C^{\prime}(\mu^{-n_{1}}+c_{5}\rho^{1-1/2k}),$

for every $n\geq n_{1}$. Since $n_{1}\geq\rho^{-1/k}$, if $\rho$ is
sufficiently small, then $\mu^{-n_{1}}\ll c_{5}\,\rho^{1-1/2k}$, and so the
estimate (9) is valid for all $n\geq 0$, concluding the first part of the
proof.

∎

######  Remark 3.3.

Some remarks relating the perturbation:

  * •

Note that since
$\|f^{\underline{\omega}}_{\underline{a}}-f_{\underline{a}}\|_{C^{0}}\lesssim
c_{5}\rho^{1+1/2k}$ then, supposing $\rho$ is small enough, we have
$G^{\underline{\omega}}(\underline{a})\subset V_{\rho}(G(\underline{a}))$.

  * •

Remember that we assume the base points $c_{a}$, $a\in\mathbb{A}$, are pre-
periodic. From this, it is easy to prove that the base points do not depend on
$\underline{\omega}$, i.e. $c^{\underline{\omega}}_{a}=c_{a}$. Indeed, let
$\underline{\alpha}\underline{\beta}\underline{\beta}...\underline{\beta}...$
be the symbolic sequence associated to the points
$c^{\underline{\omega}}_{a}$. Then we can write

$c^{\underline{\omega}}_{a}=\lim_{n\to\infty}f^{\underline{\omega}}_{\underline{\alpha}\underline{\beta}^{n}}(x),$

where $x$ is any element in $G(\beta)$, and $\beta$ is the last letter of
$\underline{\beta}$. Notice that if $n_{0}$ is big enough then
$\underline{\beta}^{n_{0}}$ contains a word of $\Sigma(\rho^{1/3k})$ repeated
twice, thus any $\underline{\gamma}\in\Sigma(\rho^{1/k})$ containing
$\underline{\beta}^{n_{0}}$ can not be in $\Sigma_{1}$. This implies that

$f^{\underline{\omega}}_{\underline{\alpha}\underline{\beta}^{n}}(x)=f_{\underline{\alpha}\underline{\beta}^{n-n_{0}}}(f^{\underline{\omega}}_{\underline{\beta}^{n_{0}}}(x)),$

for all $n>n_{0}$. Making $n$ go to infinity we conclude that
$c^{\underline{\omega}}_{a}$ does not depend on $\underline{\omega}$.

  * •

If $diam(G(\underline{a}))\geq c_{0}^{-1}\rho$ then
$|DF^{\underline{\theta}}_{\underline{a}}|\approx|DF^{\underline{\theta},\underline{\omega}}_{\underline{a}}|$
and using

$DF^{\underline{\theta},\underline{\omega}}_{\underline{a}}=Dk^{\underline{\theta},\underline{\omega}}(c_{\underline{a}})\cdot
Df^{\underline{\omega}}_{\underline{a}}(c_{a})$

one sees that
$|Df_{\underline{a}}(c_{a})|\approx|Df^{\underline{\omega}}_{\underline{a}}(c_{a})|$
and then $diam(G(\underline{a}))\approx
dim(G^{\underline{\omega}}(\underline{a}))$. On can arrive to a similar
estimate if $diam(G(\underline{a}))\geq c_{0}^{-1}\rho^{3}$, in this case we
can decompose $\underline{a}$ as a concatenation of at most $4$ words in
$\Sigma(\rho)$ and use the fact that
$diam(G(\underline{a}_{1}\underline{a}_{2}))\approx
diam(G(\underline{a}_{1}))diam(G(\underline{a}_{2}))$. Notice that with this
approach the constants get worse if we increase the power of $\rho$ in which
we are interested, $\rho^{3}$ will be enough for us.

