We say that $\widecheck{V}_{n}$ converges uniformly to $\widecheck{V}$ if the
representing functions of $V_{n}$ converge uniformly to the representing
function of $V$ when restricted to $R_{(p^{s}\wedge p^{u})^{2}}(0)$.

Notice, $V_{n}\xrightarrow[uniformly]{}V$ implies
$\widecheck{V}_{n}\xrightarrow[uniformly]{}\widecheck{V}$.

The following proposition is a very important step in our proof, especially
the fifth item, for the following reason. A symbolic coding is often a tool
which is used in order to study measures of interests, and in particular
equilibrium states. One of the most sought-after equilibrium states are SRB
measures- equilibrium states of the geometric potential. In order to take
advantage of the symbolic coding for this purpose, one needs to be able to
lift the geometric potential to the symbolic space with the (morally minimal)
property of summable variations (i.e the total variation of chains with the
same $n$ first symbols is a summable quantity in $n$). When coding a set of
points with a uniform bound on the Lyapunov exponents, one usually gets that
the coding map is Hölder continuous (in $C^{1}$ distance for stable/unstable
leaves as well), and so the geometric potential lifts to a Hölder continuous
potential as well (some arguments are needed, see [Sar13, BO18]). The Hölder
continuity of the geometric potential implies immediately the (exponentially
fast) summable variations. In our setup, there is no uniform Lyapunov exponent
for the orbits we code, and in fact these orbits may exhibit $0$ Lyapunov
exponents. This makes the continuity property of the factor map much weaker,
and not Hölder continuous. Thus proving the summable variations property
becomes a much more subtle question. This goal motivated the definition of
leaf cores, which is used in the proof below, where we are able to show that
indeed the factor map has summable variation, w.r.t $C^{1}$ distance for leaf
cores.

######  Proposition 3.14.

Suppose $(v_{i}=\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\in\mathbb{Z}}$ is a
chain of double charts, and choose arbitrary $u$-admissible manifolds
$V^{u}_{-n}$ in $v_{-n}$, and arbitrary $s$-admissible manifolds $V_{n}^{s}$
in $v_{n}$. Then:

  1. (1)

The limits $V^{u}((v_{i})_{i\leq
0}):=\lim\limits_{n\rightarrow\infty}\mathcal{F}_{u}^{n}(V^{u}_{-n}),V^{s}((v_{i})_{i\geq
0}):=\lim\limits_{n\rightarrow\infty}\mathcal{F}_{s}^{n}(V^{s}_{n})$ exist,
and are independent of the choice of $V_{-n}^{u}$ and $V_{n}^{s}$.

  2. (2)

$V^{u}((v_{i})_{i\leq 0})/V^{s}((v_{i})_{i\geq 0})$ is an $s/u$-admissible
manifold in $v_{0}$.

  3. (3)

$f[V^{s}(v_{i})_{i\geq 0}]\subset V^{s}(v_{i+1})_{i\geq 0}$ and
$f^{-1}[V^{u}(v_{i})_{i\leq 0}]\subset V^{u}(v_{i-1})_{i\leq 0}$

  4. (4)

$\displaystyle V$ ${}^{s}((v_{i})_{i\geq
0})=\\{p\in\psi_{x_{0}}[R_{p_{0}^{s}}(0)]:\forall k\geq
0,f^{k}(p)\in\psi_{x_{k}}[R_{10Q_{\epsilon}(x_{k})}(0)]\\}$ $\displaystyle V$
${}^{u}((v_{i})_{i\leq 0})=\\{p\in\psi_{x_{0}}[R_{p_{0}^{u}}(0)]:\forall k\geq
0,f^{-k}(p)\in\psi_{x_{-k}}[R_{10Q_{\epsilon}(x_{-k})}(0)]\\}$

  5. (5)

The map $(v_{i})_{i\in\mathbb{Z}}\mapsto\widecheck{V}^{u}((v_{i})_{i\leq 0})$
has summable variations: $\exists C_{1},\tau>0$ depending on the calibration
parameters (but not on $\epsilon$) s.t

$\mathrm{dist}_{C^{1}}(\widecheck{V}^{u}((v_{i})_{i\leq
0}),\widecheck{V}^{u}((w_{i})_{i\leq 0}))\leq C_{1}[I^{-N}(p^{u}_{0}\wedge
p^{s}_{0})]^{\frac{1}{\gamma}+\tau}.$

A similar statement holds for $\widecheck{V}^{s}(\cdot)$.

###### Proof.

Parts (1)–(4) are inspired by Pesin’s Stable Manifold Theorem [Pes76] .

We treat the case of $u$ manifolds, the stable case is similar.

Part 1: By the Graph Transform theorem $\mathcal{F}_{u}^{n}(V_{-n}^{u})$ is a
$u$-admissible manifold in $v_{0}$. By Proposition 3.8 for any other choice of
$u$-admissible manifolds $W_{-n}^{u}$ in $v_{-n}$:

$\displaystyle\mathrm{dist}(\mathcal{F}_{u}^{n}(V_{-n}^{u}),\mathcal{F}_{u}^{n}(W_{-n}^{u}))\leq$
$\displaystyle
e^{-\sum_{k=0}^{n-1}\frac{1-\epsilon}{S^{2}(x_{-k})}}\mathrm{dist}(V_{-n}^{u},W_{-n}^{u})\leq
2e^{-(1-\epsilon)C_{\beta,\epsilon}^{-\frac{1}{\gamma}}\sum_{k=0}^{n-1}(p^{u}_{-k})^{\frac{1}{\gamma}}}p_{-n}^{u}$
$\displaystyle=$ $\displaystyle
2e^{-(1-\epsilon)C_{\beta,\epsilon}^{-\frac{1}{\gamma}}\sum_{k=0}^{n-1}(p^{u}_{-k})^{\frac{1}{\gamma}}}(p_{-n}^{u}\wedge
p_{-n}^{s})^{2}\frac{p_{-n}^{u}}{(p_{-n}^{u}\wedge p_{-n}^{s})^{2}}$ (14)
$\displaystyle\leq$ $\displaystyle
2e^{-\left((1-\epsilon)C_{\beta,\epsilon}^{-\frac{1}{\gamma}}-\Gamma\right)\sum_{k=0}^{n-1}(p^{u}_{-k}\wedge
p^{s}_{-k})^{\frac{1}{\gamma}}}(p_{0}^{u}\wedge
p_{0}^{s})^{2}\frac{p_{-n}^{u}}{(p_{-n}^{u}\wedge p_{-n}^{s})^{2}}$
$\displaystyle\leq$ $\displaystyle
2e^{-2\cdot\frac{1}{2\Gamma}\left((1-\epsilon)C_{\beta,\epsilon}^{-\frac{1}{\gamma}}-\Gamma\right)\cdot\Gamma\sum_{k=0}^{n-1}(I^{-k}(p_{0}^{u}\wedge
p_{0}^{s}))^{\frac{1}{\gamma}}}(p_{0}^{u}\wedge
p_{0}^{s})^{2}\frac{p_{-n}^{u}}{(p_{-n}^{u}\wedge p_{-n}^{s})^{2}}.$

In the first line we needed the fact that for any admissible manifold with
representing function $F$, $\|F\|_{\infty}\leq p^{u}$ by definition. Notice
that by choosing $\epsilon>0$ arbitrarily small (and thus making
$\frac{1}{C_{\beta,\epsilon}}$ arbitrarily large), we can make the exponent on
the RHS very large. Let $\epsilon>0$ small enough so
${\frac{1}{2\Gamma}\left((1-\epsilon)C_{\beta,\epsilon}^{-\frac{1}{\gamma}}-\Gamma\right)}\geq
1$, then by (3.0.1), with the following estimates:

$\displaystyle\mathrm{dist}(\mathcal{F}_{u}^{n}(V_{-n}^{u}),\mathcal{F}_{u}^{n}(W_{-n}^{u}))\leq$
$\displaystyle
2e^{-\Gamma\sum_{k=0}^{n-1}(I^{-k}(p_{0}^{u}))^{\frac{1}{\gamma}}}p_{-n}^{u}(\because
p^{u}_{-k+1}\leq I(p^{u}_{-k}))$ $\displaystyle\leq$ $\displaystyle
2e^{-\Gamma\sum_{k=0}^{n-1}(I^{-k}(p_{0}^{u}))^{\frac{1}{\gamma}}}p^{u}_{0}\frac{p_{-n}^{u}}{p^{u}_{0}}\leq\frac{1}{p^{u}_{0}}I^{-n}(p_{0}^{u}).$
(15)

When restricting to the leaf cores,
$\mathrm{dist}(\widecheck{V}^{u}_{-n},\widecheck{V}^{u}_{-n})\leq 2\cdot
10^{-2}(p^{u}_{-n}\wedge p^{s}_{-n})^{2}$. In this case, with Corollary 3.12
replacing Proposition 3.8, the estimates which follow (3.0.1) give in addition

$\displaystyle\mathrm{dist}(\widecheck{\mathcal{F}}_{u}^{n}(V_{-n}^{u}),\widecheck{\mathcal{F}}_{u}^{n}(W_{-n}^{u}))\leq$
$\displaystyle\left[e^{-\Gamma\sum_{k=0}^{n-1}(I^{-k}(p_{0}^{u}\wedge
p_{0}^{s}))^{\frac{1}{\gamma}}}(p_{0}^{u}\wedge
p^{s}_{0})\right]^{2}\leq\left(I^{-n}(p_{0}^{u}\wedge p_{0}^{s})\right)^{2}.$
(16)

Equation (3.0.1) implies that if the limit exists then it is independent of
$V_{-n}^{u}$. By Theorem 3.6, for every $n\geq m$,
$\mathcal{F}_{u}^{n-m}(V^{u}_{-n})$ is a $u$-admissible manifold in $v_{-m}$,
and so

$\displaystyle\mathrm{dist}(\mathcal{F}_{u}^{n}(V^{u}_{-n}),\mathcal{F}_{u}^{m}(V^{u}_{-m}))=\mathrm{dist}(\mathcal{F}_{u}^{m}(\mathcal{F}_{u}^{n-m}(V^{u}_{-n})),\mathcal{F}_{u}^{m}(V^{u}_{-m}))\leq\frac{1}{p^{u}_{0}}I^{-m}(p_{0}^{u}).$

Hence, $\mathcal{F}_{u}^{n}(V_{-n}^{u})$ is a Cauchy sequence in a complete
space, and therefore converges.

Part 2: Admissibility of the limit: Write $v_{0}=\psi_{x}^{p^{s},p^{u}}$ and
let $F_{n}$ denote the functions which represent
$\mathcal{F}_{u}^{n}(V^{u}_{-n})$ in $v_{0}$. Since
$\mathcal{F}_{u}^{n}(V^{u}_{-n})$ are $u$-admissible in $v_{0}$, for every
$n$:

  * •

$\|d_{\cdot}F_{n}\|_{\beta/2}\leq\frac{1}{2}$,

  * •

$\|d_{0}F_{n}\|\leq\frac{1}{2}(p^{s}\wedge p^{u})^{\beta/2}$,

  * •

$|F_{n}(0)|_{\infty}\leq 10^{-3}(p^{s}\wedge p^{u})^{2}$,

  * •

$\sup|F_{n}|\leq p^{u}$.

Since
$\mathcal{F}_{u}^{n}(V_{-n}^{u})\xrightarrow[n\rightarrow\infty]{}V^{u}((v_{i})_{i\leq
0})$, $F_{n}\xrightarrow[n\rightarrow\infty]{}F$ uniformly, where $F$
represents $V^{u}((v_{i})_{i\leq 0})$. By the Arzela-Ascoli theorem, $\exists
n_{k}\uparrow\infty$ s.t
$d_{\cdot}F_{n_{k}}\xrightarrow[k\rightarrow\infty]{}L$ uniformly where
$\|L\|_{\beta/2}\leq\frac{1}{2}$. For each term in the sequences, the
following identity holds:

$\forall t\in R_{p^{u}}(0):\text{
}\vec{F}_{n_{k}}(t)=\vec{F}_{n_{k}}(0)+\int\limits_{0}^{1}d_{\lambda
t}\vec{F}_{n_{k}}\cdot t\text{ }d\lambda.$

Since $d_{\cdot}F_{n_{k}}$ converge uniformly, we get:

$\forall t\in R_{p^{u}}(0):\text{
}\vec{F}(t)=\vec{F}(0)+\int\limits_{0}^{1}L(\lambda t)\cdot t\text{
}d\lambda.$

The same calculations give us that $\forall t_{0}\in R_{p^{u}}(0):$

$\forall t\in R_{p^{u}}(0):\text{
}\vec{F}(t)=\vec{F}(t_{0})+\int\limits_{0}^{1}L(\lambda(t-t_{0})+t_{0})\cdot(t-t_{0})\text{
}d\lambda=$
$=\vec{F}(t_{0})+\int\limits_{0}^{1}[L(\lambda(t-t_{0})+t_{0})-L(t_{0})]\cdot(t-t_{0})\text{
}d\lambda+L(t_{0})\cdot(t-t_{0}).$

Since $L$ is $\frac{\beta}{2}$-Hölder on a compact set, and in particular
uniformly continuous, the second and third summands are $o(|t-t_{0}|)$, and
hence $F$ is differentiable and $d_{t}F=L(t)$. We also see that
$\\{d_{\cdot}F_{n}\\}$ can only have one limit point. Consequently
$d_{\cdot}F_{n}\xrightarrow[n\rightarrow\infty]{uniformly}d_{\cdot}F$ . So
$\|d_{\cdot}F\|_{\beta/2}\leq\frac{1}{2}$,
$\|d_{0}F\|\leq\frac{1}{2}(p^{s}\wedge p^{u})^{\beta/2}$, $|F(0)|_{\infty}\leq
10^{-3}(p^{s}\wedge p^{u})^{2}$, and $\sup|F|\leq p^{u}$, whence the
$u$-admissibility of $V^{u}((v_{i})_{i\leq 0})$.

Part 3: Let $V^{u}:=V^{u}((v_{i})_{i\leq
0})=\lim\mathcal{F}_{u}^{n}(V^{u}_{-n})$, and $W^{u}:=V^{u}((v_{i-1})_{i\leq
0})=\lim\mathcal{F}_{u}^{n}(V^{u}_{-n-1})$. $\forall n\geq 0$,

$\displaystyle\mathrm{dist}(V^{u},\mathcal{F}_{u}(W^{u}))\leq$
$\displaystyle\mathrm{dist}(V^{u},\mathcal{F}_{u}^{n}(V_{-n}^{u}))+\mathrm{dist}(\mathcal{F}_{u}^{n}(V_{-n}^{u}),\mathcal{F}_{u}^{n+1}(V_{-n-1}^{u}))+\mathrm{dist}(\mathcal{F}_{u}^{n+1}(V_{-n-1}^{u}),\mathcal{F}_{u}(W^{u}))$
$\displaystyle\leq$
$\displaystyle\mathrm{dist}(V^{u},\mathcal{F}_{u}^{n}(V_{-n}^{u}))+2I^{-n}(1)+e^{-\frac{1-\epsilon}{S^{2}(x_{-1})}}\mathrm{dist}(\mathcal{F}_{u}^{n}(V_{-n-1}^{u}),W^{u}).$

The first and third summands tend to zero by definition, and the second goes
to zero since by Lemma 2.4. So $V^{u}=\mathcal{F}_{u}(W^{u})\subset f[W^{u}]$.

Part 4: The inclusion $\subset$ is simple: Every $u$-admissible manifold
$W_{i}^{u}$ in $\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}}$ is contained in
$\psi_{x_{i}}[R_{p_{i}^{u}}(0)]$, because if $W_{i}^{u}$ is represented by the
function $F$ then any $p=\psi_{x_{i}}(v,w)$ in $W_{i}^{u}$ satisfies
$|w|_{\infty}\leq p_{i}^{u}$ and

$|v|_{\infty}=|F(w)|_{\infty}\leq|F(0)|_{\infty}+\max\|d_{\cdot}F\|\cdot|w|_{\infty}\leq\varphi+\epsilon\cdot|w|_{\infty}\leq(10^{-3}+\epsilon)p_{i}^{u}<p_{i}^{u}$

Applying this to $V^{u}:=V^{u}[(v_{i})_{i\leq 0}]$, we see that for every
$p\in V^{u}$, $p\in\psi_{x_{0}}[R_{p_{0}^{u}}(0)]$. Hence by part 3 for every
$k\geq 0$:

$f^{-k}(p)\in f^{-k}[V^{u}]\subset V^{u}[(v_{i-k})_{i\leq
0}]\subset\psi_{x_{-k}}[R_{p_{-k}^{u}}(0)]\subset\psi_{x_{-k}}[R_{10Q_{\epsilon}(x_{-k})}(0)].$

So now to prove $\supset$: Suppose $z\in\psi_{x_{0}}[R_{p_{0}^{u}}(0)]$ and
$f^{-k}(z)\in\psi_{x_{-k}}[R_{10Q_{\epsilon}(x_{-k})}(0)]$ for all $k\geq 0$.
Write $z=\psi_{x_{0}}(v_{0},w_{0})$. We show that $z\in V^{u}$ by proving that
$v_{0}=F(w_{0})$ where $F$ is the representing function for $V^{u}$. Set
$z^{\prime}:=\psi_{x_{0}}(F(w_{0}),w_{0})=:\psi_{x_{0}}(v^{\prime},w^{\prime})$.
For $k\geq 0$,
$f^{-k}(z),f^{-k}(z^{\prime})\in\psi_{x_{-k}}[R_{10Q_{\epsilon}(x_{-k})}(0)]$
(the first point by assumption, and the second since $f^{-k}(z^{\prime})\in
f^{-k}[V^{u}]\subset V^{u}((v_{i-k})_{i\leq 0}$). It is therefore possible to
write:

$f^{-k}(z)=\psi_{x_{-k}}(v_{-k},w_{-k})\text{ and
}f^{-k}(z^{\prime})=\psi_{x_{-k}}(v^{\prime}_{-k},w^{\prime}_{-k})\text{
}(k\geq 0)$

where
$|v_{-k}|_{\infty},|w_{-k}|_{\infty},|v^{\prime}_{-k}|_{\infty},|w^{\prime}_{-k}|_{\infty}\leq
10Q_{\epsilon}(x_{-k})$ for all $k\geq 0$.

By Proposition 2.19 for $f^{-1}$, $\forall k\geq 0$,
$f^{-1}_{x_{-k-1}x_{-k}}=\psi_{x_{-k-1}}^{-1}\circ f^{-1}\circ\psi_{x_{-k}}$
can be written as

$f_{x_{-k-1}x_{-k}}^{-1}(v,w)=(D_{s,k}^{-1}v+h_{s,k}(v,w),D_{u,k}^{-1}w+h_{u,k}(v,w)),$

where $\|D_{s,k}\|^{-1}\geq e^{\frac{1}{S^{2}(x_{-k})}},\|D_{u,k}^{-1}\|\leq
e^{-\frac{1}{U^{2}(x_{-k})}}$, and
$\max\limits_{R_{10Q_{\epsilon}(x_{-k})}(0)}\|d_{\cdot}h_{s/u,k}\|\leq\sqrt{\epsilon}(10Q_{\epsilon}(x_{-k}))^{\frac{\beta}{2}}$
(provided $\epsilon$ is small enough). Define $\Delta
v_{-k}:=v_{-k}-v^{\prime}_{-k}$, and $\Delta w_{-k}:=w_{-k}-w^{\prime}_{-k}$.
Since for every $k\leq 0$,
$(v_{-k-1},w_{-k-1})=f_{x_{-k-1}x_{-k}}^{-1}(v_{-k},w_{-k})$ and
$(v^{\prime}_{-k-1},w^{\prime}_{-k-1})=f_{x_{-k-1}x_{-k}}^{-1}(v^{\prime}_{-k},w^{\prime}_{-k})$
we get the two following bounds:

$|\Delta v_{-k-1}|_{\infty}\geq\|D_{s,k}\|^{-1}\cdot|\Delta
v_{-k}|_{\infty}-\max\|d_{\cdot}h_{s,k}\|\cdot(|\Delta
v_{-k}|_{\infty}+|\Delta w_{-k}|_{\infty})\geq
e^{\frac{1-\epsilon}{S^{2}(x_{-k})}}|v_{-k}|_{\infty}-\epsilon^{\frac{1}{3}}Q_{\epsilon}^{\frac{\beta}{2}}(x_{-k})|\Delta
w_{-k}|_{\infty},$ $|\Delta w_{-k-1}|_{\infty}\leq\|D_{u,k}^{-1}\|\cdot|\Delta
w_{-k}|_{\infty}+\max\|d_{\cdot}h_{u,k}\|\cdot(|\Delta
w_{-k}|_{\infty}+|\Delta v_{-k}|_{\infty})\leq
e^{-\frac{1-\epsilon}{U^{2}(x_{-k})}}|w_{-k}|_{\infty}+\epsilon^{\frac{1}{3}}Q_{\epsilon}^{\frac{\beta}{2}}(x_{-k})|\Delta
v_{-k}|_{\infty}.$

Denote $a_{k}:=|\Delta v_{-k}|_{\infty},b_{k}:=|\Delta w_{-k}|_{\infty}$. Then
we get

$a_{k+1}\geq
e^{\frac{1-\epsilon}{S^{2}(x_{-k})}}a_{k}-\epsilon^{\frac{1}{3}}Q_{\epsilon}^{\frac{\beta}{2}}(x_{-k})b_{k},b_{k+1}\leq
e^{-\frac{1-\epsilon}{U^{2}(x_{-k})}}b_{k}+\epsilon^{\frac{1}{3}}Q_{\epsilon}^{\frac{\beta}{2}}(x_{-k})a_{k}.$

We claim that $a_{k}\leq a_{k+1},b_{k}\leq a_{k}\forall k\geq 0$: Proof by
induction: This is true for $k=0$ since $b_{0}=0$. Assume by induction that
$a_{k}\leq a_{k+1},b_{k}\leq a_{k}$ , then:

$b_{k+1}\leq
e^{-\frac{1-\epsilon}{U^{2}(x_{-k})}}b_{k}+\epsilon^{\frac{1}{3}}Q_{\epsilon}^{\frac{\beta}{2}}(x_{-k})a_{k}\leq
e^{-\frac{1-2\epsilon}{U^{2}(x_{-k})}}a_{k}\leq a_{k}\leq a_{k+1},$
$\Rightarrow a_{k+2}\geq
e^{\frac{1-\epsilon}{S^{2}(x_{-k-1})}}a_{k+1}-\epsilon^{\frac{1}{3}}Q_{\epsilon}^{\frac{\beta}{2}}(x_{-k-1})b_{k+1}\geq
e^{\frac{1-2\epsilon}{S^{2}(x_{-k-1})}}a_{k+1}\geq a_{k+1}.$

We also see that $a_{k+1}\geq e^{\frac{1-2\epsilon}{S^{2}(x_{-k})}}a_{k}$ for
all $k\geq 0$, hence $a_{k}\geq
e^{\sum_{j=0}^{k-1}\frac{1-2\epsilon}{S^{2}(x_{-j})}}a_{0}$. Notice, for all
$\epsilon>0$ small enough

$\sum_{j=0}^{k-1}\frac{1-2\epsilon}{S^{2}(x_{-j})}\gg\sum_{j=0}^{k-1}(p^{u}_{-j}\wedge
p^{s}_{-j})^{\frac{1}{\gamma}}\geq\sum_{j=0}^{k-1}(I^{-j}(p^{u}_{0}\wedge
p^{s}_{0}))^{\frac{1}{\gamma}}.$

Thus, by Lemma 2.5, either $a_{k}\rightarrow\infty$ or $a_{0}=0$. But
$a_{k}=|v_{-k}-v^{\prime}_{-k}|_{\infty}\leq
20Q_{\epsilon}(x_{-k})<20\epsilon$, so $a_{0}=0$, and therefore
$v_{0}=v^{\prime}_{0}$, whence $F(w^{\prime}_{0})=F(w_{0})$. So

$z=\psi_{x}(F(w_{0}),w_{0})\in V^{u},$

which concludes the proof of part 4.

