9.14)
$\displaystyle\displaystyle\frac{n_{k+1}+\ell_{k+1}^{\sharp}}{t_{k}}<\varepsilon_{k},$
(9.15) $\displaystyle\displaystyle\frac{t_{k}}{t_{k+1}}<\frac{1}{C_{\rm
max}}\varepsilon_{k+1}.$ (9.16) $\displaystyle\displaystyle\frac{\log
K_{k}}{N_{k}(n_{k}+\ell_{k})}<\varepsilon_{k},$

For further reference, note that (9.13) and (9.6) imply

(9.17) $T_{k}>T_{k}^{\sharp}.$

We provide an informal description of the above-defined numbers.

######  Remark 9.6.

The number $n_{k}$, chosen large enough, enables to compensate distortion and
enables uniform separation. The number $\ell_{k}$ is a “bridging time” for
$\Lambda_{k}$ associated to the “spacing time” in the specification property
of $f$ on $\Lambda_{k}$ and to “return to a mixing piece of the spectral
decomposition of $\Lambda$” (see Proposition 8.4). During the time interval
$\\{0,\ldots,T_{k}\\}$ the orbit of any point in the image of $\Psi_{k}$ stays
close to the hyperbolic set $\Lambda_{k}$ “catching entropy and Lyapunov
exponent from the hyperbolic set”. The time $m_{k}$ is the one to “get into
the blender $\mathfrak{B}^{-}$, stay there, and get out again”.

####  9.2.2. Definition of the families $\mathcal{D}_{k}$

######  Lemma 9.7.

For every $k\in\mathbb{N}$ and
$x\in\operatorname{Im}(\Psi_{k})\subset\Lambda_{k}$, there is a center curve
$\xi(x,r_{k})$ centered at $x$ of inner radius $r_{k}$ and contained in the
stable manifold of $x$ of size $r_{k}$. Moreover, the family

$\mathcal{D}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\xi(x,r_{k})\colon
x\in\operatorname{Im}(\Psi_{k})\big{\\}}$

consists of pairwise disjoint curves.

###### Proof.

Given $x\in\operatorname{Im}(\Psi_{k})$, the existence of the center curves
follows from Remark 5.1. By (9.3), we have $r_{k}<r^{\sharp}(\Lambda_{k})$.
Hence, by item (c) in Proposition 8.4, the center curves in $\mathcal{D}_{k}$
are pairwise disjoint. ∎

######  Remark 9.8.

We have
$\operatorname{card}\mathcal{D}_{k}=\big{(}\operatorname{card}\mathcal{S}_{k}\big{)}^{N_{k}}$.

####  9.2.3. End of the proof

Given $(\mathcal{D}_{k})_{k}$ as in Lemma 9.7, we will construct the families
of curves $(\mathcal{E}_{k})_{k}$ and the maps $(\phi_{k})_{k}$. For that, we
proceed inductively using Proposition 7.1 to concatenate the center curves in
$(\mathcal{D}_{k})_{k}$. Set

$\mathcal{E}_{1}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathcal{D}_{1}=\big{\\{}\xi(x,r_{1})\colon
x\in\operatorname{Im}(\Psi_{1})\big{\\}}.$

By property (iv) in Section 9.2.1, every center curve
$\gamma_{1}=\xi_{1}\in\mathcal{E}_{1}=\mathcal{D}_{1}$ satisfies the
hypothesis (7.1) of Proposition 7.1. By definition of $\mathcal{D}_{k}$ in
Lemma 9.7, every $\xi_{2}\in\mathcal{D}_{2}$ has length
$2r_{2}<R(N(\delta/2))$, recall (9.3). By (9.17), we have
$T_{1}>T_{1}^{\sharp}$. We now apply Proposition 7.1 to the center curves
$\gamma_{1}$ and $\xi_{2}$ (the latter has length
$\operatorname{length}(\zeta_{2})=2r_{2}$) and to the number $\delta/2$ and
get the associated center curves $\widehat{\zeta}_{2}$ and
$\widetilde{\zeta}_{2}$ and the number $m_{1}$ as in (9.4). Let

$\phi_{1}(\gamma_{1},\xi_{2})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\gamma_{2},\quad\text{ where
}\quad\gamma_{2}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\widehat{\zeta}_{2}\quad\text{ and }\quad
f^{T_{1}+m_{1}}(\gamma_{2})=\widetilde{\zeta}_{2}.$

\begin{overpic}[scale={.45}]{P_cascade-2.pdf}
\put(64.0,3.0){\small$f^{T_{2}+m_{2}}$}
\put(22.0,3.0){\small$f^{T_{1}+m_{1}}=f^{t_{1}}$}
\put(18.0,25.5){\small$\gamma_{3}$}
\put(18.0,29.5){\small$\gamma_{2}=\widehat{\zeta}_{2}$}
\put(18.0,39.0){\small$\gamma_{1}=\xi_{1}$}
\put(58.0,25.0){\small$f^{T_{1}+m_{1}}(\gamma_{2})=\widetilde{\zeta}_{2}$}
\put(58.0,16.0){\small$\widehat{\zeta}_{3}$} \put(66.0,32.3){\small$\xi_{2}$}
\put(18.0,50.0){\small$\mathcal{W}^{\mathrm{uu}}(\xi_{1},\delta)$}
\put(62.0,44.0){\small$\mathcal{W}^{\mathrm{ss}}(\xi_{2},\delta/2)$}
\put(92.0,39.5){\small$\mathcal{W}^{\mathrm{ss}}(\xi_{3},\delta/4)$}
\put(94.0,30.0){\small$\xi_{3}$}
\put(90.0,26.0){\small$\widetilde{\zeta}_{3}$} \end{overpic} Figure 4. Proof
of Proposition 9.3

Moreover,

  * (i)

$\gamma_{2}$ is
$\big{(}\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\gamma_{1},\delta/2)\big{)}$-complete,

  * (ii)

$f^{i}(\gamma_{2})\subset\mathcal{W}^{\mathrm{uu}}\big{(}f^{i}(\gamma_{1}),\delta/2\big{)}$
for $i=0,\dots,T_{1}$,

  * (iii)

$f^{T_{1}+m_{1}}(\gamma_{2})\subset\mathcal{W}^{\mathrm{ss}}(\xi_{2},\delta/2)$,

  * (iv)

$f^{-(T_{1}+m_{1})}(\mathcal{W}^{\mathrm{uu}}(f^{T_{1}+m_{1}}(\gamma_{2}),\delta))\subset\mathcal{W}^{\mathrm{uu}}(\gamma_{2},\delta/2)$.

Denote by $\mathcal{E}_{2}$ the set of all curves $\gamma_{2}$ constructed in
this way,

$\mathcal{E}_{2}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\phi_{1}(\gamma_{1},\xi_{2})\colon\gamma_{1}\in\mathcal{E}_{1}=\mathcal{D}_{1},\xi_{2}\in\mathcal{D}_{2}\big{\\}}.$

Recalling (9.13), properties (i)–(iv) above prove properties (1), (2), (3),
and (4) of the assertion for $k=1$ of Proposition 9.3. Moreover, property (i)
is property (5).

Let us now proceed inductively. Assume that $k\geq 2$ and that the families
$\mathcal{E}_{1},\ldots,\mathcal{E}_{k}$ of center curves and the maps
$\phi_{1},\ldots,\phi_{k-1}$ are already defined with the numbers in (9.13)
and have the properties (1)–(4) of the proposition. Let us define
$\mathcal{E}_{k+1}$ and $\phi_{k}$. Given any $\gamma_{k}\in\mathcal{E}_{k}$,
by property (3) there is $\xi_{k}\in\mathcal{D}_{k}$ such that

$f^{t_{k-1}}(\gamma_{k})\subset\mathcal{W}^{\mathrm{ss}}(\xi_{k},\delta/2^{k-1}).$

Hence, by item (iv) in Section 9.2.1, the center curve
$f^{t_{k-1}}(\gamma_{k})$ satisfies the hypotheses (7.1) of Proposition 7.1.
By (9.17), we have $T_{k+1}>T_{k+1}^{\sharp}$. Given any
$\xi_{k+1}\in\mathcal{D}_{k+1}$ and applying Proposition 7.1 to the center
curves $\xi_{k+1}$ and $\xi_{k+1}$ and to the number $\delta/2^{k}$, we get
associated center curves $\widehat{\zeta}_{k+1}$ and $\widetilde{\zeta}_{k+1}$
and the number $m_{k+1}$ as in (9.4) having the posited properties. Let

$\phi_{k}(\gamma_{k},\xi_{k+1})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\gamma_{k+1}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}f^{-t_{k-1}}(\widehat{\zeta}_{k+1}).$

Finally, let $\mathcal{E}_{k+1}$ be the set of all center curves constructed
in this way,

$\mathcal{E}_{k+1}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\big{\\{}\phi_{k}(\gamma_{k},\xi_{k+1})\colon\gamma_{k}\in\mathcal{E}_{k},\xi_{k+1}\in\mathcal{D}_{k+1}\big{\\}}.$

As for the case $k=1$, Proposition 7.1 implies that the properties (1)–(4) of
the assertion hold for $k+1$. This proves the first part of the proposition.

It remains to show property (5). Recall that
$\mathcal{D}_{1}=\mathcal{E}_{1}$. By construction, there exist center curves
$\gamma_{i}\in\mathcal{E}_{i}$ for $1\leq i\leq k-1$ such that $\gamma_{i+1}$
is $(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\gamma_{i},\delta/2^{i}))$-complete
for $1\leq i\leq k-1.$ Therefore, for any point $x_{k}\in\gamma_{k}$, there
exist $x_{i}\in\gamma_{i}$, for $1\leq i\leq k-1$, such that
$x_{i+1}\in\mathcal{W}^{\mathrm{uu}}(x_{i},\delta/2^{i})$ for $1\leq i\leq
k-1$. Furthermore, if $x_{k}$ is an endpoint of $\gamma_{k}$, then $x_{i}$ is
an endpoint of $\gamma_{i}$ for $1\leq i\leq k-1$, due to the definition of
being
$(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\gamma_{i},\delta/2^{i}))$-complete.
This implies that $x_{k}\in\mathcal{W}^{\mathrm{uu}}(x_{1},\delta)$. By the
arbitrariness of $x_{k}$, one has
$\gamma_{k}\subset\mathcal{W}^{\mathrm{uu}}(\gamma_{1},\delta)$. Furthermore,
if $x_{k}$ is an endpoint $\gamma_{k}$, then $x_{1}$ is an endpoint of
$\gamma_{1}$. Thus, $\gamma_{k}$ is
$(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\gamma_{1},\delta))$-complete. This
proves the proposition. ∎

######  Remark 9.9.

With $\Psi_{k}$ in (9.12) and $\phi_{k}$ in Proposition 9.3, let

$\Phi_{k}\colon\mathcal{E}_{k}\times\mathcal{S}_{k+1}^{N_{k+1}}\to\mathcal{E}_{k+1},\quad\Phi_{k}(\gamma_{k},\mathfrak{X})\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\phi_{k}\big{(}\gamma_{k},\xi(\Psi_{k+1}(\mathfrak{X}),r_{k+1})\big{)}.$

By definition, $\Phi_{k}$ and $\phi_{k}$ are both onto.

The following result follows from the proof of Proposition 9.3 replacing each
$\mathcal{D}_{k}$ by just one curve $\\{\xi_{k}\\}$. It will be used in
Section 12.

######  Scholium 9.10.

Consider the sequence $(\Lambda_{k})_{k}$ in Section 9.2.1. Let
$(\xi_{k})_{k}$ be sequence of center curves $\xi_{k}=\xi(x_{k},r_{k})$
centered at $x_{k}\in\Lambda_{k}$ and of inner radius $r_{k}$. There is a
sequence $(\gamma_{k})_{k}$ satisfying the assertions of Proposition 9.3.

###  9.3. Separated properties of the center curves $(\mathcal{E}_{k})_{k}$.

We start by studying some properties of the families $\mathcal{E}_{k}$ and the
maps $\Phi_{k}$. In particular, we study the “separation” of the center
curves. In analogy to the usual definition for points (recall (3.1)), two
curves $\gamma,\gamma^{\prime}$ are _$(n,\varepsilon)$ -separated_ if for
every $x\in\gamma$ and $x^{\prime}\in\gamma^{\prime}$, one has
$d_{n}(x,x^{\prime})>\varepsilon$.