  * •

Let $\underline{\omega}_{1},\underline{\omega}_{2}\in\Omega$ and
$\underline{\theta}\in\Sigma^{-}$, suppose that

$f^{\underline{\omega}_{1}}_{\underline{\theta}_{n}}(z)=f^{\underline{\omega}_{2}}_{\underline{\theta}_{n}}(z)$

for all $n\leq N$ and all $z$ in a neighborhood of $z_{0}$, and in a
neighborhood of $c_{\theta_{0}}$. Remember that limit geometries are defined
by
$k^{\underline{\theta},\underline{\omega}}=\lim_{n\to\infty}k^{\underline{\theta},\underline{\omega}}_{n}$,
where

$k^{\underline{\theta},\underline{\omega}}_{n}(z)=Df^{\underline{\omega}}_{\underline{\theta}_{n}}(c_{\theta_{0}})^{-1}(f^{\underline{\omega}}_{\underline{\theta}_{n}}(z)-f^{\underline{\omega}}_{\underline{\theta}_{n}}(c_{\theta_{0}})).$

By our assumption we have
$k^{\underline{\theta},\underline{\omega}_{1}}_{n}(z)=k^{\underline{\theta},\underline{\omega}_{2}}_{n}(z)$,
for all $n\leq N$ and $z$ in a neighborhood of $z_{0}$. From the proof of the
existence of limit geometries (see [6]) one has that there is a constant $C$
such that

$\|k^{\underline{\theta},\underline{\omega}}-k^{\underline{\theta},\underline{\omega}}_{n}\|\leq
Cdiam(G^{\underline{\omega}}(\underline{\theta}_{n}))$

and

$\|D(k^{\underline{\theta},\underline{\omega}}\circ(k^{\underline{\theta},\underline{\omega}}_{n})^{-1})-Id\|\leq
Cdiam(G^{\underline{\omega}}(\underline{\theta}_{n})),$

the same constant $C$ works for all Cantor sets $K^{\underline{\omega}}$,
since they depend continuously on $\underline{\omega}$. It follows easily that
there is a constant $C^{\prime}$ such that

$|k^{\underline{\theta},\underline{\omega}_{1}}(z)-k^{\underline{\theta},\underline{\omega}_{2}}(z)|\leq
C^{\prime}diam(G^{\underline{\omega}_{1}}(\underline{\theta}_{N}))$

and

$|Dk^{\underline{\theta},\underline{\omega}_{1}}(z)-Dk^{\underline{\theta},\underline{\omega}_{2}}(z)|\leq
C^{\prime}diam(G^{\underline{\omega}_{1}}(\underline{\theta}_{N})),$

for all $z$ in a neighborhood of $z_{0}$. Notice that since
$diam(G^{\underline{\omega}_{j}}(\underline{\theta}_{n}))\approx|Df^{\underline{\omega}_{j}}_{\underline{\theta}_{n}}(c_{\theta_{0}})|$,
$j=1,2$, and
$Df^{\underline{\omega}_{1}}_{\underline{\theta}_{N}}(c_{\theta_{0}})=Df^{\underline{\omega}_{2}}_{\underline{\theta}_{N}}(c_{\theta_{0}})$
then $diam(G^{\underline{\omega}_{1}}(\underline{\theta}_{N}))\approx
diam(G^{\underline{\omega}_{2}}(\underline{\theta}_{N}))$.

##  4\. Proof of theorem 2.22

In this section we will define the set of relative configurations
$\mathcal{L}=\mathcal{L}_{\underline{\omega}}$, which will be a recurrent
compact set for at least one of the Cantor sets in the family of random
perturbations. We first give a primary description of $\mathcal{L}$ and prove
that assuming a probabilistic estimate, proposition 4.1, then we can prove
theorem 2.22. The proof of the probabilistic estimate will be given in later
sections.

###  4.1. The recurrent compact set

The set $\mathcal{L}=\mathcal{L}_{\underline{\omega}}$ will depend on
$\underline{\omega}$, but only the translation coordinate $t$. The image of
$\mathcal{L}_{\underline{\omega}}$ under the projection map:
$\mathcal{C}\to\mathcal{S}$ will be a subset $\tilde{\mathcal{L}}$ of
$\mathcal{S}$ independent of $\underline{\omega}$.