Part 5: If two chains $v=(v_{i})_{i\in\mathbb{Z}},w=(w_{i})_{i\in\mathbb{Z}}$
satisfy $v_{i}=w_{i}$ for $|i|\leq N$ then $\exists C_{1},\tau>0$ depending on
the calibration parameters s.t
$\mathrm{dist}_{C^{1}}(\widecheck{V}^{u}((v_{i})_{i\leq
0}),\widecheck{V}^{u}((w_{i})_{i\leq 0}))\leq C_{1}[I^{-N}(p^{u}_{0}\wedge
p^{s}_{0})]^{\frac{1}{\gamma}+\tau}$: Let $V^{u}_{k}=V^{u}((v_{i})_{i\leq-k})$
be a $u$-admissible manifold in $v_{-N+k}$, and let
$W^{u}_{k}=V^{u}((w_{i})_{i\leq-k})$ be a $u$-admissible manifold in
$w_{-k}(=v_{-k})$ for all $0\leq k\leq N$. Then $V^{u}((v_{i})_{i\leq
0})=\mathcal{F}_{u}^{k}(V^{u}_{k})$ and $V^{u}((w_{i})_{i\leq
0})=\mathcal{F}_{u}^{k}(W^{u}_{k})$ are $u$-admissible manifolds in
$w_{0}(=v_{0})$. Then by (16),
$\mathrm{dist}(\widecheck{V}^{u}_{k},\widecheck{W}^{u}_{k})\leq\left(I^{-N}(p^{u}_{0}\wedge
p^{s}_{0})\right)^{2}$. Represent $V_{-k}^{u}$ and $W_{-k}^{u}$ by $F_{k}$ and
$G_{k}$ respectively. Let $\eta:=p^{u}_{0}\wedge p^{s}_{0}$. By Corollary
3.12,

$\displaystyle\|F_{k-1}-G_{k-1}\|_{R_{\eta}(0),\infty}\leq$ $\displaystyle
e^{-\frac{1-\epsilon}{S^{2}(x_{-k})}}\|F_{k}-G_{k}\|_{R_{\eta}(0),\infty},$
(17)
$\displaystyle\|d_{\cdot}F_{k-1}-d_{\cdot}G_{k-1}\|_{R_{\eta}(0),\infty}\leq$
$\displaystyle
e^{-\frac{1-\epsilon}{S^{2}(x_{-k})}-\frac{1-\epsilon}{U^{2}(x_{-k})}}(\|d_{\cdot}F_{k}-d_{\cdot}G_{k}\|_{R_{\eta}(0),\infty}+(\sqrt{\epsilon}+\epsilon^{\beta})\|F_{k}-G_{k}\|_{R_{\eta}(0),\infty}^{\beta/2}).$

Set $d_{k}:=\|d_{\cdot}F_{k}-d_{\cdot}G_{k}\|_{R_{\eta}(0),\infty}$,
$c_{k}:=\|F_{k}-G_{k}\|_{R_{\eta}(0),\infty}$, and
$r_{k}:=\frac{1-\epsilon}{S^{2}(x_{-k})}+\frac{1-\epsilon}{U^{2}(x_{-k})}$,
then iterating (17) $N$ times we get

$\displaystyle d_{0}\leq$ $\displaystyle
e^{-r_{1}}d_{1}+(\sqrt{\epsilon}+\epsilon^{\beta})e^{-r_{1}}c_{1}^{\frac{\beta}{2}}\leq\cdots\leq
d_{N}e^{-\sum_{k=1}^{N}r_{j}}+(\sqrt{\epsilon}+\epsilon^{\beta})\sum_{k=1}^{N}c_{k}^{\frac{\beta}{2}}e^{-\sum_{j=1}^{k}r_{j}}$
$\displaystyle\leq$ $\displaystyle 2(p_{0}^{s}\wedge
p_{0}^{u})e^{\Gamma\sum_{k=0}^{N-1}(p_{-k}^{s}\wedge
p_{-k}^{u})^{\frac{1}{\gamma}}}\cdot
e^{-\sum_{k=0}^{N-1}r_{j}}+(\sqrt{\epsilon}+\epsilon^{\beta})\sum_{k=1}^{N}c_{k}^{\frac{\beta}{2}}e^{-\sum_{j=0}^{k-1}r_{j}}(\because
d_{N}\leq 2(p_{-N}^{s}\wedge p_{-N}^{u}))$ $\displaystyle\leq$ $\displaystyle
2I^{-N}(p_{0}^{u}\wedge
p_{0}^{s})+(\sqrt{\epsilon}+\epsilon^{\beta})\sum_{k=1}^{N}[e^{-\Gamma\sum_{j=0}^{N-k-1}I^{-j}(p_{-k}^{u}\wedge
p^{s}_{-k})^{\frac{1}{\gamma}}}\cdot(p_{-k}^{u}\wedge
p_{-k}^{s})]^{\beta}e^{-\sum_{j=0}^{k-1}r_{j}}(\because\eqref{link})$
$\displaystyle\leq$ $\displaystyle 2I^{-N}(p_{0}^{u}\wedge
p_{0}^{s})+(\sqrt{\epsilon}+\epsilon^{\beta})\sum_{k=1}^{N}[e^{-\Gamma\sum_{j=0}^{N-k-1}I^{-j}(p_{-k}^{u}\wedge
p^{s}_{-k})^{\frac{1}{\gamma}}}\cdot(p_{-k}^{u}\wedge
p_{-k}^{s})]^{\beta}\cdot e^{-\sum_{j=0}^{k-1}r_{j}}$ $\displaystyle\leq$
$\displaystyle 2I^{-N}(p_{0}^{u}\wedge
p_{0}^{s})+(\sqrt{\epsilon}+\epsilon^{\beta})\sum_{k=1}^{N}[e^{-\Gamma\sum_{j=0}^{N-k-1}I^{-j}(p_{-k}^{u}\wedge
p^{s}_{-k})^{\frac{1}{\gamma}}}\cdot(p_{0}^{u}\wedge
p_{0}^{s})e^{\Gamma\sum_{j=0}^{k-1}(p_{-j}^{u}\wedge
p^{s}_{-j})^{\frac{1}{\gamma}}}\cdot
e^{-\frac{1}{\beta}\sum_{j=0}^{k-1}r_{j}}]^{\beta}$ $\displaystyle\leq$
$\displaystyle 2I^{-N}(p_{0}^{u}\wedge
p_{0}^{s})+(\sqrt{\epsilon}+\epsilon^{\beta})\sum_{k=1}^{N}[e^{-\Gamma\sum_{j=0}^{N-1}I^{-j}(p_{-k}^{u}\wedge
p^{s}_{-k})^{\frac{1}{\gamma}}}\cdot(p_{0}^{u}\wedge p_{0}^{s})]^{\beta}$
$\displaystyle=$ $\displaystyle 2I^{-N}(p_{0}^{u}\wedge
p_{0}^{s})+(\sqrt{\epsilon}+\epsilon^{\beta})\cdot
N\cdot[I^{-N}(p_{0}^{u}\wedge p_{0}^{s})]^{\beta}\leq 3\cdot
N\cdot[I^{-N}(p_{0}^{u}\wedge p^{s}_{0})]^{\beta}.$

By Definition 2.3 $\gamma>\frac{2}{\beta}$, and so for
$\tau:=\frac{1}{2}(\beta-\frac{2}{\gamma})>0$,
$\beta-2(\frac{1}{\gamma}+\tau)=0$. Then

$\displaystyle d_{0}\leq
3\left(N\cdot\left(I^{-N}(1)\right)^{\frac{1}{\gamma}+\tau}\right)\cdot\left(I^{-N}(p^{u}_{0}\wedge
p^{s}_{0})\right)^{\frac{1}{\gamma}+\tau}.$ (18)

By (4) in Lemma 2.6,
$\left(I^{-N}(1)\right)^{\frac{1}{\gamma}+\tau}=O(\frac{1}{N^{1+\delta}})$ for
some $\delta>0$, and so $N\cdot\left(I^{-N}(1)\right)^{\frac{1}{\gamma}+\tau}$
is bounded. Set $C_{1}:=3\cdot\sup_{N\geq
1}\\{N\cdot\left(I^{-N}(1)\right)^{\frac{1}{\gamma}+\tau}\\}$, then $d_{0}\leq
C_{1}\cdot[I^{-N}(p^{u}_{0}\wedge p^{s}_{0})]^{\frac{1}{\gamma}+\tau}$, which
is summable by Lemma 2.6.

∎

######  Theorem 3.15.

Given a chain of double charts $(v_{i})_{i\in\mathbb{Z}}$, let
$\pi((\underline{v})):=$unique intersection point of $V^{u}((v_{i})_{i\leq
0})$ and $V^{s}((v_{i})_{i\geq 0})$. Then

  1. (1)

$\pi$ is well-defined and $\pi\circ\sigma=f\circ\pi$.

  2. (2)

$\pi:\Sigma\rightarrow M$ is uniformly continuous.

  3. (3)

$\pi[\Sigma]\supset\pi[\Sigma^{\\#}]\supset\mathrm{RST}$.

###### Proof.

Part 1: $\pi$ is well-defined thanks to Proposition 3.5. Now, write
$v_{i}=\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}}$ and $z=\pi((\underline{v}))$. We
claim that $f^{k}(z)\in\psi_{x_{k}}[R_{Q_{\epsilon}(x_{k})}(0)]\text{
}(k\in\mathbb{Z})$. For $k=0$ this is true because $z\in V^{s}((v_{i})_{i\geq
0})$, and this is an $s$-admissible manifold in
$\psi_{x_{0}}^{p_{0}^{s},p_{0}^{u}}$. For $k>0$ we use Proposition 3.14 (part
3) to see that

$f^{k}(z)\in f^{k}[V^{s}((v_{i})_{i\geq 0})]\subset V^{s}((v_{i+k})_{i\geq
0}).$

Since $V^{s}((v_{i+k})_{i\geq 0})$ is an $s$-admissible manifold in
$\psi_{x_{k}}^{p_{k}^{s},p_{k}^{u}}$,
$f^{k}(z)\in\psi_{x_{k}}[R_{Q_{\epsilon}(x_{k})}(0)]$. The case $k<0$ can be
handled the same way, using $V^{u}((v_{i})_{i\leq 0})$. Thus $z$ satisfies:

$f^{k}(z)\in\psi_{x_{k}}[R_{Q_{\epsilon}(x_{k})}(0)]\text{ }\text{ for all
}k\in\mathbb{Z}.$

Any point which satisfies this must be $z$, since by Proposition 3.14 (part 4)
it must lie in $V^{u}((v_{i})_{i\leq 0})\cap V^{s}((v_{i})_{i\geq 0})$ (and in
fact even in $\widecheck{V}^{u}((v_{i})_{i\leq
0})\cap\widecheck{V}^{s}((v_{i})_{i\geq 0})$), and this equation characterizes
$\pi(\underline{v})=z$. Hence:

$f^{k}(f(\pi(\underline{v})))=f^{k+1}(\pi(\underline{v}))\in\psi_{x_{k+1}}^{p_{k+1}^{s},p_{k+1}^{u}}\text{
}\forall k\in\mathbb{Z}$

and this is the condition that characterizes $\pi(\sigma\underline{v})$.

Part 2: We saw that $\underline{v}\mapsto\widecheck{V}^{u}((v_{i})_{i\leq 0})$
and $\underline{v}\mapsto\widecheck{V}^{s}((v_{i})_{i\geq 0})$ are uniformly
continuous (Proposition 3.14). Since the intersection of an $s$-admissible
manifold and a $u$-admissible manifold is a Lipschitz function of those
manifolds (Proposition 3.5), $\pi$ is also uniformly continuous.

Part 3: We prove $\pi[\Sigma]\supset\mathrm{RST}$: Suppose $x\in\mathrm{RST}$,
then by Proposition 2.28 there exist
$\psi_{x_{k}}^{p_{k}^{s},p_{k}^{u}}\in\mathcal{V}$ s.t
$\psi_{x_{k}}^{p_{k}^{s},p_{k}^{u}}\rightarrow\psi_{x_{k+1}}^{p_{k+1}^{s},p_{k+1}^{u}}$
for all $k$, and s.t $\psi_{x_{k}}^{p_{k}^{s}\wedge p_{k}^{u}}$ $I$-overlaps
$\psi_{f^{k}(x)}^{p_{k}^{s}\wedge p_{k}^{u}}$ for all $k\in\mathbb{Z}$. This
implies

$f^{k}(x)=\psi_{f^{k}(x)}(0)\in\psi_{f^{k}(x)}[R_{p_{k}^{s}\wedge
p_{k}^{u}}(0)]\subset\psi_{f^{k}(x)}[R_{Q_{\epsilon}(x_{k})}(0)].$

Then $x$ satisfies $f^{k}(x)\in\psi_{x_{k}}[R_{Q_{\epsilon}(x_{k})}(0)]\text{
}(k\in\mathbb{Z})$. It follows that $x=\pi(\underline{v})$ with
$\underline{v}=(\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\in\mathbb{Z}}$. Let
$\\{q(f^{n}(x))\\}_{n\in\mathbb{Z}}$ be given by the strong temperability of
$x$, then by Lemma 2.26 applied to
$\\{Q_{\epsilon}(f^{n}(x))\\}_{n\in\mathbb{Z}}$ and
$\\{q(f^{n}(x))\\}_{n\in\mathbb{Z}}$, the sequence
$\\{(p^{s}_{i},p^{u}_{i})\\}_{i\in\mathbb{Z}}$ is $I$-strongly subordinated to
$\\{Q_{\epsilon}(f^{i}(x))\\}_{i\in\mathbb{Z}}$ and satisfies $p^{s}_{i}\wedge
p^{u}_{i}\geq q(f^{i}(x))$ for all $i\in\mathbb{Z}$. By the definition of
$\mathrm{RST}$ there exist subsequences $i_{k},j_{k}\uparrow\infty$ for which
$p_{i_{k}}^{s}\wedge p_{i_{k}}^{u}\geq q(f^{i_{k}}(x))$ and
$p_{-j_{k}}^{s}\wedge p_{-j_{k}}^{u}\geq q(f^{-j_{k}}(x))$ are bounded away
from zero (see Proposition 2.26). By the discreteness property of
$\mathcal{A}$ (Proposition 2.20), $\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}}$ must
repeat some symbol infinitely often in the past, and some symbol (possibly a
different one) infinitely often in the future. Thus the above actually proves
that

$\pi[\Sigma^{\\#}]\supset\mathrm{RST},$

where $\Sigma^{\\#}=\\{\underline{v}\in\Sigma:\exists
v,w\in\mathcal{V},n_{k},m_{k}\uparrow\infty\text{ s.t
}v_{n_{k}}=v,v_{-m_{k}}=w\\}$. ∎

Remark: After we prove the Inverse Problem properties in §4, we in fact
identify the image of $\Sigma^{\\#}$, and not just a subset of it.

Summary up to here

  * •

We defined $I$-chains (Definition 2.4).

  * •

We proved the shadowing lemma: for every chain
$v=(\psi^{p_{i}^{s},p_{i}^{u}}_{x_{i}})_{i\in\mathbb{Z}}$ there exists $x\in
M$ s.t $f^{i}(x)\in\psi_{x_{i}}[R_{p^{s}_{i}\wedge p^{u}_{i}}(0)]$
$(i\in\mathbb{Z})$ [proof of Theorem 3.15].

  * •

We found a countable set of charts $\mathcal{A}$ s.t the countable Markov
shift

$\Sigma=\\{\underline{v}\in\mathcal{A}^{\mathbb{Z}}:\underline{v}\text{ is an
}I\text{-chain}\\}$

is a Markov extension of a set containing $\mathrm{RST}$ (Theorem 3.15); But
the coding $\pi:\Sigma\rightarrow M$ is not finite-to-one.

Along the way we saw that if $x$ is shadowed by $v=(v_{i})_{i\in\mathbb{Z}}$
then $V^{s}((v_{i})_{i\geq 0}),V^{u}((v_{i})_{i\leq 0})$ are local (perhaps
weakly) stable/unstable manifolds of $x$. (Proposition 3.14).

##  4\. Chains which shadow the same orbit are close

###  4.1. The inverse problem

The aim of this part is to show that the map
$\pi:\Sigma^{\\#}\rightarrow\mathrm{RST}$ from Theorem 3.15, is “almost
invertible”: if
$\pi((\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\in\mathbb{Z}})=\pi((\psi_{y_{i}}^{q_{i}^{s},q_{i}^{u}})_{i\in\mathbb{Z}})$,
then $\forall i$: $x_{i}\approx y_{i}$, $p^{s/u}_{i}\approx q^{s/u}_{i}$ and
$C_{0}(x_{i})\approx C_{0}(y_{i})$, where $\approx$ means that the respective
parameters belong to the same compact sets. The compact sets can be as small
as we wish by choosing small enough $\epsilon$. The discretization we
constructed in §1.2.3 in fact allows these compact sets to contain only a
finite number of such possible parameters for our charts. This does not mean
that $\pi$ is finite-to-one, but it will allow us to construct a finite-to-one
coding later.

####  4.1.1. Comparing orbits of tangent vectors

######  Lemma 4.1.

Let
$(\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\in\mathbb{Z}},(\psi_{y_{i}}^{q_{i}^{s},q_{i}^{u}})_{i\in\mathbb{Z}}$
be two chains s.t
$\pi((\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\in\mathbb{Z}})=\pi((\psi_{y_{i}}^{q_{i}^{s},q_{i}^{u}})_{i\in\mathbb{Z}})=p$,
then $d(x_{i},y_{i})<\frac{\sqrt{d}}{50}\max\\{p_{i}^{s}\wedge
p_{i}^{u},q_{i}^{s}\wedge q_{i}^{u}\\}^{2}$.

###### Proof.

$f^{i}(p)$ is the intersection of a $u$-admissible manifold and an
$s$-admissible manifold in $\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}}$, therefore (by
Proposition 3.5) $f^{i}(p)=\psi_{x_{i}}(v_{i},w_{i})$ where
$|v_{i}|_{\infty},|w_{i}|_{\infty}<10^{-2}(p_{i}^{s}\wedge p_{i}^{u})^{2}$.
$d(f^{i}(p),x_{i})=d(\exp_{x_{i}}(C_{0}(x_{i})(v_{i},w_{i})),x_{i})=|C_{0}(x_{i})(v_{i},w_{i})|_{2}\leq\sqrt{d}|(v_{i},w_{i})|_{\infty}\leq\frac{\sqrt{d}}{100}(p_{i}^{s}\wedge
p_{i}^{u})^{2}$. Similarly
$d(f^{i}(x),y_{i})<\frac{\sqrt{d}}{100}(q_{i}^{s}\wedge q_{i}^{u})^{2}$. It
follows that $d(x_{i},y_{i})<\frac{\sqrt{d}}{50}\max\\{q_{i}^{s}\wedge
q_{i}^{u},p_{i}^{s}\wedge p_{i}^{u}\\}^{2}$. ∎

The following definitions are taken from [Sar13].

######  Definition 4.2.

Let $V^{u}$ be a $u$-admissible manifold in the double chart
$\psi_{x}^{p^{s},p^{u}}$. We say that $V^{u}$ stays in windows if there exists
a negative chain $(\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\leq 0}$ with
$\psi_{x_{0}}^{p_{0}^{s},p_{0}^{u}}=\psi_{x}^{p^{s},p^{u}}$ and $u$-admissible
manifolds $W_{i}^{u}$ in $\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}}$ s.t
$f^{-|i|}[V^{u}]\subset W_{i}^{u}$ for all $i\leq 0$.

######  Definition 4.3.

Let $V^{s}$ be an $s$-admissible manifold in the double chart
$\psi_{x}^{p^{s},p^{u}}$. We say that $V^{s}$ stays in windows if there exists
a positive chain $(\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\geq 0}$ with
$\psi_{x_{0}}^{p_{0}^{s},p_{0}^{u}}=\psi_{x}^{p^{s},p^{u}}$ and $s$-admissible
manifolds $W_{i}^{s}$ in $\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}}$ s.t
$f^{i}[V_{i}^{u}]\subset W_{i}^{s}$ for all $i\geq 0$.

Remark: If $\underline{v}$ is a chain, then $V_{i}^{u}:=V^{u}((v_{k})_{k\leq
i})$ and $V_{i}^{s}:=V^{s}((v_{k})_{k\geq i})$ stay in windows, since
$f^{-k}[V_{i}^{u}]\subset V_{i-k}^{u}$ and $f^{k}[V_{i}^{s}]\subset
V_{i+k}^{s}$ for all $k\geq 0$ (by Proposition 3.14).

######  Proposition 4.4.

The following holds for all $\epsilon$ small enough: Let $V^{s}$ be an
admissible $s$-manifold in $\psi_{x}^{p^{s},p^{u}}$, and suppose $V^{s}$ stays
in windows.

  1. (1)

For every $y,z\in V^{s}$: $d(f^{k}(y),f^{k}(z))<4I^{-k}(p^{s})$ for all $k\geq
0$.

  2. (2)

For every $y\in V^{s}$ and $u\in T_{y}V^{s}(1)$: $|d_{y}f^{k}u|\leq 4\cdot
I^{-k}(p_{0}^{s})\cdot\frac{\|C_{0}(x)^{-1}\|}{p^{s}}$ for all $k\geq 0$.

Recall: $T_{y}V^{s}(1)$ is the unit ball in $T_{y}V^{s}$, w.r.t the Riemannian
norm.

  3. (3)

For $y\in V^{s}$, let $\omega_{s}(y)$ be a normalized volume form of
$T_{y}V^{s}$. For all $y,z\in V^{s}$ , $\forall k\geq 0$:

$|\log|d_{y}f^{k}\omega_{s}(y)|-\log|d_{z}f^{k}\omega_{s}(z)||<\epsilon^{2}(\frac{d(y,z)}{(p^{s})^{3}})^{\beta/4},$

where $|d_{y}f^{k}\omega_{s}(y)|=\bigl{|}\det
d_{y}f^{k}|_{T_{y}V^{s}}\bigr{|},|d_{z}f^{k}\omega_{s}(z)|=\bigl{|}\det
d_{z}f^{k}|_{T_{z}V^{s}}\bigr{|}$. The symmetric statement holds for
$u$-admissible manifolds which stay in windows, for “$s$” replaced by “$u$”
and $f$ by $f^{-1}$.

###### Proof.

Suppose $V^{s}$ is an $s$-admissible manifold in $\psi_{x}^{p^{s},p^{u}}$,
which stays in windows, then there is a positive chain
$(\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\geq 0}$ s.t
$\psi_{x_{0}}^{p_{0}^{s},p_{0}^{u}}=\psi_{x}^{p^{s},p^{u}}$, and there are
$s$-admissible manifolds $W_{i}^{s}$ in $\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}}$
s.t $f^{i}[V^{s}]\subset W_{i}^{s}$ for all $i\geq 0$. We write:

  * •

$V^{s}=\\{(t,F_{0}(t)):|t|_{\infty}\leq p^{s}\\}$

  * •

$W_{i}^{s}=\psi_{x_{i}}[\\{(t,F_{i}(t)):|t|_{\infty}\leq p_{i}^{s}\\}]$

  * •

$\eta_{i}:=p_{i}^{s}\wedge p_{i}^{s}$

Admissibility means that
$\|d_{\cdot}F_{i}\|_{\beta/2}\leq\frac{1}{2},\|d_{0}F_{i}\|\leq\frac{1}{2}(p_{i}^{s}\wedge
p_{i}^{u})^{\beta/2}$, $|F_{i}(0)|_{\infty}\leq 10^{-3}(p_{i}^{s}\wedge
p_{i}^{u})^{2}$ and $\mathrm{Lip}(F_{i})<\epsilon$ ($\because$ Definition
3.2).

Part 1: For every $y,z\in V^{s}$: $d(f^{k}(y),f^{k}(z))\leq
4I^{-k}(p_{0}^{s})$ for all $k\geq 0$.