The main aim of this subsection is to prove the following result.

######  Proposition 9.11.

For every $k$, one has

  * •

$\mathcal{E}_{k}$ is a family of
$\big{(}t_{k-1}+T_{k},\delta_{0}/2\big{)}$-separated center curves,

  * •

the maps $\Phi_{k}$ and $\phi_{k}$ (see Remark 9.9 and Proposition 9.3) are
bijective.

We postpone the proof of the above result to the end of this section.

The following result is now immediate.

######  Corollary 9.12.

For every $\ell>k$,

$\operatorname{card}\mathcal{E}_{\ell}=\operatorname{card}\mathcal{E}_{k}\cdot\prod_{j=k+1}^{\ell}\operatorname{card}\mathcal{D}_{j}.\quad\text{
In particular,
}\quad\operatorname{card}\mathcal{E}_{\ell}=\prod_{j=1}^{\ell}\mathcal{D}_{j}.$

The following refines Corollary 9.12, we prove it at the end of this
subsection.

######  Corollary 9.13.

For every $\ell>k$, each $\gamma_{\ell}\in\mathcal{E}_{\ell}$ has a unique
$(\ell-k)$-predecessor in $\gamma_{k}\in\mathcal{E}_{k}$, and $\gamma_{\ell}$
is
$(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\gamma_{k},\delta/2^{k-1}))$-complete.

Recall our notation (8.3) for distance between curves.

######  Lemma 9.14 (Separation of curves in $\mathcal{D}_{k}$).

For every $k\in\mathbb{N}$ and distinct curves
$\xi,\xi^{\prime}\in\mathcal{D}_{k}$ there are numbers $N$ and $j$,

(9.18) $1\leq N\leq N_{k}\quad\text{ and
}\quad(N-1)(n_{k}+\ell_{k})+a^{\sharp}_{k}<j<(N-1)(n_{k}+\ell_{k})+n_{k},$

such that for every $x\in\xi$ and $x^{\prime}\in\xi^{\prime}$, one has

$d(f^{j}(x),f^{j}(x^{\prime}))>4\delta_{0}/5-2r_{k}.$

Moreover, the curves $\xi,\xi^{\prime}$ are
$(T_{k},4\delta_{0}/5-2r_{k})$-separated.

###### Proof.

Consider the curves $\xi=\xi(x_{k},r_{k})$ and
$\xi^{\prime}=\xi(x_{k}^{\prime},r_{k})$ in $\mathcal{D}_{k}$, where
$x_{k},x_{k}^{\prime}\in\operatorname{Im}(\Psi_{k})$. By item (b2) in
Proposition 8.4, there exist $N$ and $j$ as in (9.18) such that

$d(f^{j}(x_{k}),f^{j}(x_{k}^{\prime}))>4\delta_{0}/5.$

Let $x\in\xi$ and $x^{\prime}\in\xi^{\prime}$. Note that (9.18) implies
$j>a^{\sharp}_{k}$ and hence, as we assume $r_{k}\in(0,r^{\sharp}_{k})$, by
Lemma 8.2 item (2), one has

$d(f^{j}(x),f^{j}x_{k}))<r_{k}\quad\text{~{}and~{}}\quad
d(f^{j}(x^{\prime}),f^{j}(x_{k}^{\prime}))<r_{k}.$

Therefore, one has

(9.19) $\begin{split}d(f^{j}(x),f^{j}(x^{\prime}))&\geq
d(f^{j}(x_{k}),f^{j}(x_{k}^{\prime}))-d(f^{j}(x),f^{j}(x_{k}))-d(f^{j}(x^{\prime}),f^{j}(x_{k}^{\prime}))\\\
&>4\delta_{0}/5-2r_{k},\end{split}$

that is, $\xi,\xi^{\prime}$ are $(j,4\delta_{0}/5-2r_{k})$-separated.

Finally, recall that $T_{k}=N_{k}(n_{k}+\ell_{k})$ and hence $N\leq N_{k}$
implies $T_{k}\geq j$. This together with (9.19) implies that
$\xi,\xi^{\prime}$ are $(T_{k},4\delta_{0}/5-2r_{k})$-separated. This proves
the lemma. ∎

Now, we turn to the separated properties of curves in $\mathcal{E}_{k}.$

######  Lemma 9.15 (Separation of curves in $\mathcal{E}_{k}$).

For every $k\geq 1$, the curves in $\mathcal{E}_{k}$ are pairwise

$\big{(}t_{k-1}+T_{k},4\delta_{0}/5-\sum_{j=0}^{k-2}\delta/2^{j}-2\sum_{j=1}^{k}r_{j}\big{)}\text{-separated},$

where for $k=1$ the first sum is disregarded. In particular, the curves in
$\mathcal{E}_{k}$ are pairwise $(t_{k-1}+T_{k},3\delta_{0}/5)$-separated.

###### Proof.

By definition, $\mathcal{E}_{1}=\mathcal{D}_{1}$. Then the case for $k=1$
follows directly from Lemma 9.14.

Assume that the assertion holds for $\mathcal{E}_{1},\ldots,\mathcal{E}_{k}$.
In order to check the assertion for $k+1$, consider curves
$\gamma_{k+1}=\phi_{k}(\gamma_{k},\xi_{k+1})$ and
$\gamma_{k+1}^{\prime}=\phi_{k}(\gamma_{k}^{\prime},\xi_{k+1}^{\prime})$ in
$\mathcal{E}_{k+1}$, where

$(\gamma_{k},\xi_{k+1}),(\gamma_{k}^{\prime},\xi_{k+1}^{\prime})\in\mathcal{E}_{k}\times\mathcal{D}_{k+1},$

There the following two cases to be considered.

Case $\xi_{k+1}\neq\xi_{k+1}^{\prime}$. By Lemma 9.14, there exist $N,j$ with

$1\leq N\leq N_{k+1}\quad\text{ and
}\quad(N-1)(n_{k+1}+\ell_{k+1})+a^{\sharp}_{k+1}<j<(N-1)(n_{k+1}+\ell_{k+1})+n_{k+1}$

so that for every $y\in\xi_{k+1}$ and $y^{\prime}\in\xi_{k+1}^{\prime}$ and

(9.20) $d(f^{j}(y),f^{j}(y^{\prime}))>4\delta_{0}/5-2r_{k+1}.$

Item (3) in Proposition 9.3 gives that

$f^{t_{k}}(\gamma_{k+1})\subset\mathcal{W}^{\mathrm{ss}}(\xi_{k+1},\delta/2^{k})\quad\textrm{~{}and~{}}\quad
f^{t_{k}}(\gamma_{k+1}^{\prime})\subset\mathcal{W}^{\mathrm{ss}}(\xi_{k+1}^{\prime},\delta/2^{k}).$

For any $z\in\gamma_{k+1}$ and $z^{\prime}\in\gamma_{k+1}^{\prime}$, take
$y\in\xi_{k+1}$ and $y^{\prime}\in\xi_{k+1}^{\prime}$ such that
$f^{t_{k}}(z)\subset\mathcal{W}^{\mathrm{ss}}(y,\delta/2^{k})$ and
$f^{t_{k}}(z^{\prime})\subset\mathcal{W}^{\mathrm{ss}}(y^{\prime},\delta/2^{k}).$
By the uniform contraction along the strong stable foliation
$\mathcal{W}^{\mathrm{ss}}$, one has

$\displaystyle d(f^{t_{k}+j}(z),f^{t_{k}+j}(z^{\prime}))$ $\displaystyle\geq
d(f^{j}(y),f^{j}(y^{\prime}))-d(f^{t_{k}+j}(z),f^{j}(y))-d(f^{t_{k}+j}(z^{\prime}),f^{j}(y^{\prime}))$
(by (9.20)) $\displaystyle\geq
4\delta_{0}/5-2r_{k+1}-2\delta/2^{k}=4\delta_{0}/5-2r_{k+1}-\delta/2^{k-1}.$

Note that

$4\delta_{0}/5-2r_{k+1}-\delta/2^{k-1}>4\delta_{0}/5-\sum_{j=0}^{k-1}\delta/2^{j}-2\sum_{j=1}^{k+1}r_{j}.$

This proves the lemma in this case.

Case $\gamma_{k}\neq\gamma_{k}^{\prime}$. By the induction hypothesis, the
curves $\gamma_{k}$ and $\gamma_{k}^{\prime}$ belong to $\mathcal{E}_{k}$ and
are
$\big{(}t_{k-1}+T_{k},4\delta_{0}/5-\sum_{j=0}^{k-2}\delta/2^{j}-2\sum_{j=1}^{k}r_{j}\big{)}$-separated.
By item (2) in Proposition 9.3, for any $0\leq i\leq t_{k-1}+T_{k}-1$, one has

$f^{i}(\gamma_{k+1})\subset\mathcal{W}^{\mathrm{uu}}(f^{i}(\gamma_{k}),\delta/2^{k})\quad\text{
and }\quad
f^{i}(\gamma_{k+1}^{\prime})\subset\mathcal{W}^{\mathrm{uu}}(f^{i}(\gamma_{k}^{\prime}),\delta/2^{k}).$

Hence, $\gamma_{k+1}$ and $\gamma_{k+1}^{\prime}$ are
$\big{(}t_{k-1}+T_{k},4\delta_{0}/5-\sum_{j=0}^{k-1}\delta/2^{j}-2\sum_{j=1}^{k}r_{j}\big{)}$-separated.
This proves the lemma in this case.

Note that

$\begin{split}4\delta_{0}/5-\sum_{j=0}^{k-1}\delta/2^{j}-2\sum_{j=1}^{k}r_{j}&\geq
4\delta_{0}/5-\sum_{j=0}^{\infty}\delta/2^{j}-2\sum_{j=1}^{\infty}r_{j}\\\
\text{\small{(by \eqref{eq5V} and
\eqref{fixdelta})}}\quad&>4\delta_{0}/5-2\delta-\delta_{0}/10>3\delta_{0}/5.\end{split}$

This finishes the proof of Lemma 9.15. ∎

######  Lemma 9.16.

Given two points
$\mathfrak{X}=(x_{1},\ldots,x_{N_{k}}),\mathfrak{X}^{\prime}=(x_{1}^{\prime},\ldots,x_{N_{k}}^{\prime})\in\mathcal{S}_{k}^{N_{k}}$,
let $N$ be the smallest integer such that $x_{N}\neq x_{N}^{\prime}$. Given
$\gamma_{k-1}\in\mathcal{E}_{k-1}$, let

$\gamma_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Phi_{k-1}(\gamma_{k-1},\mathfrak{X}),\quad\gamma_{k}^{\prime}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\Phi_{k-1}(\gamma_{k-1},\mathfrak{X}^{\prime}).$

Then there exists
$t_{k-1}+(N-1)(n_{k}+\ell_{k})+a^{\sharp}_{k}<j<t_{k-1}+(N-1)(n_{k}+\ell_{k})+n_{k}$
such that

$d(f^{j}(\gamma_{k}),f^{j}(\gamma_{k}^{\prime}))>3\delta_{0}/5.$

###### Proof.

By items (b2) and (c) in Proposition 8.4,
$\Psi_{k}(\mathfrak{X}),\Psi_{k}(\mathfrak{X}^{\prime})\in\mathcal{D}_{k}$ are
distinct center curves as in the assumption of Lemma 9.14. Combining with the
proof of Lemma 9.14 and arguing as in the proof of Lemma 9.15 for the case
where $\xi_{k+1}\neq\xi_{k+1}^{\prime}$, the assertion follows. ∎

###### Proof of Proposition 9.11.

The first assertion follows immediately from Lemma 9.15.