We will choose a subset of $\Sigma^{-}$ with good combinatorial properties,
this will be crucial to prove the estimate of proposition 4.1. First, let
$\Sigma_{nr}(\rho^{3})$ be the subset of $\Sigma(\rho^{3})$ formed by words
$\underline{a}$ such that:

  1. (1)

no word $\underline{b}\in\Sigma(\rho^{1/3k})$ appears twice in
$\underline{a}$;

  2. (2)

if $\underline{c}\in\Sigma(\rho^{1/6k})$ appears at the end of
$\underline{a}$, then it does not appear elsewhere in $\underline{a}$.

We next define $\Sigma_{nr}^{-}$ as the subset of $\Sigma^{-}$ formed by
$\underline{\theta}$ which end with a word in $\Sigma_{nr}(\rho^{3})$. This is
an open and closed subset in $\Sigma^{-}$.

A family of subsets $E(\underline{a},\underline{a}^{\prime})$ of $J_{R}$ , for
$(\underline{a},\underline{a}^{\prime})\in\Sigma(\rho^{1/2})\times\Sigma^{\prime}(\rho^{1/2})$
will be constructed in subsection 4.2, in relation to Marstrand’s theorem, and
it will satisfy the hypothesis

$Leb(J_{R}\setminus E(\underline{a},\underline{a}^{\prime}))\leq
c_{1}\,,\quad\forall(\underline{a},\underline{a}^{\prime}).$

Then, the Scale recurrence Lemma gives us another family
$E^{*}(\underline{a},\underline{a}^{\prime})$,
$(\underline{a},\underline{a}^{\prime})\in\Sigma(\rho^{1/2})\times\Sigma^{\prime}(\rho^{1/2})$,
with the properties indicated in the statement of the lemma.

The set $\widetilde{\mathcal{L}}$ is defined to be the subset of
$\mathcal{S}_{R}$ formed by the
$(\underline{\theta},\underline{\theta}^{\prime},s)$ such that
$\underline{\theta}\in\Sigma_{nr}^{-}$ , and there exists
$\underline{a}\in\Sigma(\rho^{1/2})$,
$\underline{a}^{\prime}\in\Sigma^{\prime}(\rho^{1/2})$ with $s\in
E^{*}(\underline{a},\underline{a}^{\prime})$ and $\underline{\theta}$,
$\underline{\theta}^{\prime}$ ending with $\underline{a}$,
$\underline{a}^{\prime}$ respectively.

For every $(\underline{\theta},\underline{\theta}^{\prime},s)$ in
$\widetilde{\mathcal{L}}$, we will define in subsection 4.3, considering the
properties given by Marstrand’s theorem, a non empty subset
$L_{\underline{\omega}}^{0}(\underline{\theta},\underline{\theta}^{\prime},s)$,
depending on $\underline{\omega}\in\Omega$, of the fiber of $\mathcal{C}$ over
$(\underline{\theta},\underline{\theta}^{\prime},s)$.

Let

$\mathcal{L}_{\underline{\omega}}^{0}=\\{(\underline{\theta},\underline{\theta}^{\prime},s,t):\,(\underline{\theta},\underline{\theta}^{\prime},s)\in\widetilde{\mathcal{L}},t\in
L_{\underline{\omega}}^{0}(\underline{\theta},\underline{\theta}^{\prime},s)\\};$

consider next the $\rho$-neighbourhood $\mathcal{L}_{\underline{\omega}}^{1}$
of $\mathcal{L}_{\underline{\omega}}^{0}$ in
$\widetilde{\mathcal{L}}\times\mathbbm{C}$:

$\displaystyle\quad\mathcal{L}_{\underline{\omega}}^{1}=\\{(\underline{\theta},\underline{\theta}^{\prime},s,t):\,(\underline{\theta},\underline{\theta}^{\prime},s)\in\widetilde{\mathcal{L}}\text{
and
}\exists\,(\underline{\theta}_{0},\underline{\theta}_{0}^{\prime},s_{0},t_{0})\in\mathcal{L}_{\underline{\omega}}^{0}$
$\displaystyle\text{with
}d(\underline{\theta},\underline{\theta}_{0})<2\rho^{5/2},d(\underline{\theta}^{\prime},\underline{\theta}_{0}^{\prime})<2\rho^{5/2},|s-s_{0}|<\rho,|t-t_{0}|<\rho\\}.$