Since $V^{s}$ stays in windows:
$f^{k}[V^{s}]\subset\psi_{x_{k}}[R_{Q_{\epsilon}(x_{k})}(0)]$ for all $k\geq
0$. Therefore $\forall y,z\in V^{s}$ one can write
$f^{k}(y)=\psi_{x_{k}}(y_{k}^{\prime})$ and
$f^{k}(z)=\psi_{x_{k}}(z_{k}^{\prime})$, where
$y_{k}^{\prime}:=(y_{k},F_{k}(y_{k})),z_{k}^{\prime}:=(z_{k},F_{k}(z_{k}))$
belong to $R_{Q_{\epsilon}(x_{k})}(0)$. For every $k$:
$y_{k+1}^{\prime}=f_{x_{k}x_{k+1}}(y_{k}^{\prime})$ and
$z_{k+1}^{\prime}=f_{x_{k}x_{k+1}}(z_{k}^{\prime})$. (Reminder:
$f_{x_{k}x_{k+1}}=\psi_{x_{k+1}}^{-1}\circ f\circ\psi_{x_{k}}$). By
Proposition 2.19,

$\displaystyle|y_{k+1}-z_{k+1}|_{\infty}\leq$
$\displaystyle\|D_{s,k}\|\cdot|y_{k}-z_{k}|_{\infty}+\sqrt{\epsilon}\eta_{k}^{\frac{\beta}{2}}(|y_{k}-z_{k}|_{\infty}+\mathrm{Lip}(F_{k})|y_{k}-z_{k}|_{\infty})$
$\displaystyle\leq$
$\displaystyle(e^{-\frac{1}{S^{2}(x_{k})}}+2\sqrt{\epsilon}\eta_{k}^{\frac{\beta}{2}})|y_{k}-z_{k}|_{\infty}<e^{-\frac{1-\epsilon^{2}}{S^{2}(x_{k})}}|y_{k}-z_{k}|_{\infty}\leq\ldots\leq
e^{-\sum_{j=0}^{k}\frac{1-\epsilon^{2}}{S^{2}(x_{j})}}|y_{0}-z_{0}|_{\infty}$
$\displaystyle\leq$ $\displaystyle
e^{-(1-\epsilon^{2})C_{\beta,\epsilon}^{-\frac{1}{\gamma}}\sum_{j=0}^{k}(p^{s}_{j})^{\frac{1}{\gamma}}}|y_{0}-z_{0}|_{\infty}=e^{-\frac{(1-\epsilon^{2})C_{\beta,\epsilon}^{-\frac{1}{\gamma}}}{\Gamma}\cdot\Gamma\sum_{j=0}^{k}(I^{-j}(p^{s}_{0}))^{\frac{1}{\gamma}}}|y_{0}-z_{0}|_{\infty}$
$\displaystyle=$ $\displaystyle
e^{-\frac{(1-\epsilon^{2})C_{\beta,\epsilon}^{-\frac{1}{\gamma}}}{\Gamma}\cdot\Gamma\sum_{j=0}^{k}(I^{-j}(p^{s}_{0}))^{\frac{1}{\gamma}}}p_{0}^{s}\frac{|y_{0}-z_{0}|_{\infty}}{p_{0}^{s}}$
$\displaystyle\leq$ $\displaystyle
e^{-\Gamma\sum_{j=0}^{k}(I^{-j}(p^{s}_{0}))^{\frac{1}{\gamma}}}p_{0}^{s}\frac{|y_{0}-z_{0}|_{\infty}}{p_{0}^{s}}=I^{-k}(p_{0}^{s})\cdot\frac{|y_{0}-z_{0}|_{\infty}}{p_{0}^{s}},$
(19)

where the last inequality is true since $\epsilon$ is so small that
$\frac{(1-\epsilon^{2})C_{\beta,\epsilon}^{-\frac{1}{\gamma}}}{\Gamma}>2\geq
1$ (see (3.0.1)). Since $y_{0}^{\prime},z_{0}^{\prime}$ are on the graph of an
$s$-admissible manifold in $\psi_{x_{0}}^{p_{0}^{u},p_{0}^{s}}$, their stable
components are in $R_{p_{0}^{s}}(0)$, so $|y_{0}-z_{0}|\leq 2p_{0}^{s}$. So we
get,

$|y_{k}-z_{k}|_{\infty}\leq 2I^{-k}(p_{0}^{s}).$ (20)

In fact, we even got

$|y_{k}-z_{k}|_{\infty}\leq 2(I^{-k}(p_{0}^{s}))^{2}\frac{1}{p_{0}^{s}}.$ (21)

Now since
$y_{k}^{\prime}=(y_{k},F_{k}(y_{k})),z_{k}^{\prime}=(z_{k},F_{k}(z_{k}))$ and
$\mathrm{Lip}(F_{k})<\epsilon$: $|y_{k}^{\prime}-z_{k}^{\prime}|_{\infty}\leq
2I^{-k}(p_{0}^{s})$. Pesin charts have Lipschitz constant less than $2$, so

$d(f^{k}(y),f^{k}(z))<4I^{-k}(p_{0}^{s}).$

For every $k\geq 0$, $y_{k+1}^{\prime}=f_{x_{k}x_{k+1}}(y_{k}^{\prime})$ and
$z_{k+1}^{\prime}=f_{x_{k}x_{k+1}}(z_{k}^{\prime})$. (Reminder:
$f_{x_{k}x_{k+1}}=\psi_{x_{k+1}}^{-1}\circ f\circ\psi_{x_{k}}$). Then,

$\displaystyle|y_{k}^{\prime}-z_{k}^{\prime}|_{\infty}\leq$ $\displaystyle
I^{-k}(p_{0}^{s})\frac{|y_{0}^{\prime}-z_{0}^{\prime}|_{\infty}}{p_{0}^{s}}=I^{-k}(p_{0}^{s})\frac{|\psi_{x}^{-1}(y)-\psi_{x}^{-1}(z)|_{\infty}}{p_{0}^{s}}$
(22) $\displaystyle\leq$ $\displaystyle 2\|C_{0}^{-1}(x)\|\cdot
I^{-k}(p_{0}^{s})\cdot\frac{d(y,z)}{p_{0}^{s}}<2\|C_{0}^{-1}(x)\|\cdot(I^{-k}(p_{0}^{s}))^{1-\frac{1}{2\gamma}}(p_{0}^{s})^{\frac{1}{2\gamma}}\cdot\frac{d(y,z)}{p_{0}^{s}}$
$\displaystyle\leq$ $\displaystyle C_{\beta,\epsilon}^{\frac{1}{\gamma}}\cdot
2(I^{-k}(p_{0}^{s}))^{1-\frac{1}{2\gamma}}\cdot\frac{d(y,z)}{p_{0}^{s}},\text{
}\forall k\geq 0.$

Similarly, by (21), we get

$|y_{k}^{\prime}-z_{k}^{\prime}|_{\infty}\leq
2\|C_{0}^{-1}(x)\|\cdot(I^{-k}(p_{0}^{s}))^{2}\cdot\frac{d(y,z)}{(p_{0}^{s})^{2}}.$
(23)

In addition
$f^{k}(y)=\psi_{x_{k}}(y_{k}^{\prime}),f^{k}(z)=\psi_{x_{k}}(z_{k}^{\prime})$,
whence

$d(f^{k}(y),f^{k}(z))\leq\mathrm{Lip}(\psi_{x_{k}})|y_{k}^{\prime}-z_{k}^{\prime}|\leq
C_{\beta,\epsilon}^{\frac{1}{\gamma}}\cdot
4(I^{-k}(p_{0}^{s}))^{1-\frac{1}{2\gamma}}\cdot\frac{d(y,z)}{p_{0}^{s}}.$ (24)

Part 2: Let $y\in V^{s}$ and let $u\in T_{y}V^{s}(1)$ some unit vector tangent
to $V^{s}$. $f^{k}(y)\in
W_{k}^{s}\subset\psi_{x_{k}}[R_{Q_{\epsilon}(x_{k})}(0)]$, so
$d_{y}f^{k}u=d_{y_{k}^{\prime}}\psi_{x_{k}}\Big{(}\begin{array}[]{c}\xi_{k}\\\
\zeta_{k}\\\ \end{array}\Big{)}$ where $\Big{(}\begin{array}[]{c}\xi_{k}\\\
\zeta_{k}\\\ \end{array}\Big{)}$ is tangent to the graph of $F_{k}$. Since
$\mathrm{Lip}(F_{k})<\epsilon$, we get that
$|\zeta_{k}|_{\infty}\leq\epsilon|\xi_{k}|_{\infty}$ for all $k$. The defining
relation in the sequence is

$\Big{(}\begin{array}[]{c}\xi_{k+1}\\\ \zeta_{k+1}\\\
\end{array}\Big{)}=\Big{[}\begin{pmatrix}D_{s,k}&\\\
&D_{u,k}\end{pmatrix}+\begin{pmatrix}d_{y_{k}^{\prime}}h_{s}^{(k)}\\\
d_{y_{k}^{\prime}}h_{u}^{(k)}\end{pmatrix}\Big{]}\Big{(}\begin{array}[]{c}\xi_{k}\\\
\zeta_{k}\\\ \end{array}\Big{)}$

where $D_{s,k},D_{u,k}$ are block matrices. So we get

$|\xi_{k+1}|_{\infty}\leq(e^{\frac{-1}{S^{2}(x_{k})}}+\sqrt{\epsilon}\eta_{k}^{\frac{\beta}{2}})|\xi_{k}|_{\infty}\leq
e^{-\frac{1-\epsilon}{S^{2}(x_{k})}}|\xi_{k}|_{\infty}\leq
e^{-\sum_{j=0}^{k}\frac{1-\epsilon}{S^{2}(x_{j})}}|\xi_{0}|_{\infty}.$

Recalling the definition
$d_{y}f^{k}u=d_{y_{k}^{\prime}}\psi_{x_{k}}\Big{(}\begin{array}[]{c}\xi_{k}\\\
\zeta_{k}\\\ \end{array}\Big{)}$, and recalling that
$\|d_{\cdot}\psi_{x_{k}}\|\leq 2$ we see that, $|d_{y}f^{k}u|_{\infty}\leq
4\cdot I^{-k}(p_{0}^{s})\cdot\frac{\|C_{0}(x_{0})^{-1}\|}{p_{0}^{s}}$ as
wished.

Part 3: Fix $n\in\mathbb{N}$, and denote
$A:=\Big{|}\log|d_{y}f^{n}\omega_{s}(y)|-\log|d_{z}f^{n}\omega_{s}(z)|\Big{|}$.
For every $p\in V^{s}$:

$d_{p}f^{n}\omega_{s}(p)=d_{f(p)}f^{n-1}d_{p}f(\omega_{s}(p))=|d_{p}f\omega_{s}(p)|\cdot
d_{f(p)}f^{n-1}\omega_{s}(f(p))=\cdots=\prod_{k=1}^{n}|d_{f^{k}(p)}f\omega_{s}(f^{k}(p))|\cdot\omega_{s}(f^{n}(p)),$

where
$\omega_{s}(f^{k}(p)):=\frac{d_{p}f^{k}\omega_{s}(p)}{|d_{p}f^{k}\omega_{s}(p)|}$
is a normalized volume form of $T_{f^{k}(p)}f^{k}[V^{s}]$. Thus:

$A:=\Big{|}\log\frac{|d_{y}f^{n}\omega_{s}(y)|}{|d_{z}f^{n}\omega_{s}(z)|}\Big{|}\leq\sum_{k=1}^{n}\Big{|}\log|d_{f^{k}(y)}f\omega_{s}(f^{k}(y))|-\log|d_{f^{k}(z)}f\omega_{s}(f^{k}(z))|\Big{|}.$

In section 1.2 we have covered $M$ by a finite collection $\mathcal{D}$ of
open sets $D$, equipped with a smooth map
$\theta_{D}:TD\rightarrow\mathbb{R}^{d}$ s.t
$\theta_{D}|_{T_{x}M}:T_{x}M\rightarrow\mathbb{R}^{d}$ is an isometry and
$\nu_{x}:=\theta_{D}^{-1}:\mathbb{R}^{d}\rightarrow TM$ has the property that
$(x,v)\mapsto\nu_{x}(v)$ is Lipschitz on $D\times B_{1}(0)$. Since $f$ is
$C^{1+\beta}$ and $M$ is compact, $d_{p}fv$ depends in a $\beta$-Hölder way on
$p$, and in a Lipschitz way on $v$. It follows that there exists a constant
$H_{0}>1$ s.t for every $D,D^{\prime}\in\mathcal{D};y^{\prime},z^{\prime}\in
D,\text{ }f(y^{\prime}),f(z^{\prime})\in D^{\prime};u,v\in\mathbb{R}^{d}(1)$:
$\Big{|}\Theta_{D^{\prime}}d_{y^{\prime}}f\nu_{y^{\prime}}(u)-\Theta_{D^{\prime}}d_{z^{\prime}}f\nu_{z^{\prime}}(v)\Big{|}<H_{0}(d(y^{\prime},z^{\prime})^{\beta}+|u-v|_{2})$.
Choose $D_{k}\in\mathcal{D}$ s.t $D_{k}\ni f^{k}(y),f^{k}(z)$. Such sets exist
provided $\epsilon$ is much smaller than the Lebesgue number of $\mathcal{D}$,
since by part 1 $d(f^{k}(y),f^{k}(z))<4\epsilon$. By writing
$Id=\Theta_{D_{k}}\circ\nu_{f^{k}(y)}$ and
$Id=\Theta_{D_{k}}\circ\nu_{f^{k}(z)}$, and recalling that $\Theta_{k}$ are
isometries, we see that:

$\displaystyle A\leq$
$\displaystyle\sum_{k=1}^{n}\Big{|}\log|\Theta_{D_{k+1}}d_{f^{k}(y)}f\nu_{f^{k}(y)}\Theta_{D_{k}}\omega_{s}(f^{k}(y))|-\log|\Theta_{D_{k+1}}d_{f^{k}(z)}f\nu_{f^{k}(z)}\Theta_{D_{k}}\omega_{s}(f^{k}(z))|\Big{|}$
(25) $\displaystyle\leq$
$\displaystyle\sum_{k=1}^{n}M_{f}^{d}\Big{|}\Theta_{D_{k+1}}d_{f^{k}(y)}f\nu_{f^{k}(y)}\Theta_{D_{k}}\omega_{s}(f^{k}(y))-\Theta_{D_{k+1}}d_{f^{k}(z)}f\nu_{f^{k}(z)}\Theta_{D_{k}}\omega_{s}(f^{k}(z))\Big{|},$

by the inequalities $|\log a-\log b|\leq\frac{|a-b|}{\min\\{a,b\\}}$ and the
following estimate:

$\displaystyle\inf_{p\in\\{y,z\\},k\geq
0}|\Theta_{D_{k+1}}d_{f^{k}(p)}f\nu_{f^{k}(p)}\Theta_{D_{k}}\omega_{s}(f^{k}(p))|=$
$\displaystyle\inf_{p\in\\{y,z\\}}|\mathrm{Jac}(d_{p}f|_{T_{p}V^{s}})|$
$\displaystyle=$
$\displaystyle\inf_{p\in\\{y,z\\}}\mathrm{Jac}(\sqrt{(d_{p}f|_{T_{p}V^{s}})^{t}d_{p}f|_{T_{p}V^{s}}})\geq
M_{f}^{-d}.$

Let $\omega_{1}$ be the standard volume form for $\mathbb{R}^{s(x)}$, and
consider the map $t\overset{G_{k}}{\mapsto}(t,F_{k}(t))$,
$G_{k}:R_{p^{s}_{k}}(0)(\subset\mathbb{R}^{s(x)})\rightarrow\mathbb{R}^{d}$.
Then

$\omega_{s}(f^{k}(y))=\pm\frac{d_{y_{k}}(\psi_{x_{k}}\circ
G_{k})\omega_{1}}{|d_{y_{k}}(\psi_{x_{k}}\circ G_{k})\omega_{1}|},\text{
}\omega_{s}(f^{k}(z))=\pm\frac{d_{z_{k}}(\psi_{x_{k}}\circ
G_{k})\omega_{1}}{|d_{z_{k}}(\psi_{x_{k}}\circ G_{k})\omega_{1}|},$

where the $\pm$ notation means the equality holds for one of the signs. Since
we are only interested in the absolute value of the volume forms, we can
assume w.l.o.g that the sign is identical for both of the volume forms.
$\forall u\in\mathbb{R}^{d}(1)$, $|d_{y_{k}/z_{k}}(\psi_{x_{k}}\circ
G_{k})u|\geq\frac{1}{2\cdot\|C_{0}^{-1}(x_{k})\|}$, then
$|d_{y_{k}/z_{k}}(\psi_{x_{k}}\circ
G_{k})\omega_{1}|\geq\frac{1}{(2\|C_{0}^{-1}(x_{k})\|)^{s(x)}}$. Recalling
that $\Theta_{D_{k}}$ are isometries, we get

$\displaystyle|\Theta_{D_{k}}\omega_{s}(f^{k}(y))-\Theta_{D_{k}}\omega_{s}(f^{k}(z))|$
$\displaystyle=\Big{|}\frac{\Theta_{D_{k}}d_{y_{k}}(\psi_{x_{k}}\circ
G_{k})\omega_{1}}{|\Theta_{D_{k}}d_{y_{k}}(\psi_{x_{k}}\circ
G_{k})\omega_{1}|}-\frac{\Theta_{D_{k}}d_{z_{k}}(\psi_{x_{k}}\circ
G_{k})\omega_{1}}{|\Theta_{D_{k}}d_{z_{k}}(\psi_{x_{k}}\circ
G_{k})\omega_{1}|}\Big{|}$ (26)
$\displaystyle\leq(2\|C_{0}^{-1}(x_{k})\|)^{s(x)}\Big{|}\Theta_{D_{k}}d_{y_{k}}(\psi_{x_{k}}\circ
G_{k})\omega_{1}-\Theta_{D_{k}}d_{z_{k}}(\psi_{x_{k}}\circ
G_{k})\omega_{1}\Big{|}.$

Substituting this back in (25) yields $\Big{(}$recall Hadamard’s inequality,
which asserts that if a volume form $\widetilde{\omega}$ is the wedge product
$\widetilde{\omega}=u_{1}\wedge...\wedge u_{l}$, then
$|\widetilde{\omega}|\leq\prod_{i=1}^{l}|u_{i}|_{2}\Big{)}$,

$\displaystyle A\leq$ $\displaystyle
M_{f}^{d}\sum_{k=1}^{n}\Big{|}\Theta_{D_{k+1}}d_{f^{k}(y)}f\nu_{f^{k}(y)}\Theta_{D_{k}}\omega_{s}(f^{k}(y))-\Theta_{D_{k+1}}d_{f^{k}(z)}f\nu_{f^{k}(z)}\Theta_{D_{k}}\omega_{s}(f^{k}(y))$
$\displaystyle+$
$\displaystyle\Theta_{D_{k+1}}d_{f^{k}(z)}f\nu_{f^{k}(z)}\Theta_{D_{k}}\omega_{s}(f^{k}(y))-\Theta_{D_{k+1}}d_{f^{k}(z)}f\nu_{f^{k}(z)}\Theta_{D_{k}}\omega_{s}(f^{k}(z))\Big{|}$
$\displaystyle\leq$ $\displaystyle
M_{f}^{d}\sum_{k=1}^{n}\|d_{f^{k}(z)}f\|^{s(x)}\cdot|\Theta_{D_{k}}\omega_{s}(f^{k}(y))-\Theta_{D_{k}}\omega_{s}(f^{k}(z))|+\|\Theta_{D_{k+1}}d_{f^{k}(y)}f\nu_{f^{k}(y)}-\Theta_{D_{k+1}}d_{f^{k}(z)}f\nu_{f^{k}(z)}\|$
$\displaystyle\leq$ $\displaystyle
M_{f}^{d}\cdot\sum_{k=1}^{n}\Big{(}M_{f}^{d}\cdot(2\|C_{0}^{-1}(x_{k})\|)^{s(x)}\cdot\Big{|}\Theta_{D_{k}}d_{y_{k}}(\psi_{x_{k}}\circ
G_{k})\omega_{1}-\Theta_{D_{k}}d_{z_{k}}(\psi_{x_{k}}\circ
G_{k})\omega_{1}\Big{|}+H_{0}d(f^{k}(y),f^{k}(z))^{\beta}\Big{)}$
$\displaystyle\leq$ $\displaystyle
M_{f}^{d}\cdot\sum_{k=1}^{n}\Big{(}M_{f}^{d}\cdot(2\|C_{0}^{-1}(x_{k})\|)^{s(x)}\cdot[\text{H\"{o}l}_{\frac{\beta}{2}}(d_{\cdot}(\psi_{x_{k}}\circ
G_{k}))|y_{k}-z_{k}|^{\frac{\beta}{2}}]^{s(x)}+H_{0}d(f^{k}(y),f^{k}(z))^{\beta}\Big{)}$
$\displaystyle\leq$ $\displaystyle
M_{f}^{d}\cdot\sum_{k=1}^{n}\Big{(}M_{f}^{d}\cdot(2\cdot\frac{1}{2})^{s(x)}(2\|C_{0}^{-1}(x_{k})\||y_{k}-z_{k}|^{\frac{\beta}{2}})^{s(x)}+H_{0}d(f^{k}(y),f^{k}(z))^{\beta}\Big{)}$
$\displaystyle\leq$ $\displaystyle
M_{f}^{d}\cdot\sum_{k=1}^{n}\Big{(}M_{f}^{d}(2C_{\beta,\epsilon}^{\frac{\beta}{4}}|y_{k}-z_{k}|^{\frac{\beta}{4}})^{s(x)}+H_{0}d(f^{k}(y),f^{k}(z))^{\beta}\Big{)}$
$\displaystyle\leq$ $\displaystyle
M_{f}^{d}\cdot\sum_{k=1}^{n}\Big{(}M_{f}^{d}\cdot(2C_{\beta,\epsilon}^{\frac{\beta}{4}}(2\|C_{0}^{-1}(x)\|\cdot(I^{-k}(p_{0}^{s}))^{2}\cdot\frac{d(y,z)}{(p_{0}^{s})^{2}})^{\frac{\beta}{4}})+H_{0}(C_{\beta,\epsilon}^{\frac{1}{\gamma}}\cdot
2(I^{-k}(p_{0}^{s}))^{1-\frac{1}{2\gamma}}\cdot\frac{d(y,z)}{p_{0}^{s}})^{\beta}\Big{)}$
$\displaystyle\leq$ $\displaystyle
M_{f}^{d}\cdot\sum_{k=1}^{n}\Big{(}M_{f}^{d}\cdot(2C_{\beta,\epsilon}^{\frac{\beta}{4}}((I^{-k}(p_{0}^{s}))^{2}\cdot\frac{d(y,z)}{(p_{0}^{s})^{3}})^{\frac{\beta}{4}})+H_{0}(C_{\beta,\epsilon}^{\frac{1}{\gamma}}\cdot
2(I^{-k}(p_{0}^{s}))^{1-\frac{1}{2\gamma}}\cdot\frac{d(y,z)}{p_{0}^{s}})^{\beta}\Big{)}$
$\displaystyle\leq$ $\displaystyle 3M_{f}^{2d}H_{0}\cdot
C_{\beta,\epsilon}^{\min\\{\frac{\beta}{4},\frac{1}{\gamma}\\}}\cdot\sum_{k=1}^{n}(I^{-k}(p_{0}^{s}))^{\frac{\beta}{2}}\cdot(\frac{d(y,z)}{(p_{0}^{s})^{3}})^{\frac{\beta}{4}}\leq\epsilon^{2}\cdot(\frac{d(y,z)}{(p_{0}^{s})^{3}})^{\frac{\beta}{4}},$
(27)

where Höl${}_{\frac{\beta}{2}}(d_{\cdot}G_{k})\leq\frac{1}{2}$, and
Lip($d_{\cdot}\psi_{x_{k}})\leq 2$ since $\|C_{0}(x_{k})\|\leq 1$. The seventh
inequality in (4.1.1) uses the fact that $M_{f}>1$, and the penultimate
inequality uses the fact that $|y_{k}-z_{k}|<1$ to replace $s(x)$ with $1$,
and uses (23).

By the choice of $Q_{\epsilon}(\cdot)$ (Definition 2.12) and the inequality
$p^{s}_{k}\leq Q_{\epsilon}(x_{k})$, and since $\gamma>\frac{2}{\beta}$, we
have
$\|C_{0}^{-1}(x_{k})\|\cdot(p_{k}^{s})^{\frac{\beta}{4}}<C_{\beta,\epsilon}^{\frac{\beta}{4}}$
for all $k\geq 0$. In addition, Lemma 2.6 tells us that $\sum_{k\geq
0}(I^{-k}(p_{0}^{s}))^{\frac{\beta}{2}}\leq\sum_{k\geq
0}(I^{-k}(1))^{\frac{\beta}{2}}<\infty$. So for small enough $\epsilon$,
(4.1.1) holds. This estimate is uniform in $n$, and Part (3) is proven.

∎

######  Corollary 4.5.

Let $V^{s}$ be an admissible $s$-manifold in $\psi_{x}^{p^{s},p^{u}}$ which
stays in windows. Then $\forall y\in V^{s}$,

$S^{2}(y)\leq 2\cdot 4\frac{\|C_{0}^{-1}(x)\|}{p^{s}}\sum_{m\geq
0}(I^{-m}(1))^{2}<\infty.$

A similar bound holds for $U^{2}(\cdot)$ on $u$-manifolds which stay in
windows.

Notice: In a setup where we are given two double charts
$\psi_{x}^{p^{s},p^{u}}\rightarrow\psi_{y}^{q^{s},q^{u}}$, if $V^{s}$ is an
$s$-admissible manifold in $\psi_{y}^{q^{s},q^{u}}$ which stays in windows,
then also $\mathcal{F}(V^{s})$ is an $s$-admissible manifold in
$\psi_{x}^{p^{s},p^{u}}$ which stays in windows. For the next lemma, recall
Definition 2.9 for $S(\cdot,\cdot)$.

######  Lemma 4.6.

Let $\psi_{x}^{p^{s},p^{u}}\rightarrow\psi_{y}^{q^{s},q^{u}}$. Let $V^{s}$ be
an $s$-admissible manifold in $\psi_{y}^{q^{s},q^{u}}$ which stays in windows.
Let $\mathcal{F}(V^{s})$ be the Graph Transform of $V^{s}$ w.r.t to the edge
$\psi_{x}^{p^{s},p^{u}}\rightarrow\psi_{y}^{q^{s},q^{u}}$. Let $q\in
f[\mathcal{F}(V^{s})]\subset V^{s}$. Then
$\exists\pi_{x}:T_{f^{-1}(q)}\mathcal{F}(V^{s})\rightarrow
H^{s}(x),\pi_{y}:T_{q}V^{s}\rightarrow H^{s}(y)$ invertible linear
transformations s.t $\forall\rho\geq e^{\Gamma
Q_{\epsilon}^{\frac{\beta}{4}-\frac{1}{4\gamma}}(x)}$: if
$\frac{S(q,\lambda)}{S(y,\pi_{y}\lambda)}\in[\rho^{-1},\rho]$ then

$\frac{S(f^{-1}(q),\xi)}{S(x,\pi_{x}\xi)}\in[e^{\Gamma
Q_{\epsilon}(x)^{\frac{\beta}{4}+\frac{3}{4\gamma}}}\rho^{-1},e^{-\Gamma
Q_{\epsilon}(x)^{\frac{\beta}{4}+\frac{3}{4\gamma}}}\rho],$

where $\lambda=d_{f^{-1}(q)}f\xi$ and $\xi\in T_{f^{-1}(q)}\mathcal{F}(V^{s})$
is arbitrary. A similar statement holds for the unstable case. We will also
see that $\frac{|\pi_{x}\xi|}{|\xi|}=e^{\pm
Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}}$ for all $\xi\in
T_{f^{-1}(q)}\mathcal{F}(V^{s})$ (similarly for $\pi_{y}$).

###### Proof.

Recall the disks $\mathcal{D}=\\{D\\}$ and the isometries
$\Theta_{D}:TD\rightarrow\mathbb{R}^{d},\nu_{a}=\Theta_{D}^{-1}|_{T_{a}M},a\in
D\subset M$ from Definition 2.16. If $\epsilon$ is smaller than the Lebesgue
number of $\mathcal{D}$, then $\exists D$ which contains
$x,f^{-1}(q),f^{-1}(y)$. Let $\Theta=\Theta_{D}$, and set $C_{1}:=\Theta\circ
C_{0}(x),C_{2}:=\Theta\circ C_{0}(f^{-1}(y))$. Let $F,G$ denote the
representing functions of $V^{s},\mathcal{F}(V^{s})$, respectively.

Let $\xi\in T_{f^{-1}(q)}\mathcal{F}(V^{s})$. Write
$t=\psi_{x}^{-1}(f^{-1}(q))=(t_{s},G(t_{s}))$, and
$\xi=d_{t}\psi_{x}\Big{(}\begin{array}[]{c}v\\\ d_{t_{s}}Gv\\\
\end{array}\Big{)}$ for a vector $v\in\mathbb{R}^{s(x)}$. We define
$\pi_{x}\xi:=d_{(0,0)}\psi_{x}\Big{(}\begin{array}[]{c}v\\\ 0\\\
\end{array}\Big{)}$ ($\pi_{y}$ is defined analogously). Denote by $E_{0}$ the
$|\cdot|_{\infty}$-Lipschitz constant of
$(x,\underline{u},\underline{v})\mapsto[\Theta_{D}\circ
d_{\underline{v}}\exp][\nu_{x}\underline{u}]$.