By construction, the maps $\Phi_{k}$ and $\phi_{k}$ are onto (recall Remark
9.9). So it is enough to prove that they are injective. By items (b) and (c)
in Proposition 8.4 and the definition of $\mathcal{D}_{k}$, the map
$\Psi_{k}:\mathcal{S}_{k}^{N_{k}}\to\mathcal{D}_{k}$ is bijective. Then it
follows from Lemma 9.16 that $\Phi_{k}$ and $\phi_{k}$ are injective. ∎

###### Proof of Corollary 9.13.

The first assertion is an immediate consequence of Proposition 9.11. By
Proposition 9.3 item (2),

$\mathcal{W}^{\mathrm{uu}}\big{(}f^{i}(\gamma_{k+1}),\delta/2^{k}\big{)}\subset\mathcal{W}^{\mathrm{uu}}\big{(}f^{i}(\gamma_{k}),\delta/2^{k-1}\big{)},\textrm{~{}for
every $0\leq i\leq t_{k-1}+T_{k}$.}$

By item (1) in Proposition 9.3, $\gamma_{\ell}$ is
$(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\gamma_{\ell-1},\delta/2^{\ell-1})$-complete.
Then one can inductively show that $\gamma_{\ell}\in\mathcal{E}_{\ell}$ is
$(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\gamma_{k},\delta/2^{k-1})$-complete. ∎

###  9.4. Proof of item (1) in Theorem 9.1

Proposition 9.3 gives the sequences $(t_{k})_{k}$, $(T_{k})_{k}$, and
$(\mathcal{E}_{k})_{k}$. This defines the set $L$. Let us show the properties
claimed in the theorem.

Assume that $\gamma_{k+1}$ is a successor of $\gamma_{k}$. By Proposition 9.3
item (2), one has

$f^{-i}\big{(}\mathcal{W}^{\mathrm{uu}}(f^{i}(\gamma_{k+1}),\delta/2^{k})\big{)}\subset
f^{-i}\big{(}\mathcal{W}^{\mathrm{uu}}(f^{i}(\gamma_{k}),\delta/2^{k-1})\big{)},\textrm{~{}for
every $0\leq i\leq t_{k-1}+T_{k}$.}$

Thus, one obtains

$B^{\mathrm{cu}}_{t_{k}+T_{k+1}}\big{(}\gamma_{k+1},\delta/2^{k}\big{)}\subset
B^{\mathrm{cu}}_{t_{k-1}+T_{k}}\big{(}\gamma_{k},\delta/2^{k-1}\big{)}.$

This proves that $L_{k+1}\subset L_{k}$ for all $k\in\mathbb{N}$. Hence, $L$
is a compact nonempty set.

By Proposition 9.11, the curves in $\mathcal{E}_{k}$ are
$(t_{k-1}+T_{k},\delta_{0}/2)$-separated. As by (9.2) we have
$\delta<\delta_{0}/20$, it follows that

$\big{\\{}B^{\mathrm{cu}}_{t_{k-1}+T_{k}}\big{(}\gamma_{k},\delta/2^{k-1}\big{)}\colon\gamma_{k}\in\mathcal{E}_{k}\big{\\}}$

is a collection of pairwise disjoint compact sets. By property (5) in
Proposition 9.3, each $\gamma_{k}$ is
$(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\xi,\delta))$-complete for some
$\xi\in\mathcal{D}_{1}$. By the uniform contraction of $f^{-1}$ along the
foliation $\mathcal{W}^{\mathrm{uu}}$, one deduces that $L$ consists of non-
degenerate compact center curves which are
$(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\xi,2\delta))$-complete. This proves
the theorem. ∎

##  10\. Proof of Theorem 9.1: Entropy estimates

Throughout this section, we assume the hypotheses of Theorem 9.1. Consider
$L=L\big{(}\delta,(t_{k})_{k},(T_{k})_{k},(\mathcal{E}_{k})_{k}\big{)}$ as in
(9.1), where $\delta$, $(t_{k})_{k}$, $(T_{k})_{k}$, and
$(\mathcal{E}_{k})_{k}$ are as in Proposition 9.3. In Section 10.3, we
complete the proof of item (2) in Theorem 9.1. To accomplish this, we employ
the entropy distribution principle in Appendix A. For that, we need the
auxiliary Proposition 10.1 below.

For each $k\in\mathbb{N}$, define the probability measure $\mu_{k}$ as

(10.1) $\mu_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{1}{\operatorname{card}\mathcal{E}_{k}}\sum_{\gamma_{k}\in\mathcal{E}_{k}}\operatorname{Leb}_{\gamma_{k}},$

where $\operatorname{Leb}_{\gamma_{k}}$ is the normalized Lebesgue measure on
$\gamma_{k}$. By definition, $\mu_{k}(L_{k})=1$.

######  Proposition 10.1.

The sequence $(\mu_{k})_{k}$ defined in (10.1) converges in the
weak$\ast$-topology to a probability measure $\mu$ such that $\mu(L)=1$.
Moreover, for every $\theta>0$ there exists $n_{0}\in\mathbb{N}$ such that for
every $n>n_{0}$ and every $x\in M$ satisfying $B_{n}(x,\delta/2)\cap
L\neq\emptyset$, one has

(10.2) $\mu(B_{n}(x,\delta/2))\leq e^{-n(\mathfrak{h}(f)-\theta)}.$

The convergence of $(\mu_{k})_{k}$ and the “local entropy estimate” of the
limit measure $\mu$ in Proposition 10.1 are proven in Sections 10.1 and 10.2,
respectively.

###  10.1. Convergence of the sequence of measures $(\mu_{k})_{k}$

To show convergence, it suffices to prove that for any continuous function
$\varphi\colon M\to\mathbb{R}$, the limit
$\lim_{k\to\infty}\int\varphi\,d\mu_{k}$ exists. In what follows, we fix
$\varphi$ and prove this assertion.

By item (5) in Proposition 9.3, each curve $\gamma_{k}\in\mathcal{E}_{k}$ is
$(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\xi,\delta))$-complete for some
$\xi\in\mathcal{D}_{1}$. In particular, all the curves in
$\bigcup_{k}\mathcal{E}_{k}$ have lengths uniformly bounded away from zero and
also from above. By the uniform continuity of the center and the strong
unstable bundles, and the uniform continuity of $\varphi$, for every
$\omega>0$, there exists $\eta>0$ such that for every pair of
$(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\xi,\delta))$-complete curves
$\gamma,\gamma^{\prime}$ (for some $\xi\in\mathcal{D}_{1}$) with
$\mathcal{W}^{\mathrm{uu}}(\gamma,\eta)\cap\gamma^{\prime}\neq\emptyset$, one
has

(10.3)
$\Big{|}\int\varphi\,d\operatorname{Leb}_{\gamma}-\int\varphi\,d\operatorname{Leb}_{\gamma^{\prime}}\Big{|}<\omega.$

Given integers $\ell>k$, by Corollary 9.13, for any curve
$\gamma\in\mathcal{E}_{\ell}$, there exists a
$\gamma^{\prime}\in\mathcal{E}_{k}$ such that $\gamma$ is
$(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\gamma^{\prime},\delta/2^{k-1}))$-complete.
On the other hand, inductively applying Lemma 9.16, it follows that every
curve $\gamma^{\prime}\in\mathcal{E}_{k}$ gives rise to exactly
$(\prod_{j=k+1}^{\ell}\operatorname{card}\mathcal{D}_{j})$ distinct curves in
$\mathcal{E}_{\ell}\cap\mathcal{W}^{\mathrm{uu}}(\gamma^{\prime},\delta/2^{k-1})$.

Consider $k\in\mathbb{N}$ sufficiently large such that $\delta/2^{k-1}<\eta$.
Thus, for every $\ell>k$, by the choice of $\eta$, one has

$\begin{split}&\Big{|}\int\varphi\,d\mu_{\ell}-\int\varphi\,d\mu_{k}\Big{|}\\\
&=\Big{|}\frac{1}{\operatorname{card}\mathcal{E}_{\ell}}\sum_{\gamma\in\mathcal{E}_{\ell}}\int\varphi\,d\operatorname{Leb}_{\gamma}-\frac{1}{\operatorname{card}\mathcal{E}_{k}}\sum_{\gamma^{\prime}\in\mathcal{E}_{k}}\int\varphi\,d\operatorname{Leb}_{\gamma^{\prime}}\Big{|}\\\
&\leq\frac{1}{\operatorname{card}\mathcal{E}_{k}}\sum_{\gamma^{\prime}\in\mathcal{E}_{k}}\frac{\operatorname{card}\mathcal{E}_{k}}{\operatorname{card}\mathcal{E}_{\ell}}\left(\sum_{\textrm{$\gamma$
is $(\ell-k)$ successor of
$\gamma^{\prime}$}}\Big{|}\int\varphi\,d\operatorname{Leb}_{\gamma}-\int\varphi\,d\operatorname{Leb}_{\gamma^{\prime}}\Big{|}\right)\\\
\text{\small{(by \eqref{refertoothiis})}}\quad&\leq\omega.\end{split}$

This proves that the limit $\mu\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\lim_{k}\mu_{k}$ indeed exists.

Recall that $(L_{k})_{k}$ is a nested sequence of compact sets and
$\mu_{k}(L_{k})=1$, thus for any $\ell>k$, one has $\mu_{\ell}(L_{k})=1$.
Taking the limit $\ell\to\infty$, one gets $\mu(L_{k})\geq 1$. As $k$ was
arbitrary, this gives $\mu(L)=1$. This proves the first part of Proposition
10.1. ∎

###  10.2. Estimates of the “local entropies” of $\mu$

We now estimate the $\mu$-measure of Bowen balls intersecting the set $L$. We
start by some preliminary estimates.

Given $\theta>0$, as $\mathfrak{h}(f)>0$, we fix
$\varepsilon^{\prime}\in(0,\theta)$ sufficiently small such that

(10.4)
$(\mathfrak{h}(f)-\varepsilon^{\prime})(1-\varepsilon^{\prime})(1-2\varepsilon^{\prime})>\mathfrak{h}(f)-\theta.$

By the choice in (9.15), (9.8), and (9.14), and by
$0\leq\ell_{i}\leq\ell_{i}^{\sharp}$ for every $i\in\mathbb{N}$, there exists
$k_{0}>0$ so that for all $i\geq k_{0}$

(10.5)
$\max\Big{\\{}|h_{i}-\varepsilon_{i}-\mathfrak{h}(f)|,\frac{\ell_{i}+m_{i}}{n_{i}+\ell_{i}+m_{i}},\frac{t_{i-1}}{t_{i}},\frac{n_{i+1}+\ell_{i+1}}{t_{i}}\Big{\\}}<\varepsilon^{\prime}.$

Fix now $n>t_{k_{0}+1}$. Consider the associated index $k\geq k_{0}+1$ such
that $t_{k}\leq n<t_{k+1}$ and let $N\in\\{0,\ldots,N_{k+1}\\}$ be the largest
number such that

$t_{k}+N(n_{k+1}+\ell_{k+1})\leq n.$

Hence we get

(10.6) $\begin{split}&t_{k_{0}}<t_{k}\leq n<t_{k+1},\\\
&t_{k}+N(n_{k+1}+\ell_{k+1})\leq
n<t_{k}+(N+1)(n_{k+1}+\ell_{k+1}).\end{split}$

In what follows, $k$ and $N$ are fixed with these properties.

######  Lemma 10.2.

For every $x\in M$, there is at most one curve in $\mathcal{E}_{k}$
intersecting $B_{n}(x,\delta)$.

###### Proof.

Arguing by contradiction, suppose that there are two curves
$\gamma,\gamma^{\prime}\in\mathcal{E}_{k}$ both intersecting
$B_{n}(x,\delta)$. Let $z\in\gamma\cap B_{n}(x,\delta)$ and
$z^{\prime}\in\gamma^{\prime}\cap B_{n}(x,\delta)$, then

$d(f^{i}(z),f^{i}(z^{\prime}))\leq
d(f^{i}(z),f^{i}(x))+d(f^{i}(x),f^{i}(z^{\prime}))<2\delta\quad\text{for
$i=0,\ldots,n-1$.}$

Recall $\delta\in(0,\delta_{0}/20)$. By Proposition 9.11, the curves in
$\mathcal{E}_{k}$ are $(t_{k-1}+T_{k},\delta_{0}/2)$-separated. Noting that
$n\geq t_{k}>t_{k-1}+T_{k}$, one gets a contradiction. ∎

######  Lemma 10.3.

Let $\ell>k$ and consider
$\gamma_{\ell},\gamma_{\ell}^{\prime}\in\mathcal{E}_{\ell}$. Let $\gamma_{k}$
and $\gamma_{k}^{\prime}$ be the $(\ell-k)$-predecessors of $\gamma_{\ell}$
and $\gamma_{\ell}^{\prime}$, respectively. If $\gamma_{\ell}$ and
$\gamma_{\ell}^{\prime}$ intersect $B_{n}(x,\delta)$ for some $x\in M$, then
$\gamma_{k}=\gamma_{k}^{\prime}$.