We begin with bounding the distortion of $\pi_{x}$ (assume w.l.o.g666Otherwise
cancel its size from both the denominator and numerator of the LHS of eq.
(44). $|\xi|=1$):

$\displaystyle|\Theta\xi-\Theta\pi_{x}\xi|=$ $\displaystyle|\Theta
d_{(t_{s},G(t_{s}))}\psi_{x}\Big{(}\begin{array}[]{c}v\\\ d_{t_{s}}Gv\\\
\end{array}\Big{)}-\Theta d_{(0,0)}\psi_{x}\Big{(}\begin{array}[]{c}v\\\ 0\\\
\end{array}\Big{)}|$ (32) $\displaystyle=$
$\displaystyle|\Theta\Big{(}d_{C_{0}(x)(t_{s},G(t_{s}))}\exp_{x}\Big{)}\nu_{x}\circ
C_{1}\Big{(}\begin{array}[]{c}v\\\ d_{t_{s}}Gv\\\
\end{array}\Big{)}-\Theta\Big{(}d_{C_{0}(x)(0,0)}\exp_{x}\Big{)}\nu_{x}\circ
C_{1}\Big{(}\begin{array}[]{c}v\\\ 0\\\ \end{array}\Big{)}|$ (37)
$\displaystyle\leq$ $\displaystyle
E_{0}\Big{(}\Big{|}C_{1}\Big{(}\Big{(}\begin{array}[]{c}v\\\ d_{t_{s}}Gv\\\
\end{array}\Big{)}-\Big{(}\begin{array}[]{c}v\\\ 0\\\
\end{array}\Big{)}\Big{)}\Big{|}+\Big{|}C_{0}(x)\Big{(}(t_{s},G(t_{s}))-(0,0)\Big{)}\Big{|}\Big{)}$
(42) $\displaystyle\leq$ $\displaystyle
E_{0}\Big{(}|v|\cdot\|d_{t_{s}}G\|+|\exp_{x}^{-1}f^{-1}(q)|\Big{)}=E_{0}\Big{(}|v|\cdot\|d_{t_{s}}G\|+d(f^{-1}(q),x)\Big{)}(\because
C_{1}\text{ is a contraction})$ $\displaystyle\leq$ $\displaystyle
3E_{0}Q_{\epsilon}(x)^{\beta/2}\|C_{0}^{-1}(x)\|\leq
3E_{0}C_{\beta,\epsilon}^{\frac{1}{2\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}\text{
}(\because|v|\leq 2\|C_{0}^{-1}(x)\|,\|d_{t_{s}}G\|\leq
Q_{\epsilon}(x)^{\beta/2}).$

Therefore $\frac{|\pi_{x}\xi|}{|\xi|}=e^{\pm
Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}}$ (similarly,
$\frac{|\pi_{y}\xi^{\prime}|}{|\xi^{\prime}|}=e^{\pm
Q_{\epsilon}(y)^{\frac{\beta}{2}-\frac{1}{2\gamma}}},\xi^{\prime}\in
T_{q}V^{s}\setminus\\{0\\}$). We continue to decompose:

$\frac{S(x,\pi_{x}\xi)}{S(f^{-1}(q),\xi)}=\frac{S(f^{-1}(y),\eta)}{S(f^{-1}(q),\xi)}\cdot\frac{S(x,\pi_{x}\xi)}{S(f^{-1}(y),\eta)},$
(43)

where $\eta:=d_{y}(f^{-1})\pi_{y}d_{f^{-1}(q)}f\xi\in H^{s}(f^{-1}(y))$
($\|\pi_{y}\|\leq e^{Q_{\epsilon}^{\frac{\beta}{2}-\frac{1}{2\gamma}}(y)}$,
hence $|\eta|\leq 2M_{f}^{2}$). We study the second factor. Denote

$u_{1}:=C_{0}^{-1}(x)\xi,u_{2}:=C_{0}^{-1}(f^{-1}(y))\eta.$

By the definitions of $S(\cdot,\cdot)$ and $C_{0}(\cdot)$ (Definition 2.9,
Theorem 2.2), the inverse of the second factor satisfies:

$\displaystyle\Big{|}\frac{S(f^{-1}(y),\eta)}{S(x,\pi_{x}\xi)}-1\Big{|}=$
$\displaystyle\Big{|}\frac{|C_{0}^{-1}(f^{-1}(y))\eta|}{|C_{0}^{-1}(x)\pi_{x}\xi|}-1\Big{|}=\Big{|}\frac{|C_{2}^{-1}\Theta\eta|}{|C_{1}^{-1}\Theta\pi_{x}\xi|}-1\Big{|}$
(44) $\displaystyle\leq$ $\displaystyle\Big{|}\frac{|C_{2}^{-1}\Theta\eta-
C_{1}^{-1}\Theta\eta|+|C_{1}^{-1}\Theta\eta-
C_{1}^{-1}\Theta\pi_{x}\xi|+|C_{1}^{-1}\Theta\pi_{x}\xi|}{|C_{1}^{-1}\Theta\pi_{x}\xi|}-1\Big{|}$
$\displaystyle\leq$
$\displaystyle\frac{1}{|C_{0}^{-1}(x)\pi_{x}\xi|}\Big{(}\|C_{2}^{-1}-C_{1}^{-1}\|\cdot|\eta|+\|C_{1}^{-1}\|\cdot|\Theta\pi_{x}\xi-\Theta\eta|\Big{)}$
$\displaystyle\leq$ $\displaystyle
2\Big{(}2M_{f}^{2}Q_{\epsilon}^{4}(x)Q_{\epsilon}^{4}(f^{-1}(y))+\|C_{0}^{-1}(x)\|\cdot|\Theta\pi_{x}\xi-\Theta\eta|\Big{)}$
$\displaystyle(\because$ $\displaystyle\|C_{0}(x)\|\leq 1\text{, Lemma
\ref{overlap}},|\eta|\leq 2M_{f}^{2},|\pi_{x}\xi|\geq
e^{-2Q_{\epsilon}^{\frac{\beta}{2}-\frac{1}{2\gamma}}(x)}\geq\frac{1}{2}).$

By the definition of $\omega_{0}$ (Lemma 2.15),
$\frac{Q_{\epsilon}(f^{-1}(y))}{Q_{\epsilon}(y)}\leq\omega_{0}$. Substituting
this gives:

$\displaystyle\Big{|}\frac{S(f^{-1}(y),\eta)}{S(x,\pi_{x}\xi)}-1\Big{|}\leq$
$\displaystyle
4M_{f}^{2}\omega_{0}Q_{\epsilon}^{4}(x)Q_{\epsilon}^{4}(y)+2\cdot\|C_{0}^{-1}(x)\|\cdot|\Theta\pi_{x}\xi-\Theta\eta|$
$\displaystyle\leq$
$\displaystyle\frac{1}{4}Q_{\epsilon}^{3}(x)Q_{\epsilon}^{3}(y)+2\|C_{0}^{-1}(x)\|\cdot|\Theta\pi_{x}\xi-\Theta\eta|,$
(45)

for small enough $\epsilon$. Next we bound $|\Theta\pi_{x}\xi-\Theta\eta|$:

Step 1: $|\Theta\xi-\Theta\pi_{x}\xi|\leq
3E_{0}C_{\beta,\epsilon}^{\frac{1}{2\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}\leq\frac{1}{2}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}$
by (32).

Step 2: Estimation of $|\Theta\xi-\Theta\eta|$. Let
$\lambda:=d_{f^{-1}(q)}f\xi$ and $\hat{\lambda}=\frac{\lambda}{|\lambda|}$
777$\hat{\lambda}$ is well defined since $f$ is a diffeomorphism and
$|\xi|\neq 0$.. Then

$\displaystyle|\Theta\eta-\Theta\xi|\equiv$ $\displaystyle|\Theta
d_{y}(f^{-1})\pi_{y}d_{f^{-1}(q)}f\xi-\Theta\xi|=|\Theta
d_{y}(f^{-1})\pi_{y}d_{f^{-1}(q)}f\xi-\Theta d_{q}(f^{-1})d_{f^{-1}(q)}f\xi|$
$\displaystyle=$ $\displaystyle|\Theta d_{y}(f^{-1})\pi_{y}\lambda-\Theta
d_{q}(f^{-1})\lambda|=|\Theta
d_{y}(f^{-1})\nu_{y}\Theta^{\prime}\pi_{y}\lambda-\Theta
d_{q}(f^{-1})\nu_{q}\Theta^{\prime}\lambda|$ $\displaystyle\leq$
$\displaystyle|\Theta d_{y}(f^{-1})\nu_{y}\Theta^{\prime}\pi_{y}\lambda-\Theta
d_{q}(f^{-1})\nu_{q}\Theta^{\prime}\pi_{y}\lambda|+|\Theta
d_{q}(f^{-1})\nu_{q}\Theta^{\prime}\pi_{y}\lambda-\Theta
d_{q}(f^{-1})\nu_{q}\Theta^{\prime}\lambda|$ $\displaystyle\leq$
$\displaystyle\|\Theta d_{y}(f^{-1})\nu_{y}-\Theta
d_{q}(f^{-1})\nu_{q}\|\cdot|\pi_{y}\lambda|+M_{f}|\Theta^{\prime}\pi_{y}\lambda-\Theta^{\prime}\lambda|$
$\displaystyle\leq$ $\displaystyle H_{0}\cdot
d(q,y)^{\beta}|\pi_{y}\lambda|+M_{f}|\Theta^{\prime}\pi_{y}\hat{\lambda}-\Theta^{\prime}\hat{\lambda}|\cdot|\lambda|,$

where $\Theta^{\prime}$ is the linear isometry for the neighborhood containing
$y$ and $q$, as in Definition 2.16, and $H_{0}$ is the bound for
Höl$(d_{\cdot}f^{-1})$ as in the proof of Proposition 4.4. $|\lambda|\leq
M_{f}$. As in (32),
$|\Theta^{\prime}\pi_{y}\hat{\lambda}-\Theta^{\prime}\hat{\lambda}|\leq
3E_{0}C_{\beta,\epsilon}^{\frac{1}{2\gamma}}Q_{\epsilon}(y)^{\frac{\beta}{2}-\frac{1}{2\gamma}}$,
and hence also
$|\pi_{y}\lambda|=|\pi_{y}\hat{\lambda}|\cdot|\lambda|\leq(1+Q_{\epsilon}(y)^{\frac{\beta}{2}-\frac{1}{2\gamma}})\cdot
M_{f}<2M_{f}$. Therefore,

$|\Theta\eta-\Theta\xi|\leq M_{f}\cdot(2H_{0}\cdot
Q_{\epsilon}(y)^{\beta}+M_{f}\cdot
3E_{0}C_{\beta,\epsilon}^{\frac{1}{2\gamma}}Q_{\epsilon}(y)^{\frac{\beta}{2}-\frac{1}{2\gamma}})<2M_{f}^{2}(H_{0}+3E_{0})C_{\beta,\epsilon}^{\frac{1}{2\gamma}}Q_{\epsilon}(y)^{\frac{\beta}{2}-\frac{1}{2\gamma}}.$

Since $\psi_{x}^{p^{s},p^{u}}\rightarrow\psi_{y}^{q^{s},q^{u}}$,
$\psi_{f(x)}^{p^{s}\wedge p^{u}}$ and $\psi_{y}^{q^{s}\wedge q^{u}}$
$I$-overlap. Hence Lemma 2.18 gives us the bound

$\displaystyle\frac{Q_{\epsilon}(y)}{Q_{\epsilon}(x)}=$
$\displaystyle\frac{Q_{\epsilon}(f(x))}{Q_{\epsilon}(x)}\cdot\frac{Q_{\epsilon}(y)}{Q_{\epsilon}(f(x))}\leq\omega_{0}\frac{Q_{\epsilon}(y)}{Q_{\epsilon}(f(x))}\leq\omega_{0}\Big{(}\frac{\|C_{0}^{-1}(y)\|}{\|C_{0}^{-1}(f(x))\|}\Big{)}^{-2\gamma}$
$\displaystyle\leq$
$\displaystyle\omega_{0}(e^{Q_{\epsilon}(f(x))Q_{\epsilon}(y)})^{2\gamma}\leq\omega_{0}e^{\epsilon}\text{
(for small enough }\epsilon\text{)}.$

Thus

$|\Theta\eta-\Theta\xi|\leq
2M_{f}^{2}(H_{0}+3E_{0})(e^{\epsilon}\omega_{0})^{\frac{\beta}{2}-\frac{1}{2\gamma}}C_{\beta,\epsilon}^{\frac{1}{2\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}\leq\frac{1}{2}C_{\beta,\epsilon}^{\frac{1}{4\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}\text{
for all }\epsilon\text{ enough}.$

Step 3: Adding the estimates in steps 1 and 2, we obtain

$\displaystyle|\Theta\pi_{x}\xi-\Theta\eta|\leq$
$\displaystyle|\Theta\pi_{x}\xi-\Theta\xi|+|\Theta\xi-\Theta\eta|$
$\displaystyle\leq$ $\displaystyle
Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}(\frac{1}{2}C_{\beta,\epsilon}^{\frac{1}{4\gamma}}+3E_{0}C_{\beta,\epsilon}^{\frac{1}{2\gamma}})\leq
C_{\beta,\epsilon}^{\frac{1}{4\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}.$

Substituting this in (4.1.1) gives (for small enough $\epsilon$):

$\Big{|}\frac{S(f^{-1}(y),\eta)}{S(x,\pi_{x}\xi)}-1\Big{|}\leq\frac{1}{4}Q^{3}_{\epsilon}(x)Q^{3}_{\epsilon}(y)+2\|C_{0}^{-1}(x)\|\cdot
C_{\beta,\epsilon}^{\frac{1}{4\gamma}}Q_{\epsilon}^{\frac{\beta}{2}-\frac{1}{2\gamma}}(x)\leq\frac{1}{2}C_{\beta,\epsilon}^{\frac{1}{2\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{\gamma}},$
(46)

and hence,

$\frac{S(f^{-1}(y),\eta)}{S(x,\pi_{x}\xi)}=e^{\pm
C_{\beta,\epsilon}^{\frac{1}{2\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{\gamma}}}.$

This concludes the study of the second factor of the RHS of (43).

It remains to study the first factor of the RHS of (43). We begin with the
identity

$\displaystyle S^{2}(f^{-1}(y),\eta)=$ $\displaystyle
2\sum_{m=0}^{\infty}|d_{f^{-1}(y)}f^{m}\eta|^{2}=2\Big{(}|\eta|^{2}+\sum_{m=1}^{\infty}|d_{y}f^{m-1}d_{f^{-1}(y)}f\eta|^{2}\Big{)}=2|\eta|^{2}+S^{2}(y,d_{f^{-1}(y)}f\eta).$

Similarly, and by the assumption of the lemma :

$\displaystyle S^{2}(f^{-1}(q),\xi)=$ $\displaystyle
2|\xi|^{2}+S^{2}(q,d_{f^{-1}(q)}f\xi)\leq
2|\xi|^{2}+\rho^{2}S^{2}(y,\pi_{y}d_{f^{-1}(q)}f\xi).$

By the choice of $\eta$, $d_{f^{-1}(y)}f\eta=\pi_{y}d_{f^{-1}(q)}f\xi$, so

$S^{2}(f^{-1}(q),\xi)\leq 2|\xi|^{2}+\rho^{2}S^{2}(y,d_{f^{-1}(y)}f\eta)$

and so,

$\displaystyle\frac{S^{2}(f^{-1}(q),\xi)}{S^{2}(f^{-1}(y),\eta)}\leq$
$\displaystyle\frac{2|\xi|^{2}+\rho^{2}S^{2}(y,d_{f^{-1}(y)}f\eta)}{2|\eta|^{2}+S^{2}(y,d_{f^{-1}(y)}f\eta)}=\rho^{2}-\frac{2(\rho^{2}-1)|\eta|^{2}+2(|\eta|^{2}-|\xi|^{2})}{2|\eta|^{2}+S^{2}(y,d_{f^{-1}(y)}f\eta)}$
$\displaystyle=$
$\displaystyle\rho^{2}-\frac{2(\rho^{2}-1)|\eta|^{2}+2(|\eta|^{2}-|\xi|^{2})}{S^{2}(f^{-1}(y),\eta)}.$
(47)

By the bound achieved in step 2, and since $|\xi|=1$, $|\eta|=e^{\pm
C_{\beta,\epsilon}^{\frac{1}{4\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}}$
(for small enough $\epsilon$). Hence $\Big{|}|\xi|^{2}-|\eta|^{2}\Big{|}\leq
2(e^{C_{\beta,\epsilon}^{\frac{1}{4\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}}-1)\leq
4C_{\beta,\epsilon}^{\frac{1}{4\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}$.
We get that for $\epsilon$ small enough, since $Q_{\epsilon}(x)\ll\epsilon$,

$\displaystyle\rho^{2}-1$ $\displaystyle\geq e^{2\Gamma
Q_{\epsilon}(x)^{\frac{\beta}{4}-\frac{1}{4\gamma}}}-1\geq 2\Gamma
Q_{\epsilon}(x)^{\frac{\beta}{4}-\frac{1}{4\gamma}}$
$\displaystyle\Rightarrow$
$\displaystyle(\rho^{2}-1)|\eta|^{2}+(|\xi|^{2}-|\eta|^{2})\geq(\rho^{2}-1)|\eta|^{2}-4C_{\beta,\epsilon}^{\frac{1}{4\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}=(\rho^{2}-1)[|\eta|^{2}-\frac{4C_{\beta,\epsilon}^{\frac{1}{4\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}}{\rho^{2}-1}]$
$\displaystyle\geq(\rho^{2}-1)[e^{-2C_{\beta,\epsilon}^{\frac{1}{4\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}}-\frac{4C_{\beta,\epsilon}^{\frac{1}{4\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}}{2\Gamma
Q_{\epsilon}(x)^{\frac{\beta}{4}-\frac{1}{4\gamma}}}]\geq(\rho^{2}-1)e^{-C_{\beta,\epsilon}^{\frac{1}{5\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{4}-\frac{1}{4\gamma}}}.$

Plugging this in (47) yields

$\displaystyle\frac{S^{2}(f^{-1}(q),\xi)}{S^{2}(f^{-1}(y),\eta)}\leq\rho^{2}-\frac{2(\rho^{2}-1)e^{-C_{\beta,\epsilon}^{\frac{1}{5\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{4}-\frac{1}{4\gamma}}}}{S^{2}(f^{-1}(y),\eta)}\leq\rho^{2}-\frac{2(\rho^{2}-1)e^{-2C_{\beta,\epsilon}^{\frac{1}{5\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{4}-\frac{1}{4\gamma}}}}{S^{2}(x,\pi_{x}\xi)},$

where the last inequality is because, as we saw above,
$\frac{S(f^{-1}(y),\eta)}{S(x,\pi_{x}\xi)}=e^{\pm
C_{\beta,\epsilon}^{\frac{1}{2\gamma}}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{\gamma}}$
. So

$\frac{S^{2}(f^{-1}(q),\xi)}{S^{2}(f^{-1}(y),\eta)}\leq\rho^{2}-\frac{2(\rho^{2}-1)e^{-2C_{\beta,\epsilon}^{\frac{1}{5\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{4}-\frac{1}{4\gamma}}}}{\|C_{0}^{-1}(x)\|^{2}\cdot|\pi_{x}\xi|^{2}}.$

Substituting $|\xi|=1$, $\frac{|\pi_{x}\xi|}{|\xi|}=e^{\pm
Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}}$ gives:

$\frac{S^{2}(f^{-1}(q),\xi)}{S^{2}(f^{-1}(y),\eta)}\leq\rho^{2}-\frac{2(\rho^{2}-1)e^{-2C_{\beta,\epsilon}^{\frac{1}{5\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{4}-\frac{1}{4\gamma}}}}{\|C_{0}^{-1}(x)\|^{2}\cdot
e^{2Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}}}\leq\rho^{2}\Big{(}1-\frac{2(1-\frac{1}{\rho^{2}})e^{-3C_{\beta,\epsilon}^{\frac{1}{5\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{4}-\frac{1}{4\gamma}}}}{\|C_{0}^{-1}(x)\|^{2}}\Big{)}$

For $\rho\geq e^{\Gamma Q_{\epsilon}^{\frac{\beta}{4}-\frac{1}{4\gamma}}(x)}$,
$1-\frac{1}{\rho^{2}}\geq 1-e^{-2\Gamma
Q_{\epsilon}^{\frac{\beta}{4}-\frac{1}{4\gamma}}(x)}\geq\Gamma
Q_{\epsilon}^{\frac{\beta}{4}-\frac{1}{4\gamma}}(x)$, so:

$\frac{S^{2}(f^{-1}(q),\xi)}{S^{2}(f^{-1}(y),\eta)}\leq\rho^{2}\Big{(}1-\frac{2\Gamma
Q_{\epsilon}^{\frac{\beta}{4}-\frac{1}{4\gamma}}(x)e^{-3C_{\beta,\epsilon}^{\frac{1}{5\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{4}-\frac{1}{4\gamma}}}}{\|C_{0}^{-1}(x)\|^{2}}\Big{)}.$

By definition,
$Q_{\epsilon}(x)\leq\frac{C_{\beta,\epsilon}}{\|C_{0}^{-1}(x)\|^{2\gamma}}$.
Hence,

$\frac{S^{2}(f^{-1}(q),\xi)}{S^{2}(f^{-1}(y),\eta)}\leq\rho^{2}(1-3\Gamma
Q_{\epsilon}(x)^{\frac{\beta}{4}-\frac{1}{4\gamma}+\frac{1}{\gamma}})\leq\rho^{2}e^{-3\Gamma
Q_{\epsilon}(x)^{\frac{\beta}{4}+\frac{3}{4\gamma}}}.$

Similarly,
$\frac{S^{2}(f^{-1}(q),\xi)}{S^{2}(f^{-1}(y),\eta)}\geq\rho^{-2}e^{3\Gamma
Q_{\epsilon}(x)^{\frac{\beta}{4}+\frac{3}{4\gamma}}}$.

In summary, $\frac{S^{2}(x,\pi_{x}\xi)}{S^{2}(f^{-1}(q),\xi)}$ is the product
of two factors (see (43)). The second factor is bounded by $e^{\pm
C_{\beta,\epsilon}^{\frac{1}{2\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{\gamma}}}$,
and the first factor takes values in $[\rho^{-2}e^{3\Gamma
Q_{\epsilon}(x)^{\frac{\beta}{4}+\frac{3}{4\gamma}}},\rho^{2}e^{-3\Gamma
Q_{\epsilon}(x)^{\frac{\beta}{4}+\frac{3}{4\gamma}}}]$. Therefore,

$\displaystyle\frac{S^{2}(f^{-1}(q),\xi)}{S^{2}(f^{-1}(y),\eta)}\leq$
$\displaystyle\rho^{2}e^{-3\Gamma
Q_{\epsilon}(x)^{\frac{\beta}{4}+\frac{3}{4\gamma}}+C_{\beta,\epsilon}^{\frac{1}{5\gamma}}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{\gamma}}}$
$\displaystyle\leq$ $\displaystyle\rho^{2}e^{-2\Gamma
Q_{\epsilon}^{\frac{\beta}{4}+\frac{3}{4\gamma}}(x)\left(\frac{3}{2}-\frac{C_{\beta,\epsilon}^{\frac{1}{5\gamma}}}{2\Gamma}Q_{\epsilon}^{\frac{\beta}{2}-\frac{1}{\gamma}-\frac{\beta}{4}-\frac{3}{4\gamma}}(x)\right)}$
$\displaystyle=$ $\displaystyle\rho^{2}e^{-2\Gamma
Q_{\epsilon}^{\frac{\beta}{4}+\frac{3}{4\gamma}}(x)\left(\frac{3}{2}-\frac{C_{\beta,\epsilon}^{\frac{1}{5\gamma}}}{2\Gamma}Q_{\epsilon}^{\frac{\beta}{4}+\frac{1}{4\gamma}}(x)\right)}\leq\rho^{2}\Big{(}e^{-\Gamma
Q_{\epsilon}(x)^{\frac{\beta}{4}+\frac{3}{4\gamma}}}\Big{)}^{2}.$

and
$\frac{S^{2}(f^{-1}(q),\xi)}{S^{2}(f^{-1}(y),\eta)}\geq\rho^{-2}\Big{(}e^{\Gamma
Q_{\epsilon}(x)^{\frac{\beta}{4}+\frac{3}{4\gamma}}}\Big{)}^{2}$ for small
enough $\epsilon$. This concludes the proof.

∎

######  Lemma 4.7.

The following holds for any $\epsilon$ small enough. For any two chains
$(\psi_{x_{i}}^{p^{s}_{i},p^{u}_{i}})_{i\in\mathbb{Z}},(\psi_{y_{i}}^{q^{s}_{i},q^{u}_{i}})_{i\in\mathbb{Z}}$
with each having some symbol repeating infinitely often in the future, if
$\pi[(\psi_{x_{i}}^{p^{s}_{i},p^{u}_{i}})_{i\in\mathbb{Z}}]=\pi[(\psi_{y_{i}}^{q^{s}_{i},q^{u}_{i}})_{i\in\mathbb{Z}}]=p$
then $\forall k\in\mathbb{Z}$, and for any $\xi\in T_{f^{k}(p)}V^{s}_{k}$,

$\frac{S(x_{k},\pi_{x,k}\xi)}{S(y_{k},\pi_{y,k}\xi)}=e^{\pm 2\Gamma
Q_{\epsilon}^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}(x_{k})},$

where $\pi_{x,k},\pi_{y,k}$ denote respectively the maps
$\pi_{x_{k}}:T_{f^{k}(p)}V^{s}_{k}\rightarrow
H^{s}(x_{k}),\pi_{y_{k}}:T_{f^{k}(p)}U^{s}_{k}\rightarrow H^{s}(y_{k})$ as in
Lemma 4.6, and $V^{s}_{k}=V^{s}((\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\geq
k}),U^{s}_{k}=V^{s}((\psi_{y_{i}}^{q_{i}^{s},q_{i}^{u}})_{i\geq k})$. A
similar statement holds for the unstable manifold.

Remark: Notice that $T_{f^{k}(p)}V^{s}_{k}=T_{f^{k}(p)}U^{s}_{k}$, which makes
the choice and use of $\xi$ well defined. This can be seen by the facts that
the orbit $(f^{k}(p))_{k\in\mathbb{Z}}$ stays on the intersection of stable
and unstable manifolds, and Proposition 4.4(2) gives lower bounds for the
contraction/expansion by the differential on the respective manifolds’ tangent
spaces. Thus, the stable space of $f^{k}(p)$ (which exists by Proposition
4.4(2)) coincides with $T_{f^{k}(p)}V^{s}_{k}$, and
$T_{f^{k}(p)}V^{s}_{k}=\text{stable space of }f^{k}(p)=T_{f^{k}(p)}U^{s}_{k}$.

###### Proof.

Denote $\vec{v}=(\psi_{x_{i}}^{p^{s}_{i},p^{u}_{i}})_{i\in\mathbb{Z}}$ and
$\vec{u}=(\psi_{y_{i}}^{q^{s}_{i},q^{u}_{i}})_{i\in\mathbb{Z}}$. $V^{s}_{k}$
and $U^{s}_{k}$ are $s$-admissible manifolds which stay in windows. We will
prove that:

$\forall\text{ }\xi\in T_{f^{k}(p)}V^{s}_{k}=T_{f^{k}(p)}U^{s}_{k}:\text{
}\frac{S(f^{k}(p),\xi)}{S(x_{k},\pi_{x,k}\xi)},\frac{S(f^{k}(p),\xi)}{S(y_{k},\pi_{y,k}\xi)}=e^{\pm\Gamma
Q_{\epsilon}^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}(x_{k})}.$
(48)

This is sufficient, since

$\frac{S(x_{k},\pi_{x,k}\xi)}{S(y_{k},\pi_{y,k}\xi)}=\frac{S(x_{k},\pi_{x,k}\xi)}{S(f^{k}(p),\xi)}\cdot\frac{S(f^{k}(p),\xi)}{S(y_{k},\pi_{y,k}\xi)}.$

We show $\frac{S(f^{k}(p),\xi)}{S(x_{k},\pi_{x,k}\xi)}=e^{\pm\Gamma
Q_{\epsilon}^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}(x_{k})}$;
the case of $\frac{S(f^{k}(p),\xi)}{S(y_{k},\pi_{y,k}\xi)}$ is identical. By
assumption, there exists a double chart $v$ and a sequence
$n_{k}\uparrow\infty$ s.t $v_{n_{k}}=v$ for all $k$. Write
$v=\psi_{x}^{p^{s},p^{u}}$.

For all $k\geq 0$, $f^{n_{k}}(p)\in V^{s}(\sigma^{n_{k}}\underline{v})$, and
by Corollary 4.5, $\exists C_{v}>0$ s.t $S(f^{n_{k}}(p))\leq C_{v}$. Set

$k_{v}:=\lceil\frac{\log C_{v}}{\Gamma
Q_{\epsilon}^{\frac{\beta}{4}+\frac{3}{4\gamma}}(x)}\rceil.$

Let $\xi\in T_{f^{n_{k_{v}}}(p)}V^{s}(\sigma^{n_{k_{v}}}\underline{v})(1)$.
Pulling back $n_{k_{v}}$ times, Lemma 4.6 tells us that for every
$j\in\\{0,\ldots,n_{k_{v}}\\}$ either
$\frac{S(f^{j}(p),d_{f^{n_{k_{v}}}}f^{-(n_{k_{v}}-j)}\xi)}{S(x_{j},\pi_{x,j}d_{f^{n_{k_{v}}}}f^{-(n_{k_{v}}-j)}\xi)}=e^{\pm\Gamma
Q_{\epsilon}^{\frac{\beta}{4}-\frac{1}{4\gamma}}(x_{j})}$, or the ratio
improves by $e^{-\Gamma Q_{\epsilon}^{\frac{\beta}{4}+\frac{3}{4\gamma}}(x)}$.
However, the ratio starts from a quantity bounded by $C_{v}$. It follow that
there must be $j\in\\{0,\ldots,n_{k_{v}}\\}$ s.t

$\frac{S(f^{j}(p),d_{f^{n_{k_{v}}}}f^{-(n_{k_{v}}-j)}\xi)}{S(x_{j},\pi_{x,j}d_{f^{n_{k_{v}}}}f^{-(n_{k_{v}}-j)}\xi)}=e^{\pm\Gamma
Q_{\epsilon}^{\frac{\beta}{4}-\frac{1}{4\gamma}}(x_{j})}.$

With every further iteration, the ratio can only improve, or remain bounded by
$e^{\pm
C_{\beta,\epsilon}^{\frac{1}{2\gamma}}Q_{\epsilon}^{\frac{\beta}{2}-\frac{1}{\gamma}}(x_{j})}$
(recall (46)). This holds for every $\xi\in
T_{f^{n_{k_{v}}}(p)}V^{s}(\sigma^{n_{k_{v}}}\underline{v})(1)$, and so
$\forall\xi^{\prime}\in T_{p}V^{s}(\underline{v})(1)$

$\frac{S(p,\xi^{\prime})}{S(x_{0},\pi_{x,0}\xi^{\prime})}=e^{\pm\max\\{C_{\beta,\epsilon}^{\frac{1}{2\gamma}}Q_{\epsilon}^{\frac{\beta}{2}-\frac{1}{\gamma}}(x_{0}),\Gamma
Q_{\epsilon}^{\frac{\beta}{4}-\frac{1}{4\gamma}}(x_{0})\\}}=e^{\pm\Gamma
Q_{\epsilon}^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}(x_{0})}.$

This proves the lemma for $k=0$. For other $k$, apply this to
$(\psi_{x_{i+k}}^{p^{s}_{i+k},p^{u}_{i+k}})_{i\in\mathbb{Z}}$,$(\psi_{y_{i+k}}^{q^{s}_{i+k},q^{u}_{i+k}})_{i\in\mathbb{Z}}$.

∎

Remark: The scheme of the proof only required that we have a uniform bound on
the $S(\cdot)$ parameter of a stable leaf infinitely often in the future of
the chain. The scheme works similarly for chains which may not have a symbol
recurring infinitely often, but lie within an irreducible component of
$\Sigma$. The principle which allows for it is replacing the role of the
recurring symbol with a concatenation arbitrarily far away with a fixed
periodic word (which is possible by irreducibility).

Our next task is to show that if
$\pi((\psi_{x_{i}}^{p^{s}_{i},p^{u}_{i}})_{i\in\mathbb{Z}})=\pi((\psi_{y_{i}}^{q^{s}_{i},q^{u}_{i}})_{i\in\mathbb{Z}})$
then
$C_{0}(x_{i})^{-1}:T_{x_{i}}M\rightarrow\mathbb{R}^{d},C_{0}(y_{i})^{-1}:T_{y_{i}}M\rightarrow\mathbb{R}^{d}$
are “approximately the same.” There are two issues in expressing this
formally:

  1. (a)

$C_{0}(x_{i})^{-1},C_{0}(y_{i})^{-1}$ are linear maps with different domains.

  2. (b)

$C_{0}(\cdot)$ is only determined up to orthogonal self maps of
$H^{s}(\cdot),H^{u}(\cdot)$. In the two-dimensional case this means choosing a
number from $\\{\pm 1\\}$, which commutes with composition of linear maps, but
we wish to deal with the higher dimensional case, which is more subtle because
of non-commutativity.

We address these issues in the following proposition.

######  Proposition 4.8.

Under the assumptions of the previous lemma:

$\exists\Xi_{i}:T_{x_{i}}M\rightarrow T_{y_{i}}M\text{ s.t }\forall\xi\in
T_{x_{i}}M(1):\frac{|C_{0}^{-1}(x_{i})\xi|}{|C_{0}^{-1}(y_{i})\Xi_{i}\xi|}=e^{\pm
2\Gamma
Q_{\epsilon}^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}(x_{i})},$

where $\Xi_{i}$ is an invertible linear transformation, and
$\|\Xi_{i}\|,\|\Xi_{i}^{-1}\|\leq
e^{Q_{\epsilon}^{\frac{\beta}{2}-\frac{1}{\gamma}}(x_{i})}$.

###### Proof.

Using the notations from the previous lemma, define
$\pi_{x,k}^{s}:=\pi_{x,k}:T_{f^{k}(p)}V^{s}_{k}\rightarrow
H^{s}(x_{k}),\pi_{y,k}^{s}:=\pi_{y,k}:T_{f^{k}(p)}U^{s}_{k}\rightarrow
H^{s}(y_{k})$. Let
$V^{u}_{k}:=V^{u}((\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\leq
k}),U^{u}_{k}:=V^{u}((\psi_{y_{i}}^{q_{i}^{s},q_{i}^{u}})_{i\leq k})$, and
define $\pi_{x,k}^{u}:=\pi_{x,k}:T_{f^{k}(p)}V^{u}_{k}\rightarrow
H^{u}(x_{k}),\pi_{y,k}^{u}:=\pi_{y,k}:T_{f^{k}(p)}U^{u}_{k}\rightarrow
H^{u}(y_{k})$ as in the unstable case of Lemma 4.6. Define
$\Xi_{k}^{s}:H^{s}(x_{k})\rightarrow H^{s}(y_{k})$,
$\Xi_{k}^{u}:H^{u}(x_{k})\rightarrow H^{u}(y_{k})$ the following way:

$\Xi_{k}^{s}:=\pi_{y,k}^{s}\circ(\pi_{x,k}^{s})^{-1}\text{ ,
}\Xi_{k}^{u}:=\pi_{y,k}^{u}\circ(\pi_{x,k}^{u})^{-1}.$

The composition of $\pi_{y,k}^{s/u}\circ(\pi_{x,k}^{s/u})^{-1}$ is well
defined by the remark after Lemma 4.7. Define $\Xi_{k}:T_{x_{k}}M\rightarrow
T_{y_{k}}M$:

$\forall\xi\in T_{x_{k}}M\text{:
}\Xi_{k}\xi:=\Xi_{k}^{s}\xi_{s}+\Xi_{k}^{u}\xi_{u}\text{, where
}\xi=\xi_{s}+\xi_{u},\xi_{s/u}\in H^{s/u}(x_{k}).$ (49)

This is well defined since $T_{x_{k}}M=H^{s}(x_{k})\oplus H^{u}(x_{k})$.
Notice that $\Xi_{k}[H^{s/u}(x_{k})]=H^{s/u}(y_{k})$. Hence, by Lemma 4.7:

$\frac{|C_{0}^{-1}(x_{k})\xi|}{|C_{0}^{-1}(y_{k})\Xi_{k}\xi|}=\sqrt{\frac{S^{2}(x_{k},\xi_{s})+U^{2}(x_{k},\xi_{u})}{S^{2}(y_{k},\Xi_{k}\xi_{s})+U^{2}(y_{k},\Xi_{k}\xi_{u})}}=e^{\pm
2\Gamma
Q_{\epsilon}^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}(x_{k})}.$
(50)

It remains to bound $\|\Xi_{k}\|$ and $\|\Xi_{k}^{-1}\|$. We begin by showing
bounds on the norms of $\Xi_{k}^{s/u}$. By the bounds for
$\|\pi_{x,k}^{s/u}\|$,$\|(\pi_{x,k}^{s/u})^{-1}\|$,
$\|\pi_{y,k}^{s/u}\|$,$\|(\pi_{y,k}^{s/u})^{-1}\|$ from Lemma 4.6 ,

$\|\Xi_{k}^{s/u}\|\leq\exp\Big{(}Q_{\epsilon}(x_{k})^{\frac{\beta}{2}-\frac{1}{2\gamma}}+Q_{\epsilon}(y_{k})^{\frac{\beta}{2}-\frac{1}{2\gamma}}\Big{)}.$
(51)

We continue to bound $\|\Xi_{k}\|$. To ease notations we will omit the ’$k$’
subscripts. Recall $\rho(M)$ from §2.2. By Lemma 4.1,
$d(x,f^{k}(p))<\frac{\sqrt{d}}{100}Q_{\epsilon}(x)<\rho(M)$ for $\epsilon$
small enough. Similarly, $d(y,f^{k}(p))<\rho(M)$. Therefore the following is
well defined: $\psi_{x}^{-1}(f^{k}(p))=:z_{x}^{\prime}$,
$\psi_{y}^{-1}(f^{k}(p))=:z_{y}^{\prime}$. The vectors of the first $s(x)$
coordinates of $z_{x}^{\prime}/z_{y}^{\prime}$ will be called $z_{x}/z_{y}$
respectively.

Part 1: Let $\xi\in T_{x}M$, $\xi=\xi_{s}+\xi_{u}$, $\xi_{s/u}\in H^{s/u}(x)$,
then $|\xi_{s}|,|\xi_{u}|\leq\|C_{0}^{-1}(x)\|\cdot|\xi|$.

Proof: w.l.o.g $|\xi|=1$. Notice that the size of $\xi_{s/u}$ can be very big,
even when $|\xi_{s}+\xi_{u}|=|\xi|=1$. This can happen when
$\sphericalangle(H^{s}(x),H^{u}(x)):=\inf\limits_{\eta_{s}\in
H^{s}(x),\eta_{u}\in H^{u}(x)}|\sphericalangle(\eta_{s},\eta_{u})|$ is very
small, and $\xi_{s}$ and $\xi_{u}$ are almost parallel, of the same size, and
are pointing to almost opposite directions. Consider the triangle created by
the tips of $\xi_{s}$ and $-\xi_{u}$, and the origin. Denote its angle at the
origin by $\alpha$ (the angle between $\xi_{s}$ and $-\xi_{u}$). The size of
the edge in front of $\alpha$ has length 1 (since
$|\xi_{s}-(-\xi_{u})|=|\xi_{s}+\xi_{u}|=1$). Angles between two non-parallel
vectors are in $(0,\pi)$, and their sine is always positive. By the sine
theorem,

$|\xi_{s}|,|\xi_{u}|\leq\frac{1}{|\sin\alpha|}\leq\frac{1}{\inf\limits_{\eta_{s}\in
H^{s}(x),\eta_{u}\in
H^{u}(x)}|\sin\sphericalangle(\eta_{s},\eta_{u})|}=\frac{1}{\sin\sphericalangle(H^{s}(x),H^{u}(x))}.$
(52)

Notice that
$\frac{1}{|\sin\sphericalangle(\eta_{s},\eta_{u})|}\leq\|C_{0}^{-1}(x)\|$,
$\forall\eta_{s/u}\in H^{s/u}(x)$. To see this, notice first the identity

$\|C_{0}^{-1}(x)\|^{2}=\sup_{\eta_{s}\in H^{s}(x),\eta_{u}\in
H^{u}(x),|\eta_{s}+\eta_{u}|=1}\\{S^{2}(x,\eta_{s})+U^{2}(x,\eta_{u})\\}.$

Choose $\eta_{s}\in H^{s}(x)(1),\eta_{u}\in H^{u}(x)(1)$ which minimize
$\sin\sphericalangle(\eta_{s},\eta_{u})$. w.l.o.g
$\cos\sphericalangle(\eta_{s},\eta_{u})>0$. Let
$\zeta:=\frac{\eta_{s}-\eta_{u}}{|\eta_{s}-\eta_{u}|}$. Since
$C_{0}^{-1}(x)\eta_{s}\perp C_{0}^{-1}(x)\eta_{u}$ and $\|C_{0}(x)\|\leq 1$,

$|C_{0}^{-1}(x)\zeta|^{2}\geq\frac{|\eta_{s}|^{2}}{|\eta_{s}-\eta_{u}|^{2}}+\frac{|\eta_{u}|^{2}}{|\eta_{s}-\eta_{u}|^{2}}=\frac{2}{2-2\cos\sphericalangle(\eta_{s},\eta_{u}))}\geq\frac{1}{1-\cos^{2}\sphericalangle(\eta_{s},\eta_{u})}=\frac{1}{\sin^{2}\sphericalangle(\eta_{s},\eta_{u})}.$

So $\|C_{0}^{-1}(x)\|\geq\frac{1}{|\sin\sphericalangle(\eta_{s},\eta_{u})|}$.
By (52), $|\xi_{s}|,|\xi_{u}|\leq\|C_{0}^{-1}(x)\|$.

QED

Part 2: $\Xi=d_{C_{0}(x)z_{x}^{\prime}}(\exp_{y}^{-1}\exp_{x})+E_{1}$, where
$E_{1}:T_{x}M\rightarrow T_{y}M$ is linear and
$\|E_{1}\|\leq\frac{1}{2}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{\gamma}}$.

Proof: We will begin by showing that the expression above is well defined for
$\epsilon<\frac{\rho(M)}{2}$. This is true because
$\exp_{x}[B_{\epsilon}(0)]\subset B_{\epsilon+d(x,y)}(y)\subset
B_{2\epsilon}(y)$; $|C_{0}(x)z_{x}^{\prime}|=d(x,f^{k}(p))<\epsilon$; so
$\exp_{y}^{-1}\exp_{x}$ is defined on an open neighborhood of
$C_{0}(x)z_{x}^{\prime}$, and hence is differentiable on it.

By definition, $\Xi^{s}=\pi_{y,k}^{s}\circ(\pi_{x,k}^{s})^{-1}$.

  * •

$\xi_{s}\in H^{s}(x)$ is mapped by $(\pi_{x,k}^{s})^{-1}$ to
$\xi_{s}^{\prime}$, a tangent vector at $f^{k}(p)$ given by
$\xi_{s}^{\prime}=d_{z_{x}^{\prime}}\psi_{x}\Big{(}\begin{array}[]{c}v\\\
d_{z_{x}}Gv\\\ \end{array}\Big{)}$, where $G$ represents $V^{s}_{k}$ in
$\psi_{x_{k}}^{p_{k}^{s},p^{u}_{k}}$ and $v=C_{0}^{-1}(x)\xi_{s}$. We write
$\xi_{s}^{\prime}$ as follows:

$\xi_{s}^{\prime}=d_{z_{x}^{\prime}}\psi_{x}\Big{(}\begin{array}[]{c}v\\\
d_{z_{x}}Gv\\\
\end{array}\Big{)}=(d_{z_{y}^{\prime}}\psi_{y})(d_{z_{y}^{\prime}}\psi_{y})^{-1}d_{z_{x}^{\prime}}\psi_{x}\Big{(}\begin{array}[]{c}v\\\
d_{z_{x}}Gv\\\
\end{array}\Big{)}=:d_{z_{y}^{\prime}}\psi_{y}\Big{(}\begin{array}[]{c}w\\\
d_{z_{y}}Fw\\\ \end{array}\Big{)},$

for some $w$, where $F$ is the representing function of $U^{s}_{k}$ in
$\psi_{y_{k}}^{q^{s}_{k},q^{u}_{k}}$. This representation is possible because
$\xi_{s}^{\prime}$ is tangent to the stable manifold $U^{s}_{k}$ in
$\psi_{y_{k}}^{q^{s}_{k},q^{u}_{k}}$ at $f^{k}(p)=\psi_{y}(z_{y}^{\prime})$.

  * •

$\xi_{s}^{\prime}=(\psi_{x,k}^{s})^{-1}\xi_{s}$ is mapped by $\pi_{y,k}^{s}$
to

$\Xi^{s}\xi_{s}=\pi_{y,k}^{s}\xi_{s}^{\prime}=d_{0}\psi_{y}\Big{(}\begin{array}[]{c}w\\\
0\\\ \end{array}\Big{)}.$

By construction,

$\Big{(}\begin{array}[]{c}w\\\ d_{z_{y}}Fw\\\
\end{array}\Big{)}=(d_{z_{y}^{\prime}}\psi_{y})^{-1}(d_{z_{x}^{\prime}}\psi_{x})\Big{(}\begin{array}[]{c}v\\\
d_{z_{x}}Gv\\\ \end{array}\Big{)}.$

We abuse notation and use $w,v$ to represent both vectors in
$\mathbb{R}^{s(x)}\times\\{0\\}$ and vectors in $\mathbb{R}^{s(x)}$. Then,

$w=(d_{z_{y}^{\prime}}\psi_{y})^{-1}d_{z_{x}^{\prime}}\psi_{x}v+\Big{[}(d_{z_{y}^{\prime}}\psi_{y})^{-1}d_{z_{x}^{\prime}}\psi_{x}\Big{(}\begin{array}[]{c}0\\\
d_{z_{x}}Gv\\\ \end{array}\Big{)}-\Big{(}\begin{array}[]{c}0\\\ d_{z_{y}}Fw\\\
\end{array}\Big{)}\Big{]}.$

Call the term in the brackets $E_{s}\xi_{s}$, then
$\Xi_{s}=C_{\chi}(y)(d_{z_{y}^{\prime}}\psi_{y})^{-1}d_{z_{x}^{\prime}}\psi_{x}C_{0}^{-1}(x)+C_{0}(y)E_{s}$.

  * •

$\Xi^{s}\xi_{s}=d_{0}\psi_{y}\Big{(}\begin{array}[]{c}w\\\ 0\\\
\end{array}\Big{)}\Rightarrow|w|=|(d_{0}\psi_{y})^{-1}\Xi^{s}\xi_{s}|=|C_{0}^{-1}(y)\Xi^{s}\xi_{s}|\leq\|C_{0}^{-1}(y)\|\cdot\|\Xi^{s}\|\cdot|\xi_{s}|\leq$

$\leq\|C_{0}^{-1}(y)\|\cdot
e^{Q_{\epsilon}(x_{k})^{\frac{\beta}{2}-\frac{1}{2\gamma}}+Q_{\epsilon}(y_{k})^{\frac{\beta}{2}-\frac{1}{2\gamma}}}|\xi_{s}|\leq
2\|C_{0}^{-1}(y)\|\cdot|\xi_{s}|(\because\text{ \eqref{shluk} })$.

  * •

$\mathrm{Lip}(G)\leq Q_{\epsilon}(x)^{\frac{\beta}{2}},\mathrm{Lip}(F)\leq
Q_{\epsilon}(y)^{\frac{\beta}{2}}$ (by the remark after Definition 3.2).

  * •

$\xi_{s}=C_{\chi}(x)v\Rightarrow|v|\leq\|C_{\chi}^{-1}(x)\|\cdot|\xi_{s}|$.