###### Proof.

By hypothesis, we can choose points $z\in\gamma_{\ell}\cap B_{n}(x,\delta)$
and ${z}^{\prime}\in\gamma_{\ell}^{\prime}\cap B_{n}(x,\delta)$. Inductively
applying item (2) in Proposition 9.3, we get

$f^{i}(\gamma_{\ell})\subset\mathcal{W}^{\mathrm{uu}}\big{(}f^{i}(\gamma_{k}),\delta/2^{k-1}\big{)}\textrm{~{}~{}and~{}~{}}f^{i}({\gamma}_{\ell}^{\prime})\subset\mathcal{W}^{\mathrm{uu}}\big{(}f^{i}({\gamma}_{k}^{\prime}),\delta/2^{k-1}\big{)}\,\text{
for $0\leq i\leq t_{k-1}+T_{k}-1$.}$

Therefore, there exist $w\in\gamma_{k}$ and
${w}^{\prime}\in\gamma_{k}^{\prime}$ such that

$f^{i}(z)\in\mathcal{W}^{\mathrm{uu}}(f^{i}(w),\delta)\textrm{~{}and~{}}f^{i}({z}^{\prime})\in\mathcal{W}^{\mathrm{uu}}(f^{i}({w}^{\prime}),\delta)\quad\textrm{for
$0\leq i\leq t_{k-1}+T_{k}-1$}.$

As $n\geq t_{k}>t_{k-1}+T_{k}$, for $0\leq i\leq t_{k-1}+T_{k}-1$, one has

$\begin{split}d(f^{i}(w),f^{i}({w}^{\prime}))&\leq
d(f^{i}(w),f^{i}(z))+d(f^{i}(z),f^{i}(x))+d(f^{i}(x),f^{i}({z}^{\prime}))+d(f^{i}({z}^{\prime}),f^{i}({w}^{\prime}))\\\
\text{{\small(by \eqref{fixdelta})}}&\leq 4\delta<\delta_{0}/2.\end{split}$

By Proposition 9.11, two curves in $\mathcal{E}_{k}$ are
$(t_{k-1}+T_{k},\delta_{0}/2)$-separated. Hence,
$\gamma_{k}={\gamma}_{k}^{\prime}$. ∎

For the next lemma, recall $N$ in (10.6) and the
$(n_{k},\delta_{0})$-separated set $\mathcal{S}_{k}$ in (9.11).

######  Lemma 10.4.

For every $\ell>k$ and every $x\in M$, one has

$\mu_{\ell}\big{(}B_{n}(x,\delta/2)\big{)}\leq\frac{1}{\operatorname{card}\mathcal{E}_{k}\cdot(\operatorname{card}\mathcal{S}_{k+1})^{N}}.$

###### Proof.

By the definition of $\mu_{\ell}$ in (10.1), one has

(10.7)
$\begin{split}\mu_{\ell}\big{(}B_{n}(x,\delta/2)\big{)}&=\frac{1}{\operatorname{card}\mathcal{E}_{\ell}}\sum_{\gamma_{\ell}\in\mathcal{E}_{\ell}}\operatorname{Leb}_{\gamma_{\ell}}\big{(}\gamma_{\ell}\cap
B_{n}(x,\delta/2)\big{)}\\\
&\leq\frac{1}{\operatorname{card}\mathcal{E}_{\ell}}\operatorname{card}\big{\\{}\gamma_{\ell}\in\mathcal{E}_{\ell}\colon\gamma_{\ell}\cap
B_{n}(x,\delta/2)\neq\emptyset\big{\\}}.\end{split}$

Thus, to estimate this measure from above, it suffices to count the curves in
$\mathcal{E}_{\ell}$ which intersect $B_{n}(x,\delta/2)$.

######  Claim 10.5.

For every $x\in M$, the number of curves in $\mathcal{E}_{k+1}$ intersecting
$B_{n}(x,\delta)$ is bounded from above by
$(\operatorname{card}\mathcal{S}_{k+1})^{N_{k+1}-N}$.

###### Proof.

It follows from Lemma 10.3 that all curves in $\mathcal{E}_{k+1}$ intersecting
$B_{n}(x,\delta)$ have the same predecessor in $\gamma\in\mathcal{E}_{k}$.
Given $\gamma_{k+1},\gamma_{k+1}^{\prime}\in\mathcal{E}_{k+1}\cap
B_{n}(x,\delta)$, take

$\mathfrak{X}=(x_{1},\ldots,x_{N_{k+1}})\in\mathcal{S}_{k+1}^{N_{k+1}}~{}\textrm{~{}~{}and~{}~{}}~{}~{}\mathfrak{X}^{\prime}=({x}_{1}^{\prime},\ldots,{x}_{N_{k+1}}^{\prime})\in\mathcal{S}_{k+1}^{N_{k+1}}$

such that $\Phi_{k}(\gamma,\mathfrak{X})=\gamma_{k+1}$ and
$\Phi_{k}(\gamma,\mathfrak{X}^{\prime})=\gamma_{k+1}^{\prime}$. By Lemma 9.16
and using the arguments in the proof of Lemma 10.3, one has that
$x_{i}={x}_{i}^{\prime}$ for every $i\leq N$. As $0\leq N\leq N_{k+1}$, this
implies the claim. ∎

By Corollary 9.12 and Remark 9.8, one has

$\operatorname{card}\mathcal{E}_{k+1}=\operatorname{card}\mathcal{E}_{k}\cdot\operatorname{card}\mathcal{S}_{k+1}^{N_{k+1}}.$

Hence, by Claim 10.5 and applying (10.7) to $\ell=k+1$, one gets that

$\mu_{k+1}\big{(}B_{n}(x,\delta/2)\big{)}\leq\frac{\operatorname{card}\mathcal{S}_{k+1}^{N_{k+1}-N}}{\operatorname{card}\mathcal{E}_{k+1}}=\frac{1}{\operatorname{card}\mathcal{E}_{k}\cdot(\operatorname{card}\mathcal{S}_{k+1})^{N}}.$

This finishes the proof for $\ell=k+1$. For $\ell>k+1$, by applying item (2)
in Proposition 9.3 inductively, each curve
$\gamma_{\ell}\in\mathcal{E}_{\ell}$ and its $(\ell-(k+1))$-predecessor
$\gamma_{k+1}\in\mathcal{E}_{k+1}$ satisfy

$f^{i}(\gamma_{\ell})\subset\mathcal{W}^{\mathrm{uu}}(f^{i}(\gamma_{k+1}),\delta/2^{k}),\quad\text{
for every $0\leq i\leq t_{k}+T_{k+1}-1$}.$

Recall that $t_{k}+N(n_{k+1}+\ell_{k+1})\leq
n<t_{k}+(N+1)(n_{k+1}+\ell_{k+1})$ with $0\leq N\leq N_{k+1}$ and $n<t_{k+1}$
(see (10.6)). Now, we have two cases to consider.

Case (1) $0\leq N<N_{k+1}$. Thus $n<t_{k}+T_{k+1}$. If
$\gamma_{\ell}\in\mathcal{E}_{\ell}$ intersects the Bowen ball
$B_{n}(x,\delta/2)$, then its (unique) $(\ell-(k+1))$-predecessor in
$\mathcal{E}_{k+1}$ intersects $B_{n}(x,\delta)$.

Case (2) $N=N_{k+1}$. Thus $t_{k}+T_{k+1}\leq n<t_{k+1}$. If
$\gamma_{\ell}\in\mathcal{E}_{\ell}$ intersects the Bowen ball
$B_{n}(x,\delta/2)$, then its (unique) $(\ell-(k+1))$-predecessor in
$\mathcal{E}_{k+1}$ intersects $B_{t_{k}+T_{k+1}}(x,\delta)$. By item (1) in
Proposition 9.11, there is at most one center curve intersecting the Bowen
ball $B_{t_{k}+T_{k+1}}(x,\delta)$.

Thus, by (10.7), one has

$\begin{split}\mu_{\ell}\big{(}B_{n}(x,\delta/2)\big{)}&\leq\frac{1}{\operatorname{card}\mathcal{E}_{\ell}}\operatorname{card}\big{\\{}\gamma_{\ell}\in\mathcal{E}_{\ell}\colon\gamma_{\ell}\cap
B_{n}(x,\delta/2)\neq\emptyset\big{\\}}\\\ \text{\small{(by Claim
\ref{claimlunch})}}\quad&\leq\frac{1}{\operatorname{card}\mathcal{E}_{\ell}}(\operatorname{card}\mathcal{S}_{k+1})^{N_{k+1}-N}\cdot\frac{\operatorname{card}\mathcal{E}_{\ell}}{\operatorname{card}\mathcal{E}_{k+1}}\\\
\text{\small{(by Corollary
\ref{garb})}}\quad&=\frac{1}{\operatorname{card}\mathcal{E}_{\ell}}(\operatorname{card}\mathcal{S}_{k+1})^{N_{k+1}-N}\cdot\prod_{i=k+2}^{\ell}\operatorname{card}\mathcal{D}_{i}\\\
&=\frac{(\operatorname{card}\mathcal{S}_{k+1})^{N_{k+1}-N}}{\operatorname{card}\mathcal{E}_{k+1}}=\frac{1}{\operatorname{card}\mathcal{E}_{k}(\operatorname{card}\mathcal{S}_{k+1})^{N}}.\end{split}$

This proves the lemma. ∎

######  Claim 10.6.

It holds

$\frac{1}{n}\log\big{(}\operatorname{card}\mathcal{E}_{k}(\operatorname{card}\mathcal{S}_{k+1})^{N}\big{)}>\mathfrak{h}(f)-\theta.$

###### Proof.

It follows from Corollary 9.12 and Remark 9.8 that

$\operatorname{card}\mathcal{E}_{k}=\prod_{i=1}^{k}\operatorname{card}\mathcal{D}_{i}=\prod_{i=1}^{k}\big{(}\operatorname{card}\mathcal{S}_{i}\big{)}^{N_{i}}.$

By our choice of $k_{0}$ in (10.5), one has

(10.8) $h_{k}-\varepsilon_{k}\geq\mathfrak{h}(f)-\varepsilon^{\prime},\quad
t_{k_{0}-1}\leq\varepsilon^{\prime}t_{k_{0}},\quad\text{ and
}\quad(n_{k+1}+\ell_{k+1})\leq\varepsilon^{\prime}t_{k}.$

Hence, one has the following estimation

$\begin{split}&\frac{1}{n}\log\big{(}\operatorname{card}\mathcal{E}_{k}(\operatorname{card}\mathcal{S}_{k+1})^{N}\big{)}=\frac{1}{n}\log\big{(}\prod_{i=1}^{k}\big{(}\operatorname{card}\mathcal{S}_{i}\big{)}^{N_{i}}\cdot(\operatorname{card}\mathcal{S}_{k+1})^{N}\big{)}\\\
\text{\small(by
\eqref{eqcSkcardinality})}&\geq\frac{1}{n}\sum_{i=1}^{k}\big{(}N_{i}n_{i}(h_{i}-\varepsilon_{i})+Nn_{k+1}(h_{k+1}-\varepsilon_{k+1})\big{)}\\\
&\geq\frac{1}{n}\sum_{i=k_{0}}^{k}\big{(}N_{i}n_{i}(h_{i}-\varepsilon_{i})+Nn_{k+1}(h_{k+1}-\varepsilon_{k+1})\big{)}\\\
\text{\small{(by~{}\eqref{eq:choice-of-
kepsilon})}}\quad&\geq\frac{1-\varepsilon^{\prime}}{n}\sum_{i=k_{0}}^{k}\bigg{(}\big{(}N_{i}(n_{i}+\ell_{i})+m_{i}\big{)}\cdot(h_{i}-\varepsilon_{i})+N(n_{k+1}+\ell_{k+1})(h_{k+1}-\varepsilon_{k+1})\bigg{)}\\\
\text{\small{(by~{}\eqref{eq:choice-of-
kepsilon})}}\quad&\geq\frac{1-\varepsilon^{\prime}}{n}(\mathfrak{h}(f)-\varepsilon^{\prime})\sum_{i=k_{0}}^{k}\bigg{(}\big{(}N_{i}(n_{i}+\ell_{i})+m_{i}\big{)}+N(n_{k+1}+\ell_{k+1})\bigg{)}\\\
&=(\mathfrak{h}(f)-\varepsilon^{\prime})(1-\varepsilon^{\prime})\frac{1}{n}\big{(}t_{k}-t_{k_{0}-1}+N(n_{k+1}+\ell_{k+1})\big{)}\\\
\end{split}$