Hence:

$\displaystyle|C_{0}(y)E_{s}\xi_{s}|\leq$
$\displaystyle\|d_{z_{y}^{\prime}}\exp_{y}^{-1}\|\cdot\|d_{z_{x}^{\prime}}\psi_{x}\|\cdot\|d_{z_{x}}G\|\cdot|v|+\|d_{z_{y}}F\|\cdot|w|$
$\displaystyle\leq$ $\displaystyle 2\cdot
2Q_{\epsilon}(x)^{\frac{\beta}{2}}\cdot\|C_{0}^{-1}(x)\|\cdot|\xi_{s}|+Q_{\epsilon}(y)^{\frac{\beta}{2}}\cdot
2\|C_{0}^{-1}(y)\|\cdot|\xi_{s}|$ $\displaystyle\leq$
$\displaystyle(Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}+Q_{\epsilon}(y)^{\frac{\beta}{2}-\frac{1}{2\gamma}})|\xi_{s}|.$
(53)

Similarly,
$\Xi_{u}=C_{0}(y)(d_{z_{y}^{\prime}}\psi_{y})^{-1}d_{z_{x}^{\prime}}\psi_{x}C_{0}^{-1}(x)+C_{0}(y)E_{u}$,
where $E_{u}:H^{u}(x)\rightarrow T_{y}M$ and

$\|C_{0}(y)E_{u}\|\leq(Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}+Q_{\epsilon}(y)^{\frac{\beta}{2}-\frac{1}{2\gamma}}).$
(54)

For a general tangent $\xi\in T_{x}M$, write $\xi=\xi_{s}+\xi_{u},\xi_{s/u}\in
H^{s/u}(x)$, then

$\displaystyle\Xi\xi=$
$\displaystyle\Xi(\xi_{s}+\xi_{u})=\Xi_{s}\xi_{s}+\Xi_{u}\xi_{u}$
$\displaystyle=$
$\displaystyle\Big{(}C_{0}(y)(d_{z_{y}^{\prime}}\psi_{y})^{-1}d_{z_{x}^{\prime}}\psi_{x}C_{0}^{-1}(x)+C_{0}(y)E_{s}\Big{)}\xi_{s}+\Big{(}C_{0}(y)(d_{z_{y}^{\prime}}\psi_{y})^{-1}d_{z_{x}^{\prime}}\psi_{x}C_{0}^{-1}(x)+C_{0}(y)E_{u}\Big{)}\xi_{u}$
$\displaystyle=$ $\displaystyle
C_{0}(y)(d_{z_{y}^{\prime}}\psi_{y})^{-1}d_{z_{x}^{\prime}}\psi_{x}C_{0}^{-1}(x)(\xi_{s}+\xi_{u})+C_{0}(y)(E_{s}\xi_{s}+E_{u}\xi_{u})$
$\displaystyle=$ $\displaystyle
C_{0}(y)(d_{z_{y}^{\prime}}\psi_{y})^{-1}d_{z_{x}^{\prime}}\psi_{x}C_{0}^{-1}(x)\xi+C_{0}(y)(E_{s}\xi_{s}+E_{u}\xi_{u}).$

Hence, since
$d_{z^{\prime}_{x}}\psi_{x}=d_{C_{0}(x)z_{x}^{\prime}}\exp_{x}\circ
C_{0}(x),d_{z^{\prime}_{y}}\psi_{y}=d_{C_{0}(y)z_{y}^{\prime}}\exp_{y}\circ
C_{0}(y)$, we get

$\Xi\xi=d_{f^{k}(p)}\exp_{y}^{-1}d_{C_{0}(x)z_{x}^{\prime}}\exp_{x}\xi+C_{0}(y)(E_{s}\xi_{s}+E_{u}\xi_{u})\equiv
d_{C_{0}(x)z_{x}^{\prime}}(\exp_{y}^{-1}\exp_{x})\xi+E_{1}\xi,$

where

$\displaystyle|E_{1}\xi|\leq$
$\displaystyle|C_{0}(y)E_{s}\xi_{s}|+|C_{0}(y)E_{u}\xi_{u}|\leq((Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}+Q_{\epsilon}(y)^{\frac{\beta}{2}-\frac{1}{2\gamma}})|)(|\xi_{s}|+|\xi_{u}|)\text{
}(\because\text{ \eqref{Es},\eqref{Eu}})$ $\displaystyle\leq$
$\displaystyle((Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}+Q_{\epsilon}(y)^{\frac{\beta}{2}-\frac{1}{2\gamma}})|)(\|C_{0}^{-1}(x)\|\cdot|\xi|+\|C_{0}^{-1}(x)\|\cdot|\xi|)\text{
}(\because\text{Part 1})$ $\displaystyle\leq$ $\displaystyle
2\|C_{0}^{-1}(x)\|(Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}+Q_{\epsilon}(y)^{\frac{\beta}{2}-\frac{1}{2\gamma}})|\xi|.$

Therefore,

$\Xi=d_{C_{0}(x)z_{x}^{\prime}}(\exp_{y}^{-1}\exp_{x})+E_{1}\Rightarrow\|\Xi\|\leq\|d_{\cdot}\exp_{y}^{-1}\|\cdot\|d_{\cdot}\exp_{x}\|+\|E_{1}\|\leq
4+2\|C_{0}^{-1}(x)\|(Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}+Q_{\epsilon}(y)^{\frac{\beta}{2}-\frac{1}{2\gamma}}).$

Now using this again in (50):

$\displaystyle\|C_{0}^{-1}(x)\|\leq$ $\displaystyle e^{2\Gamma
Q_{\epsilon}^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}(x_{k})}\|C_{0}^{-1}(y)\Xi\|\leq
e^{2\Gamma
Q_{\epsilon}^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}(x_{k})}\|C_{0}^{-1}(y)\|\cdot\|\Xi\|$
$\displaystyle\leq$ $\displaystyle e^{2\Gamma
Q_{\epsilon}^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}(x_{k})}\|C_{0}^{-1}(y)\|\Big{(}4+2\|C_{0}^{-1}(x)\|(Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}+Q_{\epsilon}(y)^{\frac{\beta}{2}-\frac{1}{2\gamma}})\Big{)}$
$\displaystyle=$ $\displaystyle\|C_{0}^{-1}(y)\|\cdot e^{2\Gamma
Q_{\epsilon}^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}(x_{k})}\Big{(}4+2\|C_{0}^{-1}(x)\|Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}\Big{)}$
$\displaystyle+$ $\displaystyle\|C_{0}^{-1}(x)\|\cdot 2e^{2\Gamma
Q_{\epsilon}^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}(x_{k})}\Big{(}\|C_{0}^{-1}(y)\|Q_{\epsilon}(y)^{\frac{\beta}{2}-\frac{1}{2\gamma}}\Big{)}$
$\displaystyle\leq$ $\displaystyle
5(1-\epsilon)\|C_{0}^{-1}(y)\|+\epsilon\|C_{0}^{-1}(x)\|\Rightarrow(1-\epsilon)\|C_{0}^{-1}(x)\|\leq(1-\epsilon)5\|C_{0}^{-1}(y)\|\Rightarrow\|C_{0}^{-1}(x)\|\leq
5\|C_{0}^{-1}(y)\|.$

By symmetry we will also get $\|C_{0}^{-1}(y)\|\leq 5\|C_{0}^{-1}(x)\|$. In
particular, $\frac{Q_{\epsilon}(y)}{Q_{\epsilon}(x)}=5^{2\gamma}$.
Substituting these in the bounds for $\|E_{1}\|$, for all $\epsilon$ small
enough:

$\displaystyle\|E_{1}\|\leq$ $\displaystyle
2\|C_{0}^{-1}(x)\|Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}+2\|C_{0}^{-1}(x)\|Q_{\epsilon}(y)^{\frac{\beta}{2}-\frac{1}{2\gamma}}$
$\displaystyle\leq$ $\displaystyle
2\|C_{0}^{-1}(x)\|Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}+2\cdot(5^{2\gamma})^{\frac{\beta}{2}-\frac{1}{2\gamma}}\|C_{0}^{-1}(x)\|Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{2\gamma}}\leq\frac{1}{2}Q_{\epsilon}(x)^{\frac{\beta}{2}-\frac{1}{\gamma}}.$

QED

Part 3: $\forall u\in R_{\epsilon}(0)$: $\exists D\in\mathcal{D}$ s.t
$\|\Theta_{D}d_{C_{0}(x)u}(\exp_{y}^{-1}\exp_{x})-\Theta_{D}\|\leq
Q_{\epsilon}(x)$ (see Definition 2.16).

Proof: Begin by choosing $\epsilon$ smaller than
$\frac{\varpi(\mathcal{D})}{2\sqrt{d}}$ (as in Definition 2.16). Then $\exists
D\in\mathcal{D}$ such that $\exp_{x}[R_{\epsilon}(0)]\subset
B_{\frac{\varpi(\mathcal{D})}{2}}(x)$, because
$d(x,\exp_{x}C_{0}(x)v)=|C_{0}(x)v|_{2}\leq\sqrt{d}|v|_{\infty}<\epsilon\sqrt{d}$,
so $\exp_{x}[R_{\epsilon}(0)]$ has diameter less than
$2\sqrt{d}\epsilon<\varpi(\mathcal{D})$. Hence the choice of $D$ and of
$\Theta_{D}$ is proper. Since

$\Theta_{D}\exp_{y}^{-1}\exp_{x}=(\Theta_{D}\exp_{y}^{-1}-\Theta_{D}\exp_{x}^{-1})\exp_{x}+\Theta_{D},$

we get that for any $u\in R_{\epsilon}(0)$:

$\Theta_{D}d_{C_{0}(x)u}(\exp_{y}^{-1}\exp_{x})=d_{C_{0}(x)u}(\Theta_{D}\exp_{y}^{-1}\exp_{x})=d_{\exp_{x}C_{0}(x)u}(\Theta_{D}\exp_{y}^{-1}-\Theta_{D}\exp_{x}^{-1})d_{C_{0}(x)u}\exp_{x}+\Theta_{D}.$

Then

$\|\Theta_{D}d_{C_{0}(x)u}(\exp_{y}^{-1}\exp_{x})-\Theta_{D}\|\leq\|\Theta_{D}\exp_{y}^{-1}-\Theta_{D}\exp_{x}^{-1}\|_{C^{2}}\cdot\|d_{C_{0}(x)u}\exp_{x}\|\leq
L_{2}d(x,y)\|d_{C_{0}(x)u}\exp_{x}\|,$

where $L_{2}$ is the uniform Lipschitz const. of
$x\mapsto\nu_{x}^{-1}\exp_{x}^{-1}$ (Proposition 2.17). This is due to the
fact that
$d(x,\exp_{x}C_{0}(x)u)<\sqrt{d}\epsilon\Rightarrow\exp_{x}C_{0}(x)u\in D$. In
addition, $\|d_{C_{0}(x)u}\exp_{x}\|\leq\|Id_{T_{x}M}\|+E_{0}|C_{0}(x)u-0|\leq
1+E_{0}\epsilon$ ($E_{0}$ is a constant introduced in Lemma 4.6). Therefore,
in total,

$\|\Theta_{D}d_{C_{0}(x)u}(\exp_{y}^{-1}\exp_{x})-\Theta_{D}\|\leq
L_{2}(1+\epsilon E_{0})\cdot d(x,y)<Q_{\epsilon}(x)\text{ }\because\text{Lemma
\ref{firstchapter2}}.$ (55)

QED

Adding the estimates of part 2 and part 3, we get

$\displaystyle\|\Xi\|=$
$\displaystyle\|\Theta_{D}d_{C_{0}(x)v}(\exp_{y}^{-1}\exp_{x})-\Theta_{D}+\Theta_{D}+\Theta_{D}E_{1}\|$
$\displaystyle\leq$
$\displaystyle\|\Theta_{D}d_{C_{0}(x)v}(\exp_{y}^{-1}\exp_{x})-\Theta_{D}\|+\|\Theta_{D}\|+\|E_{1}\|\leq
Q_{\epsilon}(x)+1+\frac{1}{2}Q_{\epsilon}^{\frac{\beta}{2}-\frac{1}{\gamma}}(x)\leq
e^{Q_{\epsilon}^{\frac{\beta}{2}-\frac{1}{\gamma}}(x)}.$

If we exchange the roles of $x_{k},y_{k}$ then we get $\Xi_{k}^{-1}$, so
$\|\Xi_{k}^{-1}\|$ is also less than
$e^{Q_{\epsilon}^{\frac{\beta}{2}-\frac{1}{\gamma}}(x_{k})}$. ∎

######  Corollary 4.9.

Under the assumptions and notations of the previous lemma:

$\forall
i\in\mathbb{Z}:\frac{\|C_{0}^{-1}(x_{i})\|}{\|C_{0}^{-1}(y_{i})\|}=e^{\pm(2\Gamma+1)Q_{\epsilon}(x_{i})^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}},$

and so

$\frac{Q_{\epsilon}(x_{i})}{Q_{\epsilon}(y_{i})}=e^{\pm
2\gamma(2\Gamma+1)Q_{\epsilon}(x_{i})^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}}.$

###### Proof.

By Proposition 4.8:

$\displaystyle\|C_{0}^{-1}(x_{i})\|\leq$ $\displaystyle e^{2\Gamma
Q_{\epsilon}(x_{i})^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}}\|C_{0}^{-1}(y_{i})\Xi_{i}\|\leq
e^{(2\Gamma+1)Q_{\epsilon}(x_{i})^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}}\|C_{0}^{-1}(y_{i})\|,$

and by symmetry we get the other inequality, and we are done. ∎

####  4.1.2. Comparing frame parameters

Remark: Notice that if $\gamma>\frac{5}{\beta}$, and $\epsilon>0$ is
sufficiently small, then Corollary 4.9 in fact tell us

$Q_{\epsilon}(y_{i})=I^{\pm 1}(Q_{\epsilon}(x_{i})),\text{ }i\in\mathbb{Z}.$
(56)

It is important that the bound we get on the resemblance of
$Q_{\epsilon}(x_{i})$ and $Q_{\epsilon}(y_{i})$ is of a form which commutes
with the map $I$. This is the reason that it is important to require
$\gamma>\frac{5}{\beta}$ to make sure the bounds of Corollary 4.9 can be
expressed in terms of $I$ ($\geq\frac{5}{\beta}$ is sufficient, if one carries
the constants more carefully). To see where this commutativity relation is
needed, see (4.1.2) below.

######  Corollary 4.10.

$\pi[\Sigma^{\\#}]=\mathrm{RST}$.

###### Proof.

In Theorem 3.15 we saw $\pi[\Sigma^{\\#}]\supseteq\mathrm{RST}$. By Corollary
4.5, for every
$\underline{u}=(\psi_{x_{i}}^{p^{s}_{i},p^{u}_{i}})_{\in\mathbb{Z}}\in\Sigma^{\\#}$,
$p:=\pi(\underline{u})\in 0$-summ. $\\{p^{s}_{i}\wedge
p^{u}_{i}\\}_{i\in\mathbb{Z}}$ is $I$-strongly subordinated to
$\\{Q_{\epsilon}(x_{i})\\}$ by Lemma 2.24. By the proof of Proposition 4.8,
the map $\pi_{x,i}:T_{f^{i}(p)}M\to T_{x_{k}}M$ satisfies
$\|\pi_{x,i}\|,\|\pi_{x,i}^{-1}\|\leq
e^{Q_{\epsilon}(x_{i})^{\frac{\beta}{2}-\frac{1}{\gamma}}}$. By the proof of
Lemma 4.7, $\frac{\|C_{0}^{-1}(f^{i}(p))\|}{\|C_{0}^{-1}(x_{i})\|}=e^{\pm
2\Gamma
Q_{\epsilon}(x_{i})^{\min\\{\frac{\beta}{2}-\frac{1}{\gamma},\frac{\beta}{4}-\frac{1}{4\gamma}\\}}}$.
As in (56), we get $Q_{\epsilon}(f^{i}(p))=I^{\pm 1}(Q_{\epsilon}(x_{i}))$.
Then it follows that

$I^{-1}(p_{i}^{s}\wedge p_{i}^{u})\leq I^{-1}(Q_{\epsilon}(x_{i}))\leq
Q_{\epsilon}(f^{i}(x)).$

Thus $p\in ST$ (with the strongly tempered kernel $I^{-1}(p_{i}^{s}\wedge
p_{i}^{u})$). Since $\limsup_{n\to\pm\infty}p_{i}^{s}\wedge p_{i}^{u}>0$,
$p\in\mathrm{RST}$. ∎

######  Definition 4.11.

(Compare with [Sar13, Definition 8.1])

  1. (1)

If $v$ is a double chart, then $p^{u/s}(v)$ means the $p^{u/s}$ in
$\psi_{x}^{p^{s},p^{u}}=v$.

  2. (2)

A negative chain $(v_{i})_{i\leq 0}$ is called $I$-maximal if it has a symbol
recurring infinitely often, and

$p^{u}(v_{0})\geq I^{-1}(p^{u}(u_{0}))$

for every regular chain $(u_{i})_{i\in\mathbb{Z}}$ for which there is a
positive regular chain $(v_{i})_{i\geq 0}$ s.t
$\pi((u_{i})_{i\in\mathbb{Z}})=\pi((v_{i})_{i\in\mathbb{Z}})$.

  3. (3)

A positive chain $(v_{i})_{i\geq 0}$ is called $I$-maximal if it has a symbol
recurring infinitely often, and

$p^{s}(v_{0})\geq I^{-1}(p^{s}(u_{0}))$

for every chain $(u_{i})_{i\in\mathbb{Z}}\in\Sigma^{\\#}$ s.t
$\pi((u_{i})_{i\in\mathbb{Z}})=\pi((v_{i})_{i\in\mathbb{Z}})$.

######  Proposition 4.12.

For every chain $(v_{i})_{i\in\mathbb{Z}}\in\Sigma^{\\#}$, $(u_{i})_{i\leq 0}$
and $(u_{i})_{i\geq 0}$ are $I$-maximal.

###### Proof.

Step 1: Let $\underline{u},\underline{v}\in\Sigma^{\\#}$ be two chains s.t
$\pi(\underline{u})=\pi(\underline{v})$. If $u_{0}=\psi_{x}^{p^{s},p^{u}}$ and
$v_{0}=\psi_{y}^{q^{s},q^{u}}$, then $Q_{\epsilon}(y)=I^{\pm
1}(Q_{\epsilon}(x))$, by (56).

Step 2: Every negative chain $(v_{i})_{i\leq 0}$ with a symbol recurring
infinitely often s.t $v_{0}=\psi_{x}^{p^{s},p^{u}}$ where
$p^{u}=Q_{\epsilon}(x)$ is $I$-maximal, and every positive chain with a symbol
recurring infinitely often $(v_{i})_{i\leq 0}$ s.t
$v_{0}=\psi_{x}^{p^{s},p^{u}}$ where $p^{s}=Q_{\epsilon}(x)$ is $I$-maximal.

Proof: Let $(v_{i})_{i\leq 0}$ be a negative chain with a symbol recurring
infinitely often, and assume $v_{0}=\psi_{x}^{p^{s},p^{u}}$ where
$p^{u}=Q_{\epsilon}(x)$. We show that it is $I$-maximal. Suppose
$(v_{i})_{i\in\mathbb{Z}}\in\Sigma^{\\#}$ is an extension of $(v_{i})_{i\leq
0}$, and let $(u_{i})_{i\in\mathbb{Z}}\in\Sigma^{\\#}$ be a chain s.t
$\pi((u_{i})_{i\in\mathbb{Z}})=\pi((v_{i})_{i\in\mathbb{Z}})$. Write
$u_{0}=\psi_{y}^{q^{s},q^{u}}$. Now, by step 1, $p^{u}=Q_{\epsilon}(x)\geq
I^{-1}(Q_{\epsilon}(y))\geq I^{-1}(q^{u})$. The second half of step 2 is shown
similarly.

Step 3: Let $(v_{i})_{i\leq 0}$ be a negative chain with a symbol recurring
infinitely often, and suppose $v_{0}\rightarrow v_{1}$. If $(v_{i})_{i\leq 0}$
is $I$-maximal, then also $(v_{i})_{i\leq 1}$ is $I$-maximal. Let
$(v_{i})_{i\geq 0}$ be a positive chain with a symbol recurring infinitely
often, and suppose $v_{-1}\rightarrow v_{0}$. If $(v_{i})_{i\geq 0}$ is
$I$-maximal, then also $(v_{i})_{i\geq-1}$ is $I$-maximal.

Proof: Let $(v_{i})_{i\leq 0}$ be an $I$-maximal negative chain with a symbol
recurring infinitely often, and suppose $v_{0}\rightarrow v_{1}$. Suppose that
$(u_{i})_{i\in\mathbb{Z}}\in\Sigma^{\\#}$ and that $(v_{i})_{i\leq 1}$ has a
symbol repeating infinitely often, and that there is an extension
$(v_{i})_{i\leq 1}$ to a chain $(v_{i})_{i\in\mathbb{Z}}\in\Sigma^{\\#}$ s.t
$\pi((u_{i+1})_{i\in\mathbb{Z}})=\pi((v_{i+1})_{i\in\mathbb{Z}})$. We write
$v_{i}=\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}}$, and
$u_{i}=\psi_{y_{1}}^{q_{i}^{s},q_{i}^{u}}$; and we show that $p_{1}^{u}\geq
I^{-1}(q_{1}^{u})$. Since
$\pi((u_{i+1})_{i\in\mathbb{Z}})=\pi((v_{i+1})_{i\in\mathbb{Z}})$ and
$\pi\circ\sigma=f\circ\pi$:

$\pi((u_{i})_{i\in\mathbb{Z}})=\pi((v_{i})_{i\in\mathbb{Z}}).$

Therefore, since $(v_{i})_{i\leq 0}$ is $I$-maximal, $p_{0}^{u}\geq
I^{-1}(q_{0}^{u})$. Also, by step 1: $Q_{\epsilon}(x_{1})\geq
I^{-1}(Q_{\epsilon}(y_{1}))$. It follows that

$\displaystyle p_{1}^{u}=$
$\displaystyle\min\\{I(p_{0}^{u}),Q_{\epsilon}(x_{1})\\}\text{ }(\because
v_{0}\rightarrow v_{1})$ $\displaystyle\geq$
$\displaystyle\min\\{I(I^{-1}(q_{0}^{u})),I^{-1}(Q_{\epsilon}(y_{1}))\\}$
$\displaystyle=$ $\displaystyle
I^{-1}(\min\\{I(q_{0}^{u}),Q_{\epsilon}(y_{1})\\})=I^{-1}(q_{1}^{u})\text{
}(\because u_{0}\rightarrow u_{1}\text{ and }I^{-1}\circ I=I\circ I^{-1}).$
(57)

This proves the case of step 3 dealing with negative chains. The proof for the
case of positive chains is shown similarly.

Step 4: Proof of the proposition: Suppose
$(v_{i})_{i\in\mathbb{Z}}\in\Sigma^{\\#}$, and write
$v_{i}=\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}}$. Since $(v_{i})_{i_{\mathbb{Z}}}$
is a chain, we get that $\\{(p_{i}^{s},p_{i}^{u})\\}_{i\in\mathbb{Z}}$ is
$I$-strongly subordinated to $\\{Q_{\epsilon}(x_{i})\\}_{i\in\mathbb{Z}}$.
Since $(v_{i})_{i\in\mathbb{Z}}\in\Sigma^{\\#}$,
$\limsup_{i\rightarrow\infty}(p_{i}^{s}\wedge p_{i}^{u})>0$. Therefore by
Lemma 2.27, $p_{n}^{u}=Q_{\epsilon}(x_{n})$ for some $n<0$, and
$p_{l}^{s}=Q_{\epsilon}(x_{l})$ for some $l>0$. By step 2, $(v_{i})_{i\leq n}$
is an $I$-maximal negative chain, and $(v_{i})_{i\geq l}$ is an $I$-maximal
positive chain. By step 3: $(v_{i})_{i\leq 0}$ is an $I$-maximal negative
chain, and $(v_{i})_{i\geq 0}$ is an $I$-maximal positive chain. ∎

######  Lemma 4.13.

Let
$(\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\in\mathbb{Z}},(\psi_{y_{i}}^{q_{i}^{s},q_{i}^{u}})_{i\in\mathbb{Z}}\in\Sigma^{\\#}$
be two chains s.t
$\pi((\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\in\mathbb{Z}})=\pi((\psi_{y_{i}}^{q_{i}^{s},q_{i}^{u}})_{i\in\mathbb{Z}})=p$.
Then for all $k\in\mathbb{Z}$, $p_{k}^{u}=I^{\pm
1}(q_{k}^{u}),p_{k}^{s}=I^{\pm 1}(q_{k}^{s})$.

###### Proof.

By Proposition 4.12, $(\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\leq 0}$ is
$I$-maximal, so $p_{0}^{u}\geq I^{-1}(q_{0}^{u})$.
$(\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\leq 0}$ is also $I$-maximal, so
$q_{0}^{u}\geq I^{-1}(p_{0}^{u})$. It follows that $p_{0}^{u}=I^{\pm
1}(q_{0}^{u})$. Similarly $p_{0}^{s}=I^{\pm 1}(q_{0}^{s})$. Working with the
shifted sequences
$(\psi_{x_{i+k}}^{p_{i+k}^{s},p_{i+k}^{u}})_{i\in\mathbb{Z}}$, and
$(\psi_{y_{i+k}}^{q_{i+k}^{s},q_{i+k}^{u}})_{i\in\mathbb{Z}}$, we obtain
$p_{k}^{u}=I^{\pm 1}(q_{k}^{u}),p_{k}^{s}=I^{\pm 1}(q_{k}^{s})$ for all
$k\in\mathbb{Z}$. ∎

####  4.1.3. Comparing Pesin charts

######  Theorem 4.14.

The following holds for all $\epsilon$ small enough. Suppose
$(\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\in\mathbb{Z}},(\psi_{y_{i}}^{q_{i}^{s},q_{i}^{u}})_{i\in\mathbb{Z}}\in\Sigma^{\\#}$
are chains s.t
$\pi((\psi_{x_{i}}^{p_{i}^{s},p_{i}^{u}})_{i\in\mathbb{Z}})=p=\pi((\psi_{y_{i}}^{q_{i}^{s},q_{i}^{u}})_{i\in\mathbb{Z}})$
then for all i:

  1. (1)

$d(x_{i},y_{i})<\epsilon$,

  2. (2)

$p_{i}^{u}=I^{\pm 1}(q_{i}^{u}),p_{i}^{s}=I^{\pm 1}(q_{i}^{s})$,

  3. (3)

$(\psi_{y_{i}}^{-1}\circ\psi_{x_{i}})(u)=O_{i}u+a_{i}+\Delta_{i}(u)$ for all
$u\in R_{\epsilon}(0)$, where $O_{i}\in O(d)$ is a $d\times d$ orthogonal
matrix, $a_{i}$ is a constant vector s.t $|a_{i}|_{\infty}\leq
10^{-1}(q_{i}^{u}\wedge q_{i}^{s})$, and $\Delta_{i}$ is a vector field s.t
$\Delta_{i}(0)=0,\|d_{v}\Delta_{i}\|<\frac{1}{2}\epsilon^{\frac{1}{3}}$.
$O_{i}$ preserves $\mathbb{R}^{s(x)}$ and $\mathbb{R}^{d-s(x)}$.888Recall, in
our notations
$\mathbb{R}^{s(x)}:=C_{0}^{-1}(x_{i})[H^{s}(x_{i})]=C_{0}^{-1}(y_{i})[H^{s}(y_{i})]\subset\mathbb{R}^{d}$
and
$\mathbb{R}^{d-s(x)}:=(\mathbb{R}^{s(x)})^{\perp}=C_{0}^{-1}(x_{i})[H^{u}(x_{i})]=C_{0}^{-1}(y_{i})[H^{u}(y_{i})]$.

###### Proof.

Parts $(1)$ and $(2)$ are the content of Lemma 4.1 and Lemma 4.13
respectively; Part $(3)$ is shown similarly to [BO18, Theorem 4.13]. ∎

###  4.2. Similar charts have similar manifolds

####  4.2.1. Comparing manifolds

######  Proposition 4.15.

The following holds for all $\epsilon$ small enough. Let $V^{s}$ (resp.
$U^{s}$) be an $s$-admissible manifold in $\psi_{x}^{p^{s},p^{u}}$ (resp.
$\psi_{y}^{q^{s},q^{u}}$). Suppose $V^{s},U^{s}$ stay in windows. If $x=y$
then either $V^{s},U^{s}$ are disjoint, or one contains the other. The same
statement holds for $u$-admissible manifolds.

Proof is similar to [BO18, Proposition 4.15].

##  5\. Markov partitions and symbolic dynamics

In sections 1,2,3 of this paper we developed the shadowing theory needed to
construct Markov partitions à la Bowen ([Bow78, Bow08]), following the
improvements of Sarig to the shadowing theory [Sar13], and its extension to
higher dimensions [BO18], while carrying it in a setup where hyperbolicity is
replaced by $0$-summability. What remains is to carry out this construction.
This is the content of this part. As in [BO18], we follow [Bow78] and [Sar13]
closely here.

###  5.1. A locally finite countable Markov partition

####  5.1.1. The cover

In §2.4 we constructed a countable Markov shift $\Sigma$ with countable
alphabet $\mathcal{V}$ and a uniformly continuous map $\pi:\Sigma\rightarrow
M$ which commutes with the left shift $\sigma:\Sigma\rightarrow\Sigma$, so
that $\pi[\Sigma]\supseteq\mathrm{RST}$. Moreover, when

$\Sigma^{\\#}=\\{u\in\Sigma:u\text{ is a regular
chain}\\}=\\{u\in\Sigma:\exists v,w\in\mathcal{V}\exists
n_{k},m_{k}\uparrow\infty\text{ s.t }u_{n_{k}}=v,u_{-m_{k}}=w\\}$

then $\pi[\Sigma^{\\#}]\supset\mathrm{RST}$. However, $\pi$ is not finite-to-
one. In this section we study the following countable cover of $\mathrm{RST}$:

######  Definition 5.1.

$\mathcal{Z}:=\\{Z(v):v\in\mathcal{V}\\}$, where
$Z(v):=\\{\pi(u):u\in\Sigma^{\\#},u_{0}=v\\}$.

######  Theorem 5.2.

For every $Z\in\mathcal{Z}$, $|\\{Z^{\prime}\in\mathcal{Z}:Z^{\prime}\cap
Z\neq\varnothing\\}|<\infty$.

###### Proof.

Fix some $Z=Z(\psi_{x}^{p^{s},p^{u}})$. If
$Z^{\prime}=Z(\psi_{y}^{q^{s},q^{u}})$ intersects $Z$ then there must exist
two chains $v,w\in\Sigma^{\\#}$ s.t
$v_{0}=\psi_{x}^{p^{s},p^{u}},w_{0}=\psi_{y}^{q^{s},q^{u}}$ and
$\pi(v)=\pi(w)$. Lemma 4.13 says that in this case $q^{u}\geq
I^{-1}(p^{u}),q^{s}\geq I^{-1}(p^{s})$. It follows that $Z^{\prime}$ belongs
to
$\\{Z(\psi_{y}^{q^{s},q^{u}}):\psi_{y}^{q^{s},q^{u}}\in\mathcal{V},q^{u}\wedge
q^{s}\geq I^{-1}(p^{u}\wedge p^{s})\\}$. By definition, the cardinality of
this set is less than or equal to:

$|\\{\psi_{y}^{\eta}\in\mathcal{A}:\eta\geq I^{-1}(p^{s}\wedge
p^{u})\\}|\times|\\{(q^{s},q^{u})\in\mathcal{I}\times\mathcal{I}:q^{s}\wedge
q^{u}\geq I^{-1}(p^{s}\wedge p^{u})\\}|.$

This is a finite number because of the discreteness of $\mathcal{A}$
(Proposition 2.20). ∎

####  5.1.2. Product structure

Suppose $x\in Z(v)\in\mathcal{Z}$, then $\exists\underline{u}\in\Sigma^{\\#}$
s.t $u_{0}=v$ and $\pi(\underline{u})=x$. Associated to $\underline{u}$ are
two admissible manifolds in $v$: $V^{u}((u_{i})_{i\leq 0})$ and
$V^{s}((u_{i})_{i\geq 0})$. These manifolds do not depend on the choice of
$\underline{u}$: If $\underline{w}\in\Sigma^{\\#}$ is another chain s.t
$w_{0}=v$ and $\pi(\underline{w})=x$ then

$V^{u}((u_{i})_{i\leq 0})=V^{u}((w_{i})_{i\leq 0})\text{ and
}V^{s}((u_{i})_{i\geq 0})=V^{s}((w_{i})_{i\geq 0}),$

because of Proposition 4.15 and the equalities
$p^{s/u}(w_{0})=p^{s/u}(u_{0})=p^{s/u}(v)$. We are therefore free to make the
following definitions:

######  Definition 5.3.

Suppose $Z=Z(v)\in\mathcal{Z}$. For any $x\in Z$:

  1. (1)

$V^{u}(x,Z):=V^{u}((u_{i})_{i\leq 0})$ for some (any)
$\underline{u}\in\Sigma^{\\#}$ s.t $u_{0}=v$ and $\pi(\underline{u})=x$.

We also set $W^{u}(x,Z):=V^{u}(x,Z)\cap Z$.

  2. (2)

$V^{s}(x,Z):=V^{s}((u_{i})_{i\geq 0})$ for some (any)
$\underline{u}\in\Sigma^{\\#}$ s.t $u_{0}=v$ and $\pi(\underline{u})=x$.

We also set $W^{s}(x,Z):=V^{s}(x,Z)\cap Z$.

######  Proposition 5.4.

Suppose $Z\in\mathcal{Z}$, then for any $x,y\in Z$: $V^{u}(x,Z),V^{u}(y,Z)$
are either equal or they are disjoint. Similarly for $V^{s}(x,Z),V^{s}(y,Z)$;
and for $W^{s}(x,Z),W^{s}(y,Z)$ and for $W^{u}(x,Z),W^{u}(y,Z)$.

###### Proof.

The statement holds for $V^{s/u}(x,Z)$ because of Proposition 4.15. The
statement for $W^{s/u}$ is an immediate corollary. ∎

######  Proposition 5.5.

Suppose $Z\in\mathcal{Z}$, then for any $x,y\in Z$: $V^{u}(x,Z)$ and
$V^{s}(y,Z)$ intersect at a unique point $z$, and $z\in Z$. Thus
$W^{u}(x,Z)\cap W^{s}(y,Z)=\\{z\\}$.

###### Proof.

See [Sar13, Proposition 10.5]. ∎

######  Definition 5.6.

The Smale bracket of two points $x,y\in Z\in\mathcal{Z}$ is the unique point
$[x,y]_{Z}\in W^{u}(x,Z)\cap W^{s}(y,Z)$.

######  Lemma 5.7.

Suppose $x,y\in Z(v_{0})$, and $f(x),f(y)\in Z(v_{1})$. If $v_{0}\rightarrow
v_{1}$ then $f([x,y]_{Z(v_{0})})=[f(x),f(y)]_{Z(v_{1})}$.

###### Proof.

This is proved as in [Sar13, Lemma 10.7] using Theorem 3.6(2). ∎

######  Lemma 5.8.

Suppose $Z,Z^{\prime}\in\mathcal{Z}$. If $Z\cap Z^{\prime}\neq\varnothing$
then for any $x\in Z,y\in Z^{\prime}$: $V^{s}(x,Z)$ and $V^{s}(y,Z^{\prime})$
intersect at a unique point.

We do not claim that this point is in $Z$ nor $Z^{\prime}$.

Proof is similar to [BO18, Lemma 5.8].

####  5.1.3. The symbolic Markov property

######  Proposition 5.9.

If $x=\pi((v_{i})_{i\in\mathbb{Z}})$, where $\underline{v}\in\Sigma^{\\#}$,
then $f[W^{s}(x,Z(v_{0}))]\subset W^{s}(f(x),Z(v_{1}))$

and $f^{-1}[W^{u}(f(x),Z(v_{1}))]\subset W^{u}(x,Z(v_{0}))$.

###### Proof.

See [Sar13, Proposition 10.9]. ∎

######  Lemma 5.10.

Suppose $Z,Z^{\prime}\in\mathcal{Z}$ and $Z\cap Z^{\prime}\neq\varnothing$,
then:

  1. (1)

If $Z=Z(\psi_{x_{0}}^{p_{0}^{s},p_{0}^{u}})$ and
$Z^{\prime}=Z(\psi_{y_{0}}^{q_{0}^{s},q_{0}^{u}})$ then
$Z\subset\psi_{y_{0}}[R_{q_{0}^{s}\wedge q_{0}^{u}}(0)]$

  2. (2)

For any $x\in Z\cap Z^{\prime}$: $W^{u}(x,Z)\subset V^{u}(x,Z^{\prime})$ and
$W^{s}(x,Z)\subset V^{s}(x,Z^{\prime})$

Proof is similar to [BO18, Lemma 5.10].

##  6\. Proof of Theorem 1.1 and of Theorem 1.2

###  6.1. Summary of the properties of the Markov extension

We present a summary of the objects we have constructed so far, and their
properties.

We have constructed a family of sets $\mathcal{Z}$ s.t

  1. (1)

$\mathcal{Z}$ is countable (Definition 5.1; see also Proposition 2.20 and
Definition 2.23).

  2. (2)

$\mathcal{Z}$ covers (exactly) $\mathrm{RST}$ (Corollary 4.10).

  3. (3)

$\mathcal{Z}$ is locally finite: $\forall
Z\in\mathcal{Z},\\#\\{Z^{\prime}\in\mathcal{Z}:Z\cap
Z^{\prime}\neq\varnothing\\}<\infty$ (Theorem 5.2).

  4. (4)

Every $Z\in\mathcal{Z}$ has product structure: There are subsets
$W^{s}(x,Z),W^{u}(x,Z)\subset Z$ $(x\in Z)$ s.t

    1. (a)

$x\in W^{s}(x,Z)\cap W^{u}(x,Z)$.

    2. (b)

$\forall x,y\in Z\exists!z\in Z$ s.t $W^{u}(x,Z)\cap W^{s}(y,Z)=\\{z\\}$, we
write $z=[x,y]_{Z}.$

    3. (c)

$\forall x,y\in Z$ $W^{u}(x,Z),W^{u}(y,Z)$ are equal or disjoint (similarly
for $W^{s}(x,Z),W^{s}(y,z)$).

(Proposition 5.4, Proposition 5.5).

  5. (5)

The symbolic Markov property: if $x\in Z(u_{0}),f(x)\in Z(u_{1})$, and
$u_{0}\rightarrow u_{1}$, then $f[W^{s}(x,Z_{0})]\subset W^{s}(f(x),Z_{1})$
and $f^{-1}[W^{u}(f(x),Z_{1})]\subset W^{u}(x,Z_{0})$ (Proposition 5.9).999To
apply Proposition 5.9, represent $x=\pi(v)$ with $v\in\Sigma^{\\#}$,
$v_{0}=u_{0}$, and $f(x)=\pi(w)$ with $w\in\Sigma^{\\#}$, $w_{0}=u_{1}$; and
note that $x=\pi(u),u\in\Sigma^{\\#}$ where $u_{i}=v_{i}$ $(i\leq 0)$,
$u_{i}=w_{i-1}$ $(i\geq 1)$.

  6. (6)

Uniform expansion/contraction on $W^{s}(x,Z),W^{u}(x,Z)$: $\forall
Z\in\mathcal{Z},\forall z\in Z$, $\forall x\in W^{s}(z,Z),y\in W^{u}(z,Z)$,
$\forall n\geq 0$, $d(f^{n}(x),f^{n}(z)),d(f^{-n}(y),f^{-n}(z))\leq
4I^{-n}(1)$ (Proposition 4.4(1)).

  7. (7)

Bowen’s property with finite degree (see [BB17, Definition 1.11]):

    1. (a)

$\forall u,v\in\Sigma:$ $\Big{(}\forall i,Z(u_{i})\cap
Z(v_{i})\neq\varnothing\Big{)}\Rightarrow\pi(u)=\pi(v)$ (Proposition 3.14(4),
proof of Proposition 4.12).

    2. (b)

$\forall u,v\in\Sigma^{\\#}$: $\pi(u)=\pi(v)\Leftrightarrow\Big{(}Z(u_{i})\cap
Z(v_{i})\neq\varnothing$ $\forall i\in\mathbb{Z}\Big{)}$ ((a), Definition
5.1).

    3. (c)

$\forall Z\in\mathcal{Z}$: $\\#\\{Z^{\prime}\in\mathcal{Z}:Z^{\prime}\cap
Z\neq\emptyset\\}<\infty$ (Theorem 5.2).

For the sake of completeness, we present below the refinement of $\mathcal{Z}$
into a partition, while the same arguments appear in verbatim in [BO18, § 6]
and beforehand in [Sar13]. This refinment is used to prove Theorem 1.1 and
Theorem 1.2.

###  6.2. Construction of a partition with a finite-to-one almost everywhere
coding

We explain how the results which are listed in §6.1 are sufficient to prove
Theorem 1.1 and Theorem 1.2.

Step 1: $\mathcal{Z}$ can be refined into a countable partition $\mathcal{R}$
with the following properties:

  1. (a)

Product structure.

  2. (b)

Symbolic Markov property.

  3. (c)

Every $Z\in\mathcal{Z}$ contains only finitely many $R\in\mathcal{R}$.

Proof: This is done exactly as in [Sar13, § 11], using the Bowen-Sinai
refinement procedure [Bow70, Bow08, Bow78], the local-finiteness property (3),
and the product structure (4) of $Z_{i}\in\mathcal{Z}$. The partition elements
are the equivalence classes of the following equivalence relation on
$\bigcup\mathcal{Z}$ (see [Sar13, Proposition 11.2]):

$x\sim y\iff\forall Z,Z^{\prime}\in\mathcal{Z},\begin{bmatrix}x\in Z\iff y\in
Z\\\ W^{u}(x,Z)\cap Z^{\prime}\neq\varnothing\iff W^{u}(y,Z)\cap
Z^{\prime}\neq\varnothing\\\ W^{s}(x,Z)\cap Z^{\prime}\neq\varnothing\iff
W^{s}(y,Z)\cap Z^{\prime}\neq\varnothing\end{bmatrix}.$

Step 2: For every finite chain $R_{0}\rightarrow\cdots\rightarrow R_{m}$,
$\bigcap_{j=0}^{m}f^{-j}[R_{j}]\neq\varnothing$.

Proof: This follows from the fact that $\mathcal{R}$ is a Markov partition,
using a classical argument due to Adler and Weiss, and Sinai. The proof is by
induction on $m$. Assume the statement holds for $m$, and prove for $m+1$:
Take $p\in f^{m}\Big{[}\bigcap_{j=0}^{m}f^{-j}[R_{j}]\Big{]}$, and take
$p^{\prime}\in R_{m}\cap f^{-1}R_{m+1}$. $p,p^{\prime}\in R_{m}$, whence
belong to some $Z\in\mathcal{Z}$; therefore $[p,p^{\prime}]_{Z}$ is well
defined by Definition 5.6 and Proposition 5.5. By the symbolic Markov
property, $f^{-m}([p,p^{\prime}]_{Z})\in\bigcap_{j=0}^{m+1}f^{-j}[R_{j}]$.

Step 3: Let $\widehat{\mathcal{G}}$ be a directed graph with set of vertices
$\mathcal{R}$ and set of edges $\\{(R,S)\in\mathcal{R}\times\mathcal{R}:R\cap
f^{-1}[S]\neq\varnothing\\}$. Let
$\widehat{\Sigma}:=\Sigma(\widehat{\mathcal{G}})$ and define
$\widehat{\pi}:\widehat{\Sigma}\rightarrow M$ by
$\\{\widehat{\pi}(\underline{R})\\}=\bigcap\limits_{n\geq
0}\overline{\bigcap\limits_{i=-n}^{n}f^{-i}[R_{i}]}$. Then $\widehat{\pi}$ is
well defined, and:

  1. (a)

$\widehat{\pi}[\widehat{\Sigma}]\supset\mathrm{RST}$.

  2. (b)

$\widehat{\pi}\circ\sigma=f\circ\widehat{\pi}$.

  3. (c)

$\widehat{\pi}$ is uniformly continuous.

  4. (d)

$\widehat{\Sigma}$ is locally compact.

Proof: This proof is a concise presentation of [Sar13, Theorem 12.3, theorem
12.5]. Step 2 guarantees that the intersection in the expression defining
$\\{\widehat{\pi}\\}$ is not empty. The intersection is a singleton, because
$\mathrm{diam}(\bigcap_{i=-n}^{n}f^{-i}[R_{i}])\leq 8I^{-n}(1)$. To see this,
fix $x,y\in\bigcap_{i=-n}^{n}f^{-i}[R_{i}]$, and fix
$\underline{u}\in\Sigma^{\\#}$ s.t $\pi(\underline{u})=x$. It follows that
$\bigcap_{i=-n}^{n}f^{-i}[R_{i}]\subset\bigcap_{i=-n}^{n}f^{-i}[Z(u_{i})]$,
whence $y\in\bigcap_{i=-n}^{n}f^{-i}[Z(u_{i})]$. Let $z:=[x,y]_{Z(u_{0})}$. By
(4) and (5), $z\in\bigcap_{i=-n}^{n}f^{-i}[Z(u_{i})]$, $f^{n}(z)\in
W^{u}(f^{n}(x),Z(u_{n}))$, $f^{-n}(z)\in W^{s}(f^{-n}(y),Z(u_{-n}))$, whence
by (6) $d(x,y)\leq
d(x,z)+d(z,y)=d(f^{-n}(f^{n}(x)),f^{-n}(f^{n}(z)))+d(f^{n}(f^{-n}(z)),f^{n}(f^{-n}(y)))\leq
8I^{-n}(1)$. Therefore $\widehat{\pi}$ is well defined.

  1. (a)

Let $x\in\mathrm{RST}$. By Theorem 3.15, $f^{i}(x)\in\pi[\Sigma^{\\#}]$ for
all $i\in\mathbb{Z}$. By definition $\mathcal{Z}$ is a cover of
$\pi[\Sigma^{\\#}]$, and $\mathcal{R}$ refines $\mathcal{Z}$; whence
$f^{i}(x)\in\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}}\displaylimits\mathcal{R}$
$\forall i\in\mathbb{Z}$. Denote by $R_{i}$ the unique partition member of
$\mathcal{R}$ which contains $f^{i}(x)$. $\underline{R}$ is an admissible
chain by definition, and $\widehat{\pi}(\underline{R})=x$. Let
$\underline{u}\in\Sigma^{\\#}$ s.t $\pi(\underline{u})=x$, then
$Z(u_{i})\supset R_{i}$ $\forall i\in\mathbb{Z}$. $\mathcal{Z}$ is locally-
finite, so by the pigeonhole principle, $\vec{R}$ must be in
$\widehat{\Sigma}^{\\#}$.

  2. (b)

$\\{(f\circ\widehat{\pi})(\underline{R})\\}=\bigcap_{n\geq
0}\bigcap_{i=-n}^{n}\overline{f^{-(i-1)}[R_{i-1+1}]}=\\{\widehat{\pi}(\sigma\underline{R})\\}\text{
}(\because f\text{ is a diffeomorphism}).$

  3. (c)

Suppose $d(\underline{R},\underline{S})=e^{-n}$, then $R_{i}=S_{i}$ for
$|i|\leq n$, whence
$\widehat{\pi}(\underline{R}),\widehat{\pi}(\underline{S})\in\overline{\bigcap_{i=-n}^{n}f^{-i}[R_{i}]}$.
We just saw in the beginning of the proof of step 3 that
$\mathrm{diam}(\bigcap_{i=-n}^{n}f^{-i}[R_{i}])\leq 8I^{-n}(1)$. So
$d(\widehat{\pi}(\underline{R}),\widehat{\pi}(\underline{S}))\leq 8\cdot
I^{\lceil\log d(\underline{R},\underline{S})\rceil}(1)$.

  4. (d)

Fix a vertex $R\in\mathcal{R}$. Assume $R\rightarrow S$. Let $u\in\mathcal{V}$
s.t $R\subset Z(u)$. Take $x\in R\cap f^{-1}[S]\subset Z(u)$, then
$\exists\underline{u}\in\Sigma^{\\#}$ s.t $u_{0}=u$ and
$\pi(\underline{u})=x$. Then $f(x)\in S\cap Z(u_{1})$. Thus, by step 1(c), and
Lemma 2.24,

$\\#\\{S\in\mathcal{R}:R\rightarrow S\\}\leq\sum_{u\in\mathcal{V}:R\subset
Z(u)}\sum_{v\in\mathcal{V}:u\rightarrow v}\\#\\{S\in\mathcal{R}:S\subset
Z(v)\\}<\infty.$

A similar calculation shows $\\#\\{S^{\prime}:S^{\prime}\rightarrow
R\\}<\infty$.

Step 4: $\widehat{\pi}[\widehat{\Sigma}^{\\#}]=\mathrm{RST}$, and
$\widehat{\pi}|_{\widehat{\Sigma}^{\\#}}$ is finite-to-one.

Proof: In Corollary 4.10 we saw
$\bigcup\mathcal{Z}=\pi[\Sigma^{\\#}]=\mathrm{RST}$. $\mathcal{R}$ is a
refinement of $\mathcal{Z}$. [BO21, Proposition 4.11] shows
$\widehat{\pi}[\widehat{\Sigma}^{\\#}]=\pi[\Sigma^{\\#}]$ (with our
$\epsilon>0$)101010The proof of [BO21, Proposition 4.11] does not depend on
$\epsilon$ at all; $\epsilon$ only appears in the statement to satisfy the
equality $\pi[\Sigma^{\\#}]=\mathrm{RST}$ which we have already seen
independently in Corollary 4.10., and completes the proof that
$\widehat{\pi}[\widehat{\Sigma}^{\\#}]=\mathrm{RST}$. The finite-to-one
property follows from Theorem 1.2, whose proof follows verbatim the proof of
[BO18, Theorem 1.3] (which follows the proof of Sarig in [Sar13], which in
turn adapts the earlier works of Bowen [Bow70, Bow71, Bow78]).

######  Proposition 6.1.

For every $x\in\widehat{\pi}[\widehat{\Sigma}]$, $T_{x}M=E^{s}(x)\oplus
E^{u}(x)$, where

$(1)\text{ }\sum_{n\geq 0}\|d_{x}f^{n}|_{E^{s}(x)}\|^{2}<\infty\text{ ,
}(2)\text{ }\sum_{n\geq 0}\|d_{x}f^{-n}|_{E^{u}(x)}\|^{2}<\infty.$

The maps $\underline{R}\mapsto E^{s/u}(\widehat{\pi}(\underline{R}))$ have
summable variations as maps from $\widehat{\Sigma}$ to $TM$.

###### Proof.

The proof is similar to [Sar13, Proposition 12.6], so as in [BO18, Proposition
6.1], we limit ourselves to a brief sketch; One first writes $x$ as the image
of a chain $\underline{u}$ in $\Sigma$ (it is possible since the partition
$\mathcal{R}$ is a refinement of the cover $\mathcal{Z}$). Then the tangent
space $T_{x}M$ splits by $T_{x}\widecheck{V}^{s}(\underline{u})\oplus
T_{x}\widecheck{V}^{u}(\underline{u})$. The rest is clear by Lemma 4.4(2) and
Proposition 3.14(5). ∎

##  7\. Absolute Continuity of Weak Foliations

In §3 we constructed local stable and unstable leaves for points in
$\mathrm{RST}$. In fact, by using a strengthened method of the Graph
Transform, we constructed local stable and unstable manifolds for every chain,
while Theorem 3.15 says that every point in $\mathrm{RST}$ can be coded by a
chain in $\Sigma^{\\#}$.

These local stable manifolds contract forwards in time, as shown in
Proposition 4.4. However, they may contract in a strictly slower than an
exponential rate, which distinguishes their behavior from the behavior of
classical local stable leaves of non-uniformly hyperbolic orbits. For that
reason, the foliations by such leaves are called weak foliations. In an
upcoming paper, we present a few examples of systems which exhibit orbits with
strictly weak foliations, while these orbits are shown to carry invariant
measures of interest (alas not probability measures, as every invariant
probability measure carried by $\mathrm{RST}$ is hyperbolic), and we show
furthermore that these orbits have full Riamannian volume.

In this section of the paper we wish to prove that weak foliations are
absolutely continuous w.r.t the holonomy map and the Riemannian
volume.111111This terminology is common- however may be slightly misleading;
since in the general case the stable leaves may not generate a foliation, and
may not exist for every point. This property was shown by Anosov and Sinai for
the stable foliations of Anosov systems in [AS67], and later extended by Pesin
to the non-uniformly hyperbolic setup (see [BP07, § 8.6] for proof). Pesin’s
proof deals with the difficulty that not every point may admit a local stable
leaf, and with the fact that the size of local stable leaves may vary greatly
in small neighborhoods. Pesin’s approach is to restrict to a regular set (i.e
Pesin block), and consider the leaves which pass through points in a fixed
such set.

The proof we present below follows the idea of Pesin, while utilizing one more
property: A careful examination of Pesin’s proof shows that a minimal
requirement for his scheme to work is that the leaves contract fast enough so
that the sequence of their diameters over a forwards-orbit is summable. In our
proof, restricting to a partition element takes the role of restricting to a
regular set. Proposition 4.4 and Lemma 2.6 show the summability property on
partition elements, which is needed for the scheme to work. Although, one
still has to show that the $\beta$-Hölder continuity of $d_{\cdot}f$ is
sufficient for summability when $\beta$ may be arbitrarily small, which was
obvious for the exponentially contracting rate in Pesin’s case.

For the simplicity of presentation we prove the absolute continuity of the
holonomy map between two unstable leaves of chains in a fixed chart; but the
proof easily extends for two admissible $u$-manifolds in the chart similarly
(recall Definition 3.2). The principle which is used is that the contraction
on the stable leaves overpowers any possible contraction in the transversals
in a summable manner.

###  7.1. Bounded leaf measures

Let $\mathcal{V}$ be the collection of double charts from Definition 2.23. Fix
$u\in\mathcal{V}$, and the set $A(u):=\pi[\Sigma\cap[u]]$. Notice, $A(u)$ is
compact and $A(u)\supseteq Z(u)$ which is covered by finitely many partition
elements (recall §6.2). Fix $(u_{i}^{1})_{i\leq 0}$ and $(u_{i}^{2})_{i\leq
0}$ two negative chains s.t $u^{1}_{0}=u^{2}_{0}=u=\psi_{x}^{p^{s},p^{u}}$.
Denote the respective representative functions of $V^{u}((u_{i}^{1})_{i\leq
0})$ and $V^{u}((u_{i}^{2})_{i\leq 0})$ by $F^{1}$ and $F^{2}$.