By (10.6), we get

$\begin{split}\frac{1}{n}\log\big{(}\operatorname{card}\mathcal{E}_{k}(\operatorname{card}\mathcal{S}_{k+1})^{N}\big{)}&\geq(\mathfrak{h}(f)-\varepsilon^{\prime})(1-\varepsilon^{\prime})\frac{1}{n}\big{(}n-t_{k_{0}-1}-(n_{k+1}+\ell_{k+1})\big{)}\\\
\text{\small(by
\eqref{eqsomethingnewer})}\quad&\geq(\mathfrak{h}(f)-\varepsilon^{\prime})(1-\varepsilon^{\prime})\frac{1}{n}\big{(}n-\varepsilon^{\prime}t_{k_{0}}-\varepsilon^{\prime}t_{k}\big{)}\\\
\text{\small{(by \eqref{eq:choice-of-j0}, \eqref{eq:choice-of-
eprime})}}\,&>(\mathfrak{h}(f)-\varepsilon^{\prime})(1-\varepsilon^{\prime})(1-2\varepsilon^{\prime})>\mathfrak{h}(f)-\theta.\end{split}$

This proves the claim. ∎

By Lemma 10.4, for every $\ell>k$ and $x\in M$, one gets
$\mu_{\ell}(B_{n}(x,\delta/2))<e^{-n(\mathfrak{h}(f)-\theta)}$. This implies

$\liminf_{\ell\to\infty}\mu_{\ell}(B_{n}(x,\delta/2))\leq
e^{-n(\mathfrak{h}(f)-\theta)}.$

As $B_{n}(x,\delta/2)$ is open and $\mu_{\ell}$ converges to $\mu$, by
Portmanteau’s theorem one has

$\mu(B_{n}(x,\delta/2))\leq\liminf_{\ell\to\infty}\mu_{\ell}(B_{n}(x,\delta/2))\leq
e^{-n(\mathfrak{h}(f)-\theta)},$

proving the second part of Proposition 10.1. ∎

###  10.3. Proof of Theorem 9.1 item (2)

Let $\theta\in(0,\mathfrak{h}(f))$. Let $\delta$ be sufficiently small as in
Proposition 9.3 and consider the asociated sequences $(t_{k})_{k}$,
$(T_{k})_{k}$, and $(\mathcal{E}_{k})_{k}$. Let
$L_{\delta}=L\big{(}\delta,(t_{k})_{k},(T_{k})_{k},(\mathcal{E}_{k})_{k}\big{)}$
as in (9.1) (in the following arguments, the only relevant parameter is
$\delta$). Let $\mu_{\delta}$ be the probability measure and
$n_{0}\in\mathbb{N}$ provided by Proposition 10.1 such that
$\mu_{\delta}(L_{\delta})=1$ and that for every $n>n_{0}$ and $x\in M$
satisfying $B_{n}(x,\delta/2)\cap L_{\delta}\neq\emptyset$, it holds

$\mu_{\delta}(B_{n}(x,\delta/2))\leq e^{-n(\mathfrak{h}(f)-\theta)}.$

Hence, by Theorem A.4 we get

$h_{\rm top}(f,L_{\delta},\delta/2)\geq\mathfrak{h}(f)-\theta.$

Lemmas A.1 and A.2 together imply that

$h_{\rm top}(f,L_{\delta})=\lim_{\varepsilon\to 0}h_{\rm
top}(f,L_{\delta},\varepsilon)\geq h_{\rm
top}(f,L_{\delta},\delta/2)\geq\mathfrak{h}(f)-\theta.$

As $\theta\in(0,\mathfrak{h}(f))$ was arbitrary, this implies item (2) in
Theorem 9.1. ∎

###  10.4. Comments on Scholium 9.2

Let us briefly explain how this scholium is derived. In Equation (10.1),
instead of using the Lebesgue measure on the curves
$\gamma_{k}\in\mathcal{E}_{k}$, one can also define a probability measure on
$L$ in the following way. Fix $\xi\in\mathcal{E}_{1}$ and a point $x\in\xi$.
Note that $\gamma_{k}\in\mathcal{E}_{k}$ intersects
$\mathcal{W}^{\mathrm{uu}}(x,2\delta)$ in at most one point. Then one can
define a probability measure

$\widehat{\mu}_{k}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\frac{\operatorname{card}\mathcal{E}_{1}}{\operatorname{card}\mathcal{E}_{k}}\sum_{\gamma_{k}\in\mathcal{E}_{k}}\delta_{\gamma_{k}\cap\mathcal{W}^{\mathrm{uu}}(x,2\delta)}.$

Analogously as in the proof of Proposition 10.1, one gets that
$\widehat{\mu}_{k}$ converges to a probability measure $\widehat{\mu}$ such
that

  * •

$\widehat{\mu}(L\cap\mathcal{W}^{\mathrm{uu}}(x,2\delta))=1$;

  * •

for every $\theta>0$ there exists $n_{0}\in\mathbb{N}$ such that for every
$n>n_{0}$ and every $z\in M$ satisfying
$B_{n}(z,\delta/2)\cap\big{(}L\cap\mathcal{W}^{\mathrm{uu}}(x,2\delta)\big{)}\neq\emptyset$,
one has

$\widehat{\mu}(B_{n}(z,\delta/2))\leq e^{-n(\mathfrak{h}(f)-\theta)}.$

Arguing as in Section 10.3 and using Theorem A.4, one can conclude. ∎

##  11\. End of the proof of Theorem 9.1: Forward exponents

Throughout this section, we assume the hypotheses of Theorem 9.1. Consider
$L=L\big{(}\delta,(t_{k})_{k},(T_{k})_{k},(\mathcal{E}_{k})_{k}\big{)}$ as in
(9.1), where $\delta$, $(t_{k})_{k}$, $(T_{k})_{k}$, and
$(\mathcal{E}_{k})_{k}$ are as in Proposition 9.3. The following proposition
is item (3) in Theorem 9.1.

######  Proposition 11.1.

For every $x\in L$,
$\lim_{n\to+\infty}\frac{1}{n}\log\|D^{\mathrm{c}}f^{n}(x)\|=0$ and this
convergence is uniform on $L.$

###### Proof.

Recall the quantifiers in Section 9.2.1. Recall $C_{\rm max}$ in (9.5) and
note that $|\chi_{i}|\leq C_{\rm max}/2$ for every $i\in\mathbb{N}$.

Fix $\varepsilon>0$. By item (ii) in Section 9.2.1, $\chi_{i}$ and
$\varepsilon_{i}$ are chosen monotonically tending to zero. Hence there exists
$k_{0}\in\mathbb{N}$ such that

(11.1) $|\chi_{i}|+6\varepsilon_{i}<\varepsilon,\quad\textrm{~{} for every
$i\geq k_{0}$.}$

Recall that (9.7) and (9.15) imply that for all $i\in\mathbb{N}$

(11.2) $m_{i}C_{\rm
max}<n_{i}\varepsilon_{i}<T_{i}\varepsilon_{i}<t_{i}\varepsilon_{i}\quad\text{
and }\quad t_{i}C_{\rm max}<t_{i+1}\varepsilon_{i+1}.$

Fix $n>t_{k_{0}+1}$. As in (10.6), we consider the associated index $k\geq
k_{0}+1$ and $N\in\\{0,\ldots,N_{k+1}\\}$ such that

(11.3) $\begin{split}&t_{k}\leq n<t_{k+1},\\\ &t_{k}+N(n_{k+1}+\ell_{k+1})\leq
n<t_{k}+(N+1)(n_{k+1}+\ell_{k+1}).\end{split}$

From now on, $k$ and $N$ are fixed with the above properties.

Observe that

(11.4)
$\begin{split}\log\|D^{\mathrm{c}}f^{n}(x)\|=\log\|D^{\mathrm{c}}f^{t_{k}}(x)\|+\log\|D^{\mathrm{c}}f^{n-t_{k}}(f^{t_{k}}(x))\|.\end{split}$

For the next two lemmas, recall the distortion control-time
$\ell_{k}^{\flat}=(\delta,\varepsilon_{k})$ in Lemma 5.4.

######  Lemma 11.2.

For every $x\in L$, one has

$\big{|}\log\|D^{\mathrm{c}}f^{t_{k}}(x)\|\big{|}<t_{k}(|\chi_{k_{0}}|+5\varepsilon_{k_{0}}).$

###### Proof.

Fix $x\in L$. Recalling the definition of $t_{k}$ in (9.13),

$t_{k}=t_{k_{0}-1}+\sum_{j=k_{0}}^{k}(T_{j}+m_{j}),$

let us first write

(11.5)
$\begin{split}\log\|D^{\mathrm{c}}f^{t_{k}}(x)\|&=\log\|D^{\mathrm{c}}f^{t_{k_{0}-1}}(x)\|\\\
&+\sum_{j=k_{0}}^{k}\Big{(}\log\|D^{\mathrm{c}}f^{T_{j}}(f^{t_{j-1}}(x))\|+\log\|D^{\mathrm{c}}f^{m_{j}}(f^{t_{j}-m_{j}}(x))\|\Big{)}.\end{split}$

As a first step, for every $k_{0}\leq j\leq k+1$, we choose points
$y_{j}\in\Lambda_{j}$ that have controlled finite-time Lyapunov exponents
approximating the one of $x$. By the definition of $L$ in (9.1), for every
$k_{0}\leq j\leq k+1$, there exists a (unique) center curve
$\gamma_{j}\in\mathcal{E}_{j}$ such that

$x\in B^{\mathrm{cu}}_{t_{j-1}+T_{j}}\big{(}\gamma_{j},\delta/2^{j-1}\big{)},$

and thus there exists a point $y_{j}\in\gamma_{j}$ such that

(11.6)
$~{}f^{i}(x)\in\mathcal{W}^{\mathrm{uu}}\big{(}f^{i}(y_{j}),\delta/2^{j-1}\big{)}\subset\mathcal{W}^{\mathrm{uu}}\big{(}f^{i}(\gamma_{j}),\delta/2^{j-1}\big{)}\quad\textrm{~{}~{}for
$t_{j-1}\leq i\leq t_{j-1}+T_{j}-1$}.$

On the other hand, by item (3) in Proposition 9.3, there exists a curve
$\xi_{j}\in\mathcal{D}_{j}$ such that
$f^{t_{j-1}}(\gamma_{j})\subset\mathcal{W}^{\mathrm{ss}}(\xi_{j},\delta/2^{j-1})$
and in particular,

$d(f^{i}(\gamma_{j}),f^{i}(\xi_{j}))\leq\delta/2^{j-1}\quad\textrm{~{}~{}for
$t_{j-1}\leq i\leq t_{j-1}+T_{j}-1$}.$

By the definition of $\mathcal{D}_{j}$ (see Lemma 9.7) and item (iv) in
Section 9.2.1 for $\Lambda_{j}$ and the point
$f^{t_{j-1}}(y_{j})\in\mathcal{W}^{\mathrm{ss}}(\xi_{j},\delta/2^{j-1})$, one
has

(11.7)
$\big{|}\log\|D^{\mathrm{c}}f^{T_{j}}(f^{t_{j-1}}(y_{j}))\|-T_{j}\chi_{j}\big{|}\leq\log
K_{j}+T_{j}\varepsilon_{j}.$

By (9.6), we have $\ell_{j}^{\flat}\leq n_{j}<T_{j}$ and hence, together with
(11.6), we can apply Lemma 5.4 to the points $f^{t_{j-1}}(x)$ and
$f^{t_{j-1}}(y_{j})$, and get

(11.8)
$\begin{split}&\big{|}\log\|D^{\mathrm{c}}f^{T_{j}}(f^{t_{j-1}}(x))\|-T_{j}\chi_{j}\big{|}\\\
&\leq\big{|}\log\|D^{\mathrm{c}}f^{T_{j}}(f^{t_{j-1}}(x))\|-\log\|D^{\mathrm{c}}f^{T_{j}}(f^{t_{j-1}}(y_{j}))\|\big{|}\\\
&\hskip
8.53581pt+\big{|}\log\|D^{\mathrm{c}}f^{T_{j}}(f^{t_{j-1}}(y_{j}))\|-T_{j}\chi_{j}\big{|}\\\
\text{\small{(by Lemma~{}\ref{lemsomecount} and \eqref{thiseq})}}\quad&\leq
T_{j}\varepsilon_{j}+\big{(}T_{j}\varepsilon_{j}+\log K_{j}\big{)}\\\
&=2T_{j}\varepsilon_{j}+\log K_{j}.\end{split}$

In particular, recalling that $T_{j}=N_{j}(n_{j}+\ell_{j})$, we get

(11.9)
$\begin{split}\big{|}\frac{1}{T_{j}}\log\|D^{\mathrm{c}}f^{T_{j}}(f^{t_{j-1}}(x))\|-\chi_{j}\big{|}&\leq
2\varepsilon_{j}+\frac{\log K_{j}}{T_{j}}=2\varepsilon_{j}+\frac{\log
K_{j}}{N_{j}(n_{j}+\ell_{j})}\\\ \text{\small{(by
\eqref{eqexp1bisbis})}}\quad&\leq 3\varepsilon_{j}.\end{split}$

Hence, recalling (11.5) and putting all steps together, one gets

$\begin{split}\big{|}\log\|D^{\mathrm{c}}f^{t_{k}}(x)\|\big{|}&=\big{|}\log\|D^{\mathrm{c}}f^{t_{k_{0}-1}}(x)\|\\\
&+\sum_{j=k_{0}}^{k}\big{(}\log\|D^{\mathrm{c}}f^{T_{j}}(f^{t_{j-1}}(x))\|+\log\|D^{\mathrm{c}}f^{m_{j}}(f^{t_{j}-m_{j}}(x))\|\big{)}\big{|}\\\
\text{\small{(by \eqref{defCmax} and \eqref{eqB})}}\quad&\leq
t_{k_{0}-1}C_{\rm
max}+\sum_{j=k_{0}}^{k}\big{(}T_{j}(|\chi_{j}|+3\varepsilon_{j})+m_{j}C_{\rm
max}\big{)}\\\ \text{\small{(by \eqref{useful})}}\quad&\leq
t_{k_{0}}\varepsilon_{k_{0}}+\sum_{j=k_{0}}^{k}T_{j}(|\chi_{j}|+4\varepsilon_{j})\\\
&\leq t_{k}(|\chi_{k_{0}}|+5\varepsilon_{k_{0}}),\end{split}$

where the last inequality follows from
$t_{k_{0}}+\sum_{j=k_{0}}^{k}T_{j}<t_{k}$, and
$\varepsilon_{j}\leq\varepsilon_{k_{0}}$ and $|\chi_{j}|\leq|\chi_{k_{0}}|$
for every $j\geq k_{0}$ (see the property (ii) in Section 9.2.1). This proves
the lemma. ∎

With Lemma 11.2 at hand, we are now ready to estimate
$\log\|D^{\mathrm{c}}f^{n}(x)\|$ in (11.4). There are three cases to consider,
according to the position of $n$ in (11.3).

Case (1) $t_{k}\leq n<t_{k}+\ell_{k+1}^{\flat}$. The fact that $n>t_{k_{0}+1}$
together with our choices in (9.10) and property (ii) in Section 9.2.1 imply

$\begin{split}\frac{1}{n}\big{|}\log\|D^{\mathrm{c}}f^{n-t_{k}}(f^{t_{k}}(x))\|\big{|}\leq\frac{\ell_{k_{0}+1}^{\flat}}{n}C_{\rm
max}\leq\frac{\ell_{k_{0}+1}^{\flat}}{t_{k_{0}+1}}C_{\rm
max}<\varepsilon_{k_{0}}.\end{split}$

Hence, with Lemma 11.2 we get

$\begin{split}\frac{1}{n}\big{|}\log\|D^{\mathrm{c}}f^{n}(x)\|\big{|}&\leq\frac{1}{n}\Big{(}\log\|D^{\mathrm{c}}f^{t_{k}}(x)\|+\log\|D^{\mathrm{c}}f^{n-t_{k}}(f^{t_{k}}(x))\|\Big{)}\\\
&\leq\frac{1}{n}\Big{(}t_{k}(|\chi_{k_{0}}|+5\varepsilon_{k_{0}})+\log\|D^{\mathrm{c}}f^{n-t_{k}}(f^{t_{k}}(x))\|\Big{)}<|\chi_{k_{0}}|+6\varepsilon_{k_{0}}<\varepsilon,\end{split}$

where the last inequality follows from (11.1).

Case (2) $t_{k}+\ell_{k+1}^{\flat}\leq n<t_{k}+T_{k+1}$. Recall again that
$\varepsilon_{j}\searrow 0$ monotonically and $t_{k}\geq
N_{k}(n_{k}+\ell_{k})$ due to (9.13). Note that $\ell^{\flat}_{k+1}\leq
n-t_{k}$. Hence, arguing as in (11.8) for $j=k$, we get

$\begin{split}\big{\lvert}\log\|D^{\mathrm{c}}f^{n-t_{k}}(f^{t_{k}}(x))\|-(n-t_{k})\chi_{k+1}\big{\rvert}&\leq
2(n-t_{k})\varepsilon_{k+1}+\log K_{k+1}\\\ \text{\small{(by
\eqref{eqexp1bis})}}\quad&\leq
2(n-t_{k})\varepsilon_{k+1}+n_{k}\varepsilon_{k}\\\
&<2(n-t_{k})\varepsilon_{k_{0}}+t_{k}\varepsilon_{k_{0}}.\end{split}$

Hence, as $|\chi_{k}|\searrow 0$ monotonically, one has

$\big{|}\log\|D^{\mathrm{c}}f^{n-t_{k}}(f^{t_{k}}(x))\|\big{|}\leq(n-t_{k})(|\chi_{k+1}|+2\varepsilon_{k_{0}})+t_{k}\varepsilon_{k_{0}}<(n-t_{k})(|\chi_{k_{0}}|+2\varepsilon_{k_{0}})+t_{k}\varepsilon_{k_{0}}.$

Combining with Lemma 11.2 and (11.1), one gets

$\frac{1}{n}\log\|D^{\mathrm{c}}f^{n}(x)\|<\frac{1}{n}\big{(}t_{k}(|\chi_{k_{0}}|+5\varepsilon_{k_{0}})+(n-t_{k})(|\chi_{k_{0}}|+2\varepsilon_{k_{0}})+t_{k}\varepsilon_{k_{0}}\big{)}<|\chi_{k_{0}}|+6\varepsilon_{k_{0}}<\varepsilon.$

Case (3) $t_{k}+T_{k+1}\leq n<t_{k+1}$. Note that in this case Equation (9.13)
implies

(11.10) $n-t_{k}-T_{k+1}<m_{k+1}.$

First observe that

$\begin{split}&\frac{1}{n}\log\|D^{\mathrm{c}}f^{n-t_{k}}(f^{t_{k}}(x))\|\\\
&=\frac{1}{n}\log\|D^{\mathrm{c}}f^{T_{k+1}}(f^{t_{k}}(x))\|+\frac{1}{n}\log\|D^{\mathrm{c}}f^{n-t_{k}-T_{k+1}}(f^{t_{k}+T_{k+1}}(x))\|.\end{split}$

Arguing as in (11.9) for $j=k+1$ and using (11.10), we continue to estimate

$\begin{split}\frac{1}{n}\log\|D^{\mathrm{c}}f^{n-t_{k}}(f^{t_{k}}(x))\|&\leq\frac{T_{k+1}}{n}(\chi_{k+1}+3\varepsilon_{k+1})+\frac{m_{k+1}}{n}C_{\rm
max}\\\ \text{\small{(as we are in Case
(3))}}\quad&\leq\frac{n-t_{k}}{n}(|\chi_{k+1}|+3\varepsilon_{k+1})+\frac{m_{k+1}}{t_{k}}C_{\rm
max}\\\ \text{\small{(by \eqref{useful}, $|\chi_{j+1}|<|\chi_{j}|$, and
$\varepsilon_{j+1}<\varepsilon_{j}$)}}\quad&<\frac{n-t_{k}}{n}(|\chi_{k_{0}}|+3\varepsilon_{k_{0}})+\varepsilon_{k_{0}}.\end{split}$

By Lemma 11.2, one gets

$\begin{split}\frac{1}{n}\log\|D^{\mathrm{c}}f^{n}(x)\|&<\frac{1}{n}\big{(}t_{k}(|\chi_{k_{0}}|+5\varepsilon_{k_{0}})+\log\|D^{\mathrm{c}}f^{n-t_{k}}(f^{t_{k}}(x))\|\big{)}\\\
&\leq\frac{1}{n}\big{(}t_{k}(|\chi_{k_{0}}|+5\varepsilon_{k_{0}})+(n-t_{k})(|\chi_{k_{0}}|+3\varepsilon_{k_{0}})\big{)}+\varepsilon_{k_{0}}\\\
\text{\small(by \eqref{eq:bound-LE-
varepsilon})}\quad&\leq|\chi_{k_{0}}|+6\varepsilon_{k_{0}}<\varepsilon.\end{split}$

To summarize, for every $n>t_{k_{0}+1}$, one has
$\frac{1}{n}\log\|D^{\mathrm{c}}f^{n}(x)\|<\varepsilon$ which implies that the
point $x$ has zero center Lyapunov exponent and the convergence is uniform.
This finishes the proof of the proposition. ∎

##  12\. Fractal sets in $\mathcal{L}(0)$: Proof of Theorem 3

By Theorem 9.1, we have an estimate of the _forward_ Lyapunov exponent and the
topological entropy of the fractal set
$L=L\big{(}\delta,(t_{k})_{k},(T_{k})_{k},(\mathcal{E}_{k})_{k}\big{)}=\bigcup\gamma_{\infty}$
in (9.1). In Section 12.1, we construct a set
$\widehat{L}\subset\mathcal{W}^{\mathrm{ss}}_{\rm loc}(L)\cap\mathcal{L}(0)$
which has the same entropy as $L$. The proof of Theorem 3 is completed in
Section 12.2.

###  12.1. The set $\widehat{L}$: Backward concatenations

######  Proposition 12.1.

Let $L=L\big{(}\delta,(t_{k})_{k},(T_{k})_{k},(\mathcal{E}_{k})_{k}\big{)}$ be
the set provided by Theorem 9.1. For every curve $\gamma_{\infty}$ of $L$,
there is a center curve $\widehat{\gamma}_{\infty}$ such that

  * •

$\widehat{\gamma}_{\infty}\subset\mathcal{W}^{\mathrm{ss}}(\gamma_{\infty},2\delta)$
and is
$(\mathrm{c},\mathcal{W}^{\mathrm{ss}}(\gamma_{\infty},2\delta))$-complete,

  * •

$\lim_{n\to\pm\infty}\frac{1}{n}\log\|D^{\mathrm{c}}f^{n}(x)\|=0$ and this
convergence is uniform on
$\widehat{L}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\bigcup\widehat{\gamma}_{\infty}$.

In particular, $\widehat{L}\subset\mathcal{L}(0)$. Moreover, $h_{\rm
top}(f,\widehat{L})=h_{\rm top}(f,L)$.