Notice: every point in $V^{u}((u_{i}^{j})_{i\leq 0})\cap A(u)$ can be coded by
a chain $\underline{u}\in\Sigma$ s.t $u_{i}^{j}=u_{i}$ for all $i\leq 0$ and
$j=1,2$.

######  Lemma 7.1.

Let $x\in V^{u}((u_{i}^{1})_{i\leq 0})\cap Z(u)$, and let
$\underline{u}\in\Sigma$ s.t $u_{i}^{j}=u_{i}$ for all $i\leq 0$ and
$\pi(\underline{u})=x$. Write $y:=\pi((u_{i}^{2})_{i\leq 0}\cdot(u_{i})_{i\geq
0})$, where $\cdot$ denotes an admissible concatenation. Write
$u_{i}=\psi_{x_{i}}^{p^{s}_{i},p^{u}_{i}}$, $i\in\mathbb{Z}$.

For every $n\geq 0$, let $\underline{u}^{1,n}:=(u_{i}^{1})_{i\leq
0}\cdot(u_{j})_{j=0}^{n}$ and $\underline{u}^{2,n}:=(u_{i}^{2})_{i\leq
0}\cdot(u_{j})_{j=0}^{n}$ $F_{n}^{1}$, and let $F_{n}^{2}$ be the
representative functions of $V^{u}(\underline{u}^{1,n})$ and
$V^{u}(\underline{u}^{2,n})$, respectively. Write
$\widetilde{F}^{j}_{n}(v):=(F^{j}_{n}(v),v)$, $n\geq 0$.

Then for all $n\geq 0$, for all $\eta\in(0,(I^{-n}(p^{s}\wedge p^{u}))^{2})$,

$\left|\frac{\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{1}_{n}[R_{\eta}(0)])}{\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{2}_{n}[R_{\eta}(0)])}-1\right|\leq\epsilon^{2}(I^{-n}(p^{s}\wedge
p^{u}))^{\frac{1}{\gamma}+\tau^{\prime}},$

where $\tau^{\prime}=\frac{1}{4\gamma}>0$, $\mathrm{Vol}$ denotes the
intrinsic volume of a submanifold with an induced Riemannian volume, and
$R_{\eta}(0)$ is an $\|\cdot\|_{\infty}$-norm ball of radius $\eta$ around $0$
in $\mathbb{R}^{u(x)}$.

###### Proof.

We fix $n\geq 0$, and omit the subscript in the proof where it is clear, as
all analysis is in a fixed chart $u_{n}=\psi_{x_{n}}^{p^{s}_{n},p^{u}_{n}}$.

Step 1: Define

$I:\psi_{x_{n}}(F^{1}(t),t)\mapsto\psi_{x_{n}}(F^{2}(t),t).$

Equivalently,
$I=\psi_{x_{n}}\circ\tilde{F}^{2}\circ\pi_{u}\circ\psi_{x_{n}}^{-1}$, and
$\tilde{F}^{1/2}$ are diffeomorphisms onto their images, where $\pi_{u}$ is
the projection onto the $u$-coordinates. So $I$ is a diffeomorphism from
$\widecheck{V}^{u}(\underline{u}^{1,n})$ to
$\widecheck{V}^{u}(\underline{u}^{2,n})$. $I$ satisfies

$d(I(x^{\prime}),x^{\prime})\leq\|\psi_{x_{n}}\|d_{C^{0}}(F^{1},F^{2})\leq
2d_{C^{0}}(F^{1},F^{2}).$ (58)

The identity map of $\widecheck{V}^{u}(\underline{u}^{1,n})$ can be written as
$\mathrm{Id}=\psi_{x_{n}}\circ\tilde{F}^{1}\circ\pi_{u}\circ\psi_{x_{n}}^{-1}:\widecheck{V}^{u}(\underline{u}^{1,n})\rightarrow\widecheck{V}^{u}(\underline{u}^{1,n})$.
Let $z\in\widecheck{V}^{u}(\underline{u}^{1,n})$, and let
$\Theta_{D}:TD\rightarrow\mathbb{R}^{d}$ be a local isometry as in Definition
2.16 s.t. $D$ is a neighborhood which contains $z,I(z)$. Then,

$\displaystyle\|\Theta_{D}d_{z}I-\Theta_{D}d_{z}\mathrm{Id}\|$
$\displaystyle=\|\Theta_{D}d_{z}(\psi_{x_{n}}\circ\tilde{F}^{2}\circ\pi_{u}\circ\psi_{x_{n}}^{-1})-\Theta_{D}d_{z}(\psi_{x_{n}}\circ\tilde{F}^{1}\circ\pi_{u}\circ\psi_{x_{n}}^{-1})\|$
$\displaystyle=\|[\Theta_{D}d_{t}(\psi_{x_{n}}\circ\tilde{F}^{\underline{v}})-\Theta_{D}d_{t}(\psi_{x_{n}}\tilde{F}^{\underline{u}})]\pi_{u}\circ
d_{z}(\psi_{x_{n}}^{-1}))\|,\hskip 7.11317pt\text{where
}t=\pi_{u}\psi_{x_{n}}^{-1}(z),$ $\displaystyle\leq
2\|C_{0}^{-1}(x_{n})\|\cdot\|\Theta_{D}d_{t}(\psi_{x_{n}}\circ\tilde{F}^{2})-\Theta_{D}d_{t}(\psi_{x_{n}}\circ\tilde{F}^{1})\|$
$\displaystyle=2\|C_{0}^{-1}(x_{n})\|\cdot\|\Theta_{D}d_{\tilde{F}^{2}(t)}\psi_{x_{n}}d_{t}\tilde{F}^{2}-\Theta_{D}d_{\tilde{F}^{1}(t)}\psi_{x_{n}}d_{t}\tilde{F}^{1}\|$
$\displaystyle=2\|C_{0}^{-1}(x_{n})\|\cdot\|\Theta_{D}d_{\tilde{F}^{\underline{v}}(t)}\psi_{x_{n}}(d_{t}\tilde{F}^{2}-d_{t}\tilde{F}^{1})+(\Theta_{D}d_{\tilde{F}^{2}(t)}\psi_{x_{n}}-\Theta_{D}d_{\tilde{F}^{1}(t)}\psi_{x_{n}})d_{t}\tilde{F}^{1}\|$
$\displaystyle\leq
2\|C_{0}^{-1}(x_{n})\|\cdot\left[d_{C^{1}}(\tilde{F}^{1},\tilde{F}^{2})\|d_{\cdot}\psi_{x_{n}}\|+\mathrm{Lip}(d_{\cdot}\psi_{x_{n}})d_{C^{0}}(\tilde{F}^{1},\tilde{F}^{2})\cdot\|d_{\cdot}\tilde{F}^{1}\|\right]$
$\displaystyle\leq 8\|C_{0}^{-1}(x_{n})\|\cdot
d_{C^{1}}(\widecheck{V}^{u}(\underline{u}^{1,n}),\widecheck{V}^{u}(\underline{u}^{2,n}))\leq
8(C_{\beta,\epsilon}^{-1}\cdot p_{n}^{s}\wedge
p_{n}^{u})^{\frac{-1}{2\gamma}}\cdot
d_{C^{1}}(\widecheck{V}^{u}(\underline{u}^{1,n}),\widecheck{V}^{u}(\underline{u}^{2,n}))$
$\displaystyle\leq$ $\displaystyle 8(C_{\beta,\epsilon}^{-1}\cdot
I^{-n}(p^{s}\wedge p^{u}))^{\frac{-1}{2\gamma}}\cdot
d_{C^{1}}(\widecheck{V}^{u}(\underline{u}^{1,n}),\widecheck{V}^{u}(\underline{u}^{2,n})).$
(59)

It follows that,

$\displaystyle|\mathrm{Jac}(d_{z}I)-1|=$
$\displaystyle|\mathrm{Jac}(d_{z}I)-\mathrm{Jac}(d_{z}\mathrm{Id})|\leq
C_{1}\|\Theta_{D}d_{z}I-\Theta_{D}d_{z}\mathrm{Id}\|$ $\displaystyle\leq$
$\displaystyle 8C_{1}(C_{\beta,\epsilon}^{-1}\cdot I^{-n}(p^{s}\wedge
p^{u}))^{\frac{-1}{2\gamma}}\cdot
d_{C^{1}}(\widecheck{V}^{u}(\underline{u}^{1,n}),\widecheck{V}^{u}(\underline{u}^{2,n})),$
(60)

where $C_{1}$ is the Lipschitz constant for the absolute value of the
determinant on the ball of $(d\times d)$-matrices with an operator norm
bounded by $2$, where $\|d_{z}I\|\leq 2$ by (7.1).121212Write
$u=dimV^{u}(\underline{R})$, and let $A,B$ be $u\times u$ matrices s.t.
$|a_{ij}-b_{ij}|\leq\delta$ for all $i,j\leq u$. So $\det A=\sum_{\sigma\in
S_{u}}\mathrm{sgn}(\sigma)\prod_{i=1}^{u}a_{i\sigma(i)}$. Then $|\det A-\det
B|\leq\sum_{\sigma\in
S_{u}}|\prod_{i=1}^{u}a_{i\sigma(i)}-\prod_{i=1}^{u}b_{i\sigma(i)}|\leq|S_{u}|\cdot
u\|B\|_{Fr}^{u-1}\delta$. Take maximum over $u\leq d-1$.

By Proposition 3.14,
$d_{C^{1}}(\widecheck{V}^{u}(\underline{u}^{1,n}),\widecheck{V}^{u}(\underline{u}^{2,n}))\leq
C(I^{-n}(p^{s}\wedge p^{u}))^{\frac{1}{\gamma}+\tau}$, for $C,\tau>0$
constants depending on the calibration parameters (but not on $\epsilon$).
Furthermore, by (18) in Proposition 3.14, when $\gamma>\frac{5}{\beta}$,
$\beta-\frac{2}{\gamma}>\frac{3}{\gamma}=(\frac{1}{\gamma}+\tau^{\prime})+(\frac{1}{\gamma}+\frac{1}{2\gamma}+\tau^{\prime})$,
where $\tau^{\prime}:=\frac{1}{4\gamma}>0$. Then for $C^{\prime}:=3\sup_{N\geq
0}\\{N\cdot(I^{-N}(1))^{\frac{1}{\gamma}+\tau^{\prime}}\\}<\infty$, we get

$d_{C^{1}}(\widecheck{V}^{u}(\underline{u}^{1,n}),\widecheck{V}^{u}(\underline{u}^{2,n}))\leq
C^{\prime}(I^{-n}(p^{s}\wedge
p^{u}))^{\frac{1}{\gamma}+\frac{1}{2\gamma}+\tau^{\prime}}.$

Substituting this in (7.1) gives, for all sufficiently small $\epsilon>0$,

$|\mathrm{Jac}(d_{z}I)-1|\leq
8C_{1}C_{\beta,\epsilon}^{\frac{1}{2\gamma}}\cdot
C^{\prime}(I^{-n}(p^{s}\wedge
p^{u}))^{\frac{1}{\gamma}+\tau^{\prime}}\leq\epsilon^{2}(I^{-n}(p^{s}\wedge
p^{u}))^{\frac{1}{\gamma}+\tau^{\prime}}.$ (61)

Step 2:

$\displaystyle|\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{1}[R_{\eta}(0)])-\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{2}[R_{\eta}(0)])|=$
$\displaystyle|\int\limits_{R_{\eta}(0)}\mathrm{Jac}(d_{t}(\psi_{x_{n}}\tilde{F}^{1}))d\mathrm{Leb}(t)-\int_{R_{\eta}(0)}\mathrm{Jac}(d_{t}(\psi_{x_{n}}\tilde{F}^{2}))d\mathrm{Leb}(t)|$
$\displaystyle\leq$
$\displaystyle\int_{R_{\eta}(0)}|\mathrm{Jac}(d_{t}(\psi_{x_{n}}\tilde{F}^{1}))-\mathrm{Jac}(d_{t}(\psi_{x_{n}}\tilde{F}^{2}))|d\mathrm{Leb}(t)$
$\displaystyle=$
$\displaystyle\int\limits_{R_{\eta}(0)}\mathrm{Jac}(d_{t}(\psi_{x_{n}}\widetilde{F}^{1}))\cdot|1-\mathrm{Jac}(d_{t}(\psi_{x_{n}}\tilde{F}^{2}))$
$\displaystyle\cdot\mathrm{Jac}(d_{\psi_{x_{n}}\tilde{F}^{1}(t)}((\psi_{x_{n}}\tilde{F}^{1})^{-1}))|d\mathrm{Leb}(t)$
$\displaystyle=$
$\displaystyle\int\limits_{R_{\eta}(0)}\mathrm{Jac}(d_{t}(\psi_{x_{n}}\widetilde{F}^{1}))\cdot|1-\mathrm{Jac}(d_{\psi_{x_{n}}\tilde{F}^{1}(t)}I)|d\mathrm{Leb}(t).$

By (61),

$\displaystyle\left|\frac{\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{2}[R_{\eta}(0)])}{\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{1}[R_{\eta}(0)])}-1\right|=$
$\displaystyle\frac{|\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{1}[R_{\eta}(0)])-\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{2}[R_{\eta}(0)])|}{\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{1}[R_{\eta}(0)])}$
$\displaystyle\leq$ $\displaystyle\frac{\epsilon^{2}(I^{-n}(p^{s}\wedge
p^{u}))^{\frac{1}{\gamma}+\tau^{\prime}}\int_{R_{\eta}(0)}\mathrm{Jac}(d_{t}(\psi_{x_{n}}\circ\widetilde{F}^{1}[R_{\eta}(0)])\mathrm{Leb}(t)}{\int_{R_{\eta}(0)}\mathrm{Jac}(d_{t}(\psi_{x_{n}}\circ\widetilde{F}^{1}[R_{\eta}(0)])\mathrm{Leb}(t)}$
$\displaystyle=$ $\displaystyle\epsilon^{2}(I^{-n}(p^{s}\wedge
p^{u}))^{\frac{1}{\gamma}+\tau^{\prime}}.$ (62)

∎

######  Lemma 7.2.

In the notation of Lemma 7.1, $\forall n\geq 0$

$\frac{|d_{f^{n}(x)}f^{-n}\omega_{u}(f^{n}(x))|}{|d_{f^{n}(y)}f^{-n}\omega_{u}(f^{n}(y))|}=e^{\pm\epsilon^{2}(\frac{d(x,y)}{(p^{s}\wedge
p^{u})^{3}})^{\frac{\beta}{4}}},$

where $\omega_{u}(\cdot)$ is the normalized volume form of
$T_{\cdot}V^{u}(\cdot)$.

###### Proof.

Showing
$\frac{|d_{x}f^{n}\omega_{u}(x)|}{|d_{y}f^{n}\omega_{u}(y)|}=e^{\pm\epsilon^{2}(\frac{d(x,y)}{(p^{s}\wedge
p^{u})^{3}})^{\frac{\beta}{4}}}$ is enough. Proposition 4.4 treats the case
for the normalized volume forms of $T_{\cdot}V^{s}(\cdot)$. The case for the
normalized volume forms of $T_{\cdot}V^{u}(\cdot)$ is similar. ∎

######  Corollary 7.3.

In the notation of Lemma 7.1, $\forall n\geq 0$ sufficiently large (w.r.t
$d(x,y)$ and $p^{s}\wedge p^{u}$), $\forall\eta\in(0,(I^{-n}(p^{s}\wedge
p^{u}))^{3+\frac{4}{\beta}(\frac{1}{\gamma}+\tau^{\prime})}]$,

$\frac{\mathrm{Vol}(f^{-n}[\psi_{x_{n}}\circ\widetilde{F}^{1}_{n}[R_{\eta}(0)]])}{\mathrm{Vol}(f^{-n}[\psi_{x_{n}}\circ\widetilde{F}^{2}_{n}[R_{\eta}(0)]])}=e^{\pm
2\epsilon^{2}(\frac{d(x,y)}{(p^{s}\wedge p^{u})^{3}})^{\frac{\beta}{4}}},$

where $\tau^{\prime}=\frac{1}{4\gamma}>0$.

###### Proof.

For any $t_{1},t_{2}\in R_{\eta}(0)$,
$d(\psi_{x_{n}}(t_{1}),\psi_{x_{n}}(t_{2}))\leq\frac{3}{2}\eta$. We have,

$\mathrm{Vol}(f^{-n}[\psi_{x_{n}}\circ\widetilde{F}^{1}_{n}[R_{\eta}(0)]])=\int_{\psi_{x_{n}}\circ\widetilde{F}^{1}_{n}[R_{\eta}(0)])}|d_{f^{n}(t)}f^{-n}\omega_{u}(f^{n}(t))|d\mathrm{Vol}_{V^{u}(f^{n}(x))}(t).$

By Proposition 4.4, part (3), since $\eta\leq I^{-n}(p^{s}\wedge
p^{u}))^{3+\frac{4}{\beta}(\frac{1}{\gamma}+\tau^{\prime})}$ and
$(p^{s}_{n})^{3}\geq(p^{s}_{n}\wedge p^{u}_{n})^{3}\geq(I^{-n}(p^{s}\wedge
p^{u}))^{3}$,

$\mathrm{Vol}(f^{-n}[\psi_{x_{n}}\circ\widetilde{F}^{1}_{n}[R_{\eta}(0)]])=\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{1}_{n}[R_{\eta}(0)])\cdot|d_{f^{n}(x)}f^{-n}\omega_{u}(f^{n}(x))|e^{\pm\epsilon^{2}(\frac{3}{2})^{\frac{\beta}{4}}(I^{-n}(p^{s}\wedge
p^{u}))^{\frac{1}{\gamma}+\tau^{\prime}}}.$

Similarly,

$\mathrm{Vol}(f^{-n}[\psi_{x_{n}}\circ\widetilde{F}^{2}_{n}[R_{\eta}(0)]])=\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{2}_{n}[R_{\eta}(0)])\cdot|d_{f^{n}(y)}f^{-n}\omega_{u}(f^{n}(y))|e^{\pm\epsilon^{2}(\frac{3}{2})^{\frac{\beta}{4}}(I^{-n}(p^{s}\wedge
p^{u}))^{\frac{1}{\gamma}+\tau^{\prime}}}.$

By Lemma 7.2,

$\frac{\mathrm{Vol}(f^{-n}[\psi_{x_{n}}\circ\widetilde{F}^{2}_{n}[R_{\eta}(0)]])}{\mathrm{Vol}(f^{-n}[\psi_{x_{n}}\circ\widetilde{F}^{1}_{n}[R_{\eta}(0)]])}=\frac{\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{2}_{n}[R_{\eta}(0)])}{\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{1}_{n}[R_{\eta}(0)])}e^{\pm
2\cdot(\frac{3}{2})^{\frac{\beta}{4}}\epsilon^{2}(I^{-n}(p^{s}\wedge
p^{u}))^{\frac{1}{\gamma}+\tau^{\prime}}}e^{\pm\epsilon^{2}(\frac{d(x,y)}{(p^{s}\wedge
p^{u})^{3}})^{\frac{\beta}{4}}}.$

By Lemma 7.1, for all $\epsilon$ sufficiently small,
$\frac{\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{2}[R_{\eta}(0)])}{\mathrm{Vol}(\psi_{x_{n}}\circ\widetilde{F}^{1}[R_{\eta}(0)])}=e^{\pm(1+(3-2(\frac{3}{2})^{\frac{\beta}{4}}))\epsilon^{2}(I^{-n}(p^{s}\wedge
p^{u}))^{\frac{1}{\gamma}+\tau^{\prime}}}$. Thus in total,
$\frac{\mathrm{Vol}(f^{-n}[\psi_{x_{n}}\circ\widetilde{F}^{2}_{n}[R_{\eta}(0)]])}{\mathrm{Vol}(f^{-n}[\psi_{x_{n}}\circ\widetilde{F}^{1}_{n}[R_{\eta}(0)]])}=e^{\pm
2\epsilon^{2}(\frac{d(x,y)}{(p^{s}\wedge p^{u})^{3}})^{\frac{\beta}{4}}}$ for
all $n$ sufficiently large. ∎

###  7.2. Covering lemmas and the proof of the absolute continuity of weak
foliations

######  Lemma 7.4 (Besicovitch-Pesin).

For all sufficiently large $n\geq 0$, there exist
$\\{x^{(i)}\\}_{i=1}^{\ell}\subseteq A(u)$, and sets
$\\{D_{i}^{1}\\}_{i=1}^{\ell}$ s.t

  1. (1)

$x^{(i)}=\pi(\underline{u}^{(i)})$ where $u^{(i)}_{j}=u^{1}_{j}$, $j\geq 0$;
and $u_{j}^{(i)}=\psi_{x_{j}^{(i)}}^{p^{s,i}_{j},p^{u,i}_{j}}$.

  2. (2)

$D_{i}^{1}\subseteq\psi_{x_{n}^{(i)}}\circ\widetilde{F}^{1,i}_{n}[R_{\eta}(0)]$
where $F^{1,i}$ is the representative function of $V^{u}((u^{(i)}_{j})_{j\leq
n})$ in $u_{n}^{(i)}$, and $\eta\in(0,(I^{-n}(p^{s}\wedge
p^{u}))^{3+\frac{4}{\beta}(\frac{1}{\gamma}+\tau^{\prime})}]$,
$\tau^{\prime}:=\frac{1}{4\gamma}>0$.

  3. (3)

$\forall i\neq i^{\prime}$, $\mathrm{Vol}(D_{i}^{1}\cap D_{i^{\prime}}^{1})=0$
($\mathrm{Vol}$ is the intrinsic $f^{n}[V^{u}((u^{1}_{i})_{i\leq
0})]$-volume).

  4. (4)

$A(u)\cap V^{u}((u^{1}_{i})_{i\leq
0})\subseteq\bigcup_{i=1}^{\ell}f^{-n}[D_{i}^{1}]$.

  5. (5)

There exist $R_{n}$, and a constant $c_{m}>0$ depending on the dimension of
$M$ s.t for $\eta_{n}:=(I^{-n}(p^{s}\wedge
p^{u}))^{3+\frac{4}{\beta}(\frac{1}{\gamma}+\tau^{\prime})}$, for all $1\leq
i\leq\ell$,

$B(x^{(i)},c_{m}R_{n})\subseteq D_{i}^{1}\subseteq
f^{-n}[\psi_{x_{n}^{(i)}}\circ\widetilde{F}^{1,i}_{n}[R_{\eta_{n}}(0)]])\subseteq
B(x^{(i)},R_{n}).$

The collection depends on the leaf $V^{u}((u^{1}_{i})_{i\leq 0})$, but a
collection $\\{D_{i}^{2}\\}_{i=1}^{\ell}$ with similar properties exists for
$V^{u}((u^{2}_{i})_{i\leq 0})$, and constitutes a cover of $A(u)\cap
V^{u}((u^{2}_{i})_{i\leq 0})$. In particular, the two covers share the same
multiplicity $\ell<\infty$. In addition, the collections
$\\{D_{i}^{1/2}\\}_{i=1}^{\ell}$ can be chosen in such a way that $\forall
1\leq i\leq\ell$,

$\frac{\mathrm{Vol}(D_{i}^{1})}{\mathrm{Vol}(D_{i}^{2})}=e^{\pm\epsilon^{2}(I^{-n}(p^{s}\wedge
p^{u}))^{\frac{1}{\gamma}+\tau^{\prime}}}.$ (63)

For proof see [BP07, Lemma 8.6.3]. For the estimate of
$\eqref{afterniniloulou}$ use the Jacobian bounds of the map $I$ from Lemma
7.1.

Remark: The collections $\\{D_{i}^{1/2}\\}_{i=1}^{\ell}$ depends on $n$. To
ease notation we omit notating it, when dependence is clear by context. In
addition, similarly.

######  Corollary 7.5.

For all $n\geq 0$ sufficiently large, for all $1\leq i\leq\ell$,

$\frac{\mathrm{Vol}(f^{-n}[D_{i}^{1}])}{\mathrm{Vol}(f^{-n}[D_{i}^{2}])}=e^{\pm
2\epsilon^{2}(\frac{d(x,y)}{(p^{s}\wedge p^{u})^{3}})^{\frac{\beta}{4}}}.$

Shown similarly as in Corollary 7.3.

Now that we have useful covers in our arsenal, and good estimates on the leaf
measures of the cover elements, we wish to begin the proof of the absolute
continuity property. If the leaf volume of $A(u)$ is $0$ in both leaves, we
are done. So we may assume that the volume in one of them is positive- w.l.o.g
$V^{u}((u_{i}^{1})_{i\leq 0})$\- and get estimates for the measure in the
other. We start by recalling a useful measure theoretical lemma:

######  Lemma 7.6.

Given $\delta>0$, let $U_{\delta}^{1}$ be a $\delta$-neighborhood of
$V^{u}((u_{i}^{1})_{i\leq 0})\cap A(u)$ in the leaf $V^{u}((u_{i}^{1})_{i\leq
0})$. Then $\exists\delta_{0}>0$ s.t $\forall\delta\in(0,\delta_{0}]$

$\frac{\mathrm{Vol}(A(u))}{\mathrm{Vol}(U_{\delta}^{1})}\geq
e^{-2\epsilon^{2}(\frac{2d_{C_{0}}(V^{u}((u_{i}^{1})_{i\leq
0}),V^{u}((u_{i}^{2})_{i\leq 0}))}{(p^{s}\wedge
p^{u})^{3}})^{\frac{\beta}{4}}},$

where $\mathrm{Vol}$ is the intrinsic $V^{u}((u_{i}^{1})_{i\leq 0})$-volume.

In particular, $\delta_{0}$ does not depend on $n$. Therefore,
$\forall\delta\in(0,\delta_{0})$ we may choose $n_{\delta}$ s.t $\forall n\geq
n_{\delta}$,