###### Proof.

Recall the choice of the “pivotal” point $q_{0}$ and the numbers
$r^{\mathrm{c}},r^{\mathrm{s}},r^{\mathrm{u}}_{1},r^{\mathrm{u}}_{2}$ in
Remark 6.11. By hypothesis (B), $f$ has an unstable blender-horseshoe
$\mathfrak{B}^{+}$. Given $r>0$, let $N(r)\in\mathbb{N}$ and $R(N(r))>0$ be
its associated constants as in Lemma 6.6 (with respect to $\mathfrak{B}^{+}$).

Recall that, by item (1) in Theorem 9.1, each center curve $\gamma_{\infty}$
in $L$ is $(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\xi,2\delta))$-complete for
some curve $\xi\in\mathcal{D}_{1}$. Now we are ready to construct for each
$\gamma_{\infty}\subset L$ an auxiliary curve $\gamma_{\infty}^{\prime}$.

As $\gamma_{\infty}$ is
$(\mathrm{c},\mathcal{W}^{\mathrm{uu}}(\xi,2\delta))$-complete and
$2\delta<r^{u}$ due to Equation (9.2), by Lemma 6.10, there is a center curve
$\gamma_{\infty}^{\prime}\subset\mathcal{W}^{\mathrm{u}}(q,r^{\mathrm{u}}_{1})$
which is
$(\mathrm{c},\mathcal{W}^{\mathrm{ss}}(\gamma_{\infty},r^{\mathrm{s}}))$-complete.
The point $q_{0}$ is now used to start a cascade of concatenations of center
curves, each of which has center exponents uniformly converging to zero (for
$f^{-1}$).

As $q_{0}$ is a hyperbolic periodic point with exponent
$\chi^{\mathrm{c}}(q_{0})>0$ and with unstable manifold
$\mathcal{W}^{\mathrm{u}}(q_{0},r^{\mathrm{u}}_{1})$ of inner radius
$r^{\mathrm{u}}_{1}$, fixing any constant
$\varepsilon_{1}\in(0,\chi^{\mathrm{c}}(q_{0}))$ there is $C_{1}\in\mathbb{N}$
such that for every $x\in\gamma_{\infty}^{\prime}$ and every $n\in\mathbb{N}$
one has

$C_{1}^{-1}\cdot
e^{-n(\chi^{\mathrm{c}}(q_{0})+\varepsilon_{1})}\leq\|D^{\mathrm{c}}f^{-n}(x)\|\leq
C_{1}\cdot e^{-n(\chi^{\mathrm{c}}(q_{0})-\varepsilon_{1})}.$

Analogously as in Section 9.2.1, there exists a sequence $(\Gamma_{k})_{k}$ of
basic sets of expanding type such that $\chi^{\mathrm{c}}(\nu_{k})\to 0$ as
$k\to\infty$ uniformly for every sequence of invariant measures
$\nu_{k}\in\mathcal{M}(f|_{\Gamma_{k}})$. Moreover, we get a sequence
$r_{k}\searrow 0$, a sequences of positive numbers $(C_{k})_{k}$, and
sequences $\chi_{k}\searrow 0$, and $\varepsilon_{k}\searrow 0$. In each set
$\Lambda_{k}$, we choose one center curve $\xi_{k}$ centered at some point in
$\Lambda_{k}$ of length $r_{k}<\frac{1}{2}R(N(\delta/2^{k}))$ such that for
every $y\in\xi_{k}$ and $n\in\mathbb{N}$, one has

$C_{k}^{-1}e^{-n(\chi_{k}+\varepsilon_{k})}\leq\|D^{\mathrm{c}}f^{-n}(y)\|\leq
C_{k}e^{-n(\chi_{k}-\varepsilon_{k})}.$

Consider now the sequence of center curves
$\gamma_{\infty}^{\prime},\xi_{2},\xi_{3},\ldots$. By Scholium 9.10 (applied
to $f^{-1}$), we get a sequence of center curves $(\gamma_{k})_{k}$ such that
each $\gamma_{k}$ is
$(\mathrm{c},\mathcal{W}^{\mathrm{ss}}(\gamma_{\infty}^{\prime},\delta))$-complete.
Now we let

$\widehat{\gamma}_{\infty}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\lim_{k\to\infty}\gamma_{k}.$

By construction and arguing as in Section 11, we get that for every
$x\in\widehat{\gamma}_{\infty}$ we have

  * •

$\widehat{\gamma}_{\infty}\subset\mathcal{W}^{\mathrm{ss}}(\gamma_{\infty}^{\prime},2\delta)$;

  * •

$\lim_{n\to-\infty}\frac{1}{n}\log\,\lVert D^{\mathrm{c}}f^{n}(y)\rVert=0,$
for every $y\in\widehat{\gamma}_{\infty}$.

Note that, _a priori_ , the speed of the latter limit may depend on the curve
$\gamma_{\infty}$ chosen. However, arguing as in Section 11, we get that it is
uniform. Also, by construction,
$\widehat{L}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\bigcup\widehat{\gamma}_{\infty}$ is a collection of pairwise
disjoint center curves.

Observe finally that, by construction,
$\widehat{L}\subset\mathcal{W}^{\mathrm{ss}}(L,2\delta)$, and conversely one
has $L\subset\mathcal{W}^{\mathrm{ss}}(\widehat{L},2\delta)$. By Theorem A.5,
one has that $h_{\rm top}(f,\widehat{L})=h_{\rm top}(f,L)$. This proves the
proposition. ∎

###  12.2. End of the proof of Theorem 3

Let $L_{\delta}=L(\delta,(t_{k})_{k},(T_{k})_{k},(\mathcal{E}_{k})_{k}\big{)}$
as in Theorem 9.1 (here, the only relevant parameter is $\delta$). Let
$\widehat{L}_{\delta}\subset\mathcal{L}(0)$ be the associated set of non-
degenerate center curves provided by Proposition 12.1, then $h_{\rm
top}(f,\widehat{L}_{\delta})=h_{\rm top}(f,L_{\delta})$. Hence, $h_{\rm
top}(f,\widehat{L}_{\delta})\geq\mathfrak{h}(f)$. As
$\widehat{L}_{\delta}\subset\mathcal{L}(0)$, the monotonicity of entropy
(Remark A.3) implies

$h_{\rm top}(f,\mathcal{L}(0))\geq h_{\rm
top}(f,\widehat{L}_{\delta})\geq\mathfrak{h}(f).$

By Theorem 1, we proved already $h_{\rm
top}(f,\mathcal{L}(0))\leq\mathfrak{h}(f)$. Hence, we get equality. Finally,
$\mathfrak{h}(f)\geq\mathcal{H}(0)>0$ was already shown in Remark 4.3. This
proves the theorem. ∎

##  Appendix A Topological entropy

###  A.1. Definition of topological entropy and a distribution principle

Let $(X,d)$ be a compact metric space, $f\colon X\to X$ be a continuous map,
and $Y\subset X$. We recall the definition of topological entropy of $f$ on
$Y$. For that, we rely on [P97]. We also introduce its capacitive versions
$\underline{Ch}_{\rm top}$ and $\overline{Ch}_{\rm top}$.

Recall the definition of a set of $(n,\varepsilon)$-separated points in (3.1)
and the definition of a Bowen ball in (8.1). Denote by $r_{n}(Y,\varepsilon)$
the smallest cardinality of a family of sets
$\\{B_{n}(x_{i},\varepsilon)\\}_{i}$ whose union covers $Y$ and denote by
$s_{n}(Y,\varepsilon)$ the maximum cardinality of a
$(n,\varepsilon)$-separated set in $Y$. Recall that $r_{n}(Y,\varepsilon)\leq
s_{n}(Y,\varepsilon)$. Given $h\geq 0$ let

$\begin{split}m_{\varepsilon,h}(Y)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\lim_{N\to\infty}\Big{(}\inf\Big{\\{}\sum_{i}e^{-hn_{i}}\colon
Y\subset\bigcup_{i}B_{n_{i}}(x_{i},\varepsilon),n_{i}\geq N\text{ for every
}i\Big{\\}}\Big{)},\\\
\underline{Cm}_{\varepsilon,h}(Y)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\liminf_{n\to\infty}~{}e^{-nh}\cdot
r_{n}(Y,\varepsilon),\quad\overline{Cm}_{\varepsilon,h}(Y)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\limsup_{n\to\infty}e^{-nh}\cdot s_{n}(Y,\varepsilon).\end{split}$

Fixing $\varepsilon$, this function jumps from $\infty$ to $0$ at a unique
critical parameter. Let

(A.1) $\begin{split}h_{\rm
top}(f,Y,\varepsilon)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\inf\big{\\{}h\colon m_{\varepsilon,h}(Y)=0\big{\\}},\\\
\overline{Ch}_{\rm
top}(f,Y,\varepsilon)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\inf\big{\\{}h\colon\underline{Cm}_{\varepsilon,h}(Y)=0\big{\\}},\quad\underline{Ch}_{\rm
top}(f,Y,\varepsilon)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\inf\big{\\{}h\colon\overline{Cm}_{\varepsilon,h}(Y)=0\big{\\}}.\end{split}$

######  Lemma A.1.

For every $Y\subset X$, $h>0$, and $0<\varepsilon^{\prime}<\varepsilon$, it
holds $m_{\varepsilon,h}(Y)\leq m_{\varepsilon^{\prime},h}(Y)$. Hence, in
particular, $h_{\rm top}(f,Y,\varepsilon)\leq h_{\rm
top}(f,Y,\varepsilon^{\prime})$.

###### Proof.

Note that if $Y\subset\bigcup_{i}B_{n_{i}}(x_{i},\varepsilon^{\prime})$, then
also $Y\subset\bigcup_{i}B_{n_{i}}(x_{i},\varepsilon)$. Thus,

$\inf\Big{\\{}\sum_{i}e^{-hn_{i}}\colon
Y\subset\bigcup_{i}B_{n_{i}}(x_{i},\varepsilon),n_{i}\geq
N\Big{\\}}\leq\inf\Big{\\{}\sum_{i}e^{-hn_{i}}\colon
Y\subset\bigcup_{i}B_{n_{i}}(x_{i},\varepsilon^{\prime}),n_{i}\geq
N\Big{\\}}.$

Taking the limit $N\to\infty$, we get $m_{\varepsilon,h}(Y)\leq
m_{\varepsilon^{\prime},h}(Y)$. This proves the lemma. ∎

Denote by $\widehat{h}_{\rm top}(f,Y)$ the _topological entropy_ of $f$ on $Y$
as defined in [B73].

######  Lemma A.2 ([P97, Chapter 4 Section 11]).

For every $Y\subset X$, one has that

  * (1)

the following limit exists:

$\begin{split}h_{\rm top}(f,Y)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\lim_{\varepsilon\to 0}h_{\rm top}(f,Y,\varepsilon).,\\\
\overline{Ch}_{\rm top}(f,Y)&\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\lim_{\varepsilon\to 0}\overline{Ch}_{\rm
top}(f,Y,\varepsilon),\quad\underline{Ch}_{\rm
top}(f,Y)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\lim_{\varepsilon\to 0}\underline{Ch}_{\rm
top}(f,Y,\varepsilon).\end{split}$

  * (2)

$\overline{Ch}_{\rm top}(f,Y)\geq\underline{Ch}_{\rm
top}(f,Y)\geq\widehat{h}_{\rm top}(f,Y)=h_{\rm top}(f,Y)$.

######  Remark A.3.

We recall some further properties which are relevant in our context:

  * (E1)

(monotonicity) if $Z\subset Y\subset X$, then $h_{\rm top}(f,Z)\leq h_{\rm
top}(f,Y)$,

  * (E2)

(countable stability) $h_{\rm top}(f,\bigcup_{i}Y_{i})=\sup_{i}h_{\rm
top}(f,Y_{i})$.

The following can be considered as an analogy with the classical _Mass
Distribution Principle_ to estimate Hausdorff dimension (see, for example,
[P97, Chapter 2]). For completeness, we add its short proof.

######  Theorem A.4 (Entropy Distribution Principle, [TV03, Theorem 3.6]).

Let $f$ be a continuous map on a metric space $X$ and $Y\subset X$. Assume
that there exist constants $h\geq 0$, $\delta>0$, and a Borel probability
measure $\mu_{\delta}$ such that

  * •

$\mu_{\delta}(Y)>0$;

  * •

there is $n_{0}\in\mathbb{N}$ such that $\mu_{\delta}(B_{n}(x,\delta))\leq
e^{-nh}$ for every $n\geq n_{0}$ and every Bowen ball $B_{n}(x,\delta)$
intersecting the set $Y$.

Then $h_{\rm top}(f,Y,\delta)\geq h.$

###### Proof.

By definition, it suffices to show that $\widehat{m}_{\delta,h}(Y)>0$.
Consider any countable family of Bowen balls
$\\{B_{n_{i}}(x_{i},\delta)\\}_{i}$, $n_{i}\geq n_{0}$, which covers $Y$. Up
to removing some Bowen balls, one can assume that each Bowen ball
$B_{n_{i}}(x_{i},\delta)$ intersects $Y$. Then

$0<\mu_{\delta}(Y)\leq\mu_{\delta}\big{(}\bigcup_{i}B_{n_{i}}(x_{i},\delta)\big{)}\leq\sum_{i}e^{-n_{i}h}.$

Thus,

$m_{\delta,h}(Y)=\lim_{N\to\infty}\inf\Big{\\{}\sum_{i}e^{-hn_{i}}\colon
Y\subset\bigcup_{i}B_{n_{i}}(x_{i},\delta),n_{i}\geq N\text{ for every
}i\Big{\\}}\geq\mu_{\delta}(Y)>0.$

This implies $h_{\rm top}(f,Y,\delta)\geq h.$ ∎

###  A.2. “Stable shearing” preserves topological entropy

By definition, the concept of entropy only relies on the _forward_ iterates of
a set, and, indeed, one can show that for a partially hyperbolic
diffeomorphism, if one “shears” a set along its strong stable leaves, the
entropy is unchanged. Let us make this more precise. Given a subset $Y\subset
M$ and $\ell>0$, in analogy to our notation in (5.1), let

$\mathcal{W}^{\mathrm{ss}}(Y,\ell)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\bigcup_{Y\in Y}\mathcal{W}^{\mathrm{ss}}(Y,\ell).$

######  Theorem A.5.

Let $f$ be a $C^{1}$-partially hyperbolic diffeomorphism. Assume that
$Y,Y^{\prime}\subset M$ are two subsets and $\ell>0$ such that
$Y^{\prime}\subset\mathcal{W}^{\mathrm{ss}}(Y,\ell)$ and
$Y\subset\mathcal{W}^{\mathrm{ss}}(Y^{\prime},\ell)$. Then

$h_{\rm top}(f,Y)=h_{\rm top}(f,Y^{\prime}).$

###### Proof.

As $Y^{\prime}\subset\mathcal{W}^{\mathrm{ss}}(Y,\ell)$ and
$Y\subset\mathcal{W}^{\mathrm{ss}}(Y^{\prime},\ell)$, by monotonicity of
entropy, it suffices to show that $h_{\rm top}(f,Y)\geq h_{\rm
top}(f,\mathcal{W}^{\mathrm{ss}}(Y,\ell))$.

By the uniform contraction of $f$ along the strong stable bundle, for every
$\varepsilon>0,$ there is $k_{\varepsilon}\in\mathbb{N}$ such that

(A.2)
$f^{k}(\mathcal{W}^{\mathrm{ss}}(x,\ell))\subset\mathcal{W}^{\mathrm{ss}}(f^{k}(x),{\varepsilon/2})\quad\text{
for every $x\in M$ and $k\geq k_{\varepsilon}$.}$

Given any $\varepsilon>0$ and $N\in\mathbb{N}$, fix now a countable collection
of Bowen balls $\\{B_{n_{i}}(x_{i},\varepsilon)\\}$ whose union covers $Y$ and
which satisfy $n_{i}\geq N+k_{\varepsilon}$ for each $i$.

######  Claim A.6.

$f^{k_{\varepsilon}}(\mathcal{W}^{\mathrm{ss}}(Y,\ell))\subset\bigcup_{i}B_{n_{i}-k_{\varepsilon}}\big{(}f^{k_{\varepsilon}}(x_{i}),3\varepsilon/2\big{)}$.

###### Proof.

For every $x\in\mathcal{W}^{\mathrm{ss}}(Y,\ell)$, there exists $y\in Y$ such
that $x\in\mathcal{W}^{\mathrm{ss}}(y,\ell)$. Since there exists $x_{i}$ such
that $y\in B_{n_{i}}(x_{i},\varepsilon)$, thus for every
$n_{i}-k_{\varepsilon}>j\geq 0$, one has that

$d(f^{j+k_{\varepsilon}}(x),f^{j+k_{\varepsilon}}(x_{i}))\leq
d(f^{j+k_{\varepsilon}}(x),f^{j+k_{\varepsilon}}(y))+d(f^{j+k_{\varepsilon}}(y),f^{j+k_{\varepsilon}}(x_{i}))<3\varepsilon/2.$

This proves the claim. ∎

Fixing some quantifiers, let

(A.3) $C\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\sup_{x\in
M}\big{\\{}\|Df(x)\|,\|Df^{-1}(x)\|\big{\\}}.$

The following fact is straightforward to prove.

######  Claim A.7.

There is a constant $c>0$ such that for every $x\in M$, there are $c\cdot
C^{k_{\varepsilon}\cdot\dim(M)}$-many points which are $\varepsilon\cdot
C^{-k_{\varepsilon}}$-dense in $B(x,3\varepsilon/2)$.

######  Claim A.8.

There exists a collection of sets $\\{Y_{i}\\}$ such that

$\mathcal{W}^{\mathrm{ss}}(Y,\ell)\subset\bigcup_{i}\bigcup_{y\in
Y_{i}}B_{n_{i}}(f^{-k_{\varepsilon}}(y),3\varepsilon)\quad\text{ and
}\quad\operatorname{card}(Y_{i})\leq c\cdot C^{k_{\varepsilon}\cdot\dim(M)}.$

###### Proof.

Note that for each $x_{i}$, one has

$B_{n_{i}-k_{\varepsilon}}\big{(}f^{k_{\varepsilon}}(x_{i}),3\varepsilon/2\big{)}\subset
B\big{(}f^{k_{\varepsilon}}(x_{i}),3\varepsilon/2\big{)}.$

Hence, by Claim A.6,

$\mathcal{W}^{\mathrm{ss}}(Y,\ell)\subset\bigcup_{i}f^{-k_{\varepsilon}}\big{(}B_{n_{i}-k_{\varepsilon}}(f^{k_{\varepsilon}}(x_{i}),3\varepsilon/2)\big{)}\subset\bigcup_{i}f^{-k_{\varepsilon}}\big{(}B(f^{k_{\varepsilon}}(x_{i}),3\varepsilon/2)\big{)}.$

By Claim A.7, for each $i$, we can choose a finite set of points $Y_{i}\subset
B(f^{k_{\varepsilon}}(x_{i}),3\varepsilon/2)$ which is $\varepsilon\cdot
C^{-k_{\varepsilon}}$-dense in $B(f^{k_{\varepsilon}}(x_{i}),3\varepsilon/2)$
and has cardinality at most $c\cdot C^{k_{\varepsilon}\cdot\dim(M)}$. Without
loss of generality, we can assume that $Y_{i}\subset
B_{n_{i}-k_{\varepsilon}}(f^{k_{\varepsilon}}(x_{i}),3\varepsilon/2)$.

Now fix $x\in\mathcal{W}^{\mathrm{ss}}(Y,\ell)$, then there exists $x_{i}$
such that

(A.4) $f^{k_{\varepsilon}}(x)\in
B_{n_{i}-k_{\varepsilon}}(f^{k_{\varepsilon}}(x_{i}),3\varepsilon/2).$

Then take $y\in Y_{i}$ such that $d(f^{k_{\varepsilon}}(x),y)<\varepsilon\cdot
C^{-k_{\varepsilon}}.$ Now, we have the following estimates: By Equation
(A.3), for every $0\leq j\leq k_{\varepsilon}$, one has

$d(f^{-j}(y),f^{k_{\varepsilon}-j}(x))\leq
C^{j}d(y,f^{k_{\varepsilon}}(x))\leq C^{j}\cdot\varepsilon\cdot
C^{-k_{\varepsilon}}\leq\varepsilon.$

Moreover, as $y\in Y_{i}\subset
B_{n_{i}-k_{\varepsilon}}(f^{k_{\varepsilon}}(x_{i}),3\varepsilon/2)$ and
$f^{k_{\varepsilon}}(x)\in
B_{n_{i}-k_{\varepsilon}}(f^{k_{\varepsilon}}(x_{i}),3\varepsilon/2)$, for
$0\leq j<n_{i}-k_{\varepsilon}$, one has

$\begin{split}d\big{(}f^{j}(y),f^{j+k_{\varepsilon}}(x)\big{)}\leq
d\big{(}f^{j}(y),f^{j+k_{\varepsilon}}(x_{i})\big{)}+d\big{(}f^{j+k_{\varepsilon}}(x_{i}),f^{j+k_{\varepsilon}}(x)\big{)}\leq
3\varepsilon/2+3\varepsilon/2=3\varepsilon.\end{split}$

To summarize, one has

$d(f^{j-k_{\varepsilon}}(y),f^{j}(x))\leq 3\varepsilon,\textrm{ for $0\leq
j<n_{i}$},$

proving the claim. ∎

For every $h>h_{\rm top}(f,Y)$, by definition, there exists
$\varepsilon_{0}>0$ such that for every $\varepsilon<\varepsilon_{0}$, one has
$m_{\varepsilon,h}(Y)=0$. By definition, we have

$m_{3\varepsilon,h}(\mathcal{W}^{\mathrm{ss}}(Y,\ell))=\lim_{N\to\infty}\inf\Big{\\{}\sum_{j}e^{-hn_{j}}\colon\mathcal{W}^{\mathrm{ss}}(Y,\ell)\subset\bigcup_{j}B_{n_{j}}(y_{j},3\varepsilon),n_{j}\geq
N+k_{\varepsilon}\text{ for every }j\Big{\\}}.$

Given now any countable cover $\\{B_{n_{i}}(x_{i},\varepsilon)\\}$ of $Y$, by
Claim A.8, we get a particular cover of $\mathcal{W}^{\mathrm{ss}}(Y,\ell)$
and thus we can estimate as follows:

$\begin{split}&m_{3\varepsilon,h}(\mathcal{W}^{\mathrm{ss}}(Y,\ell))\\\
&\leq\lim_{N\to\infty}\inf\Big{\\{}\sum_{i}\sum_{y\in Y_{i}}e^{-hn_{i}}\colon
Y\subset\bigcup_{i}B_{n_{i}}(x_{i},\varepsilon),n_{i}\geq
N+k_{\varepsilon}\text{ for every }i\Big{\\}}\\\
&=\lim_{N\to\infty}\inf\Big{\\{}\sum_{i}\operatorname{card}(Y_{i})e^{-hn_{i}}\colon
Y\subset\bigcup_{i}B_{n_{i}}(x_{i},\varepsilon),n_{i}\geq
N+k_{\varepsilon}\text{ for every }i\Big{\\}}\\\ &\leq c\cdot
C^{k_{\varepsilon}\cdot\dim(M)}\cdot\lim_{N\to\infty}\inf\Big{\\{}\sum_{i}e^{-hn_{i}}\colon
Y\subset\bigcup_{i}B_{n_{i}}(x_{i},\varepsilon),n_{i}\geq
N+k_{\varepsilon}\text{ for every }i\Big{\\}}\\\ &=c\cdot
C^{k_{\varepsilon}\cdot\dim(M)}\cdot m_{\varepsilon,h}(Y)=0.\end{split}$

As $\varepsilon\in(0,\varepsilon_{0})$ is arbitrary, we get $h_{\rm
top}(f,\mathcal{W}^{\mathrm{ss}}(Y,\ell))\leq h$. As $h>h_{\rm top}(f,Y)$ is
arbitrary, this implies that $h_{\rm
top}(f,\mathcal{W}^{\mathrm{ss}}(Y,\ell))\leq h_{\rm top}(f,Y).$ ∎

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