Fix
$u=(\underline{\theta},\underline{\theta}^{\prime},s,t)\in\widetilde{\mathcal{L}}\times\mathbbm{C}$.
We define two subsets $\Omega^{0}(u)$, $\Omega^{1}(u)$ of $\Omega$. First,

$\Omega^{1}(u)=\\{\underline{\omega}\in\Omega:\,(\underline{\theta},\underline{\theta}^{\prime},s,t)\in\mathcal{L}_{\underline{\omega}}^{1}\\}.$

Second, $\Omega^{0}(u)$ is the set of $\underline{\omega}\in\Omega$ such that
there exists $\underline{b}\in\Sigma(\rho)$,
$\underline{b}^{\prime}\in\Sigma^{\prime}(\rho)$, with $b_{0}=\theta_{0}$,
$b_{0}^{\prime}=\theta_{0}^{\prime}$ and the image
$T^{\underline{\omega}}_{\underline{b}}T^{\prime}_{\underline{b}^{\prime}}(u)=(\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s},\tilde{t}$)
satisfies:

  * (i)

for any $\tilde{s}_{1}$ with
$|\tilde{s}-\tilde{s}_{1}|<\dfrac{1}{2}c_{4}\rho^{1/2}$, we have
$(\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s}_{1})\in\widetilde{\mathcal{L}}$;

  * (ii)

$\tilde{t}\in
L_{\underline{\omega}}^{0}(\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s})$.

The following crucial estimate will be proven in section 6.

######  Proposition 4.1.

Assume that $c_{5}$ is chosen conveniently large. Then there exists $c_{7}>0$,
such that, for any $u\in{\widetilde{\mathcal{L}}}\times\mathbbm{C}$, one has

$\mathbb{P}(\Omega^{1}(u)\setminus\Omega^{0}(u))\leq\exp\big{(}-c_{7}\rho^{-\frac{1}{2k}(d+d^{\prime}-2)}\big{)}.$

Using the previous estimate, we will prove that for some $\underline{\omega}$
the pair $K^{\underline{\omega}},\,K$ has a recurrent compact set, thus
obtaining theorem 2.22. We proceed as in [1]. We discretize the set
$\tilde{\mathcal{L}}\times\\{t:|t|<4(1+e^{R})\\}$, i.e. we choose subsets
$\Delta_{i}$, $i\in I$, such that

$\tilde{\mathcal{L}}\times\\{t:|t|<4(1+e^{R})\\}=\bigcup_{i\in I}\Delta_{i},$

and for all
$(\underline{\theta}_{1},\underline{\theta}^{\prime}_{1},s_{1},t_{1}),\,(\underline{\theta}_{2},\underline{\theta}^{\prime}_{2},s_{2},t_{2})\in\Delta_{i}$
one has

$d(\underline{\theta}_{1},\underline{\theta}_{2})<\rho^{5/2},\,d(\underline{\theta}^{\prime}_{1},\underline{\theta}^{\prime}_{2})<\rho^{5/2},\,|s_{1}-s_{2}|<\rho^{3},\,|t_{1}-t_{2}|<\rho^{3}.$

It is not difficult to see that this can be done in such a way that $\\#I$ is
polynomial in $\rho^{-1}$. For each $i\in I$, choose $u_{i}\in\Delta_{i}$. If
$\rho$ is small enough, we have

$\mathbb{P}(\cup_{i\in
I}\Omega^{1}(u_{i})\setminus\Omega^{0}(u_{i}))\leq\\#I\cdot\exp\left(-c_{7}\rho^{-\frac{1}{2k}(d+d^{\prime}-2)}\right)<1.$

Therefore, the set $\cup_{i\in I}\Omega^{1}(u_{i})\setminus\Omega^{0}(u_{i})$
is not the whole $\Omega$ and we can choose $\underline{\omega}_{0}$ outside
of it. Observe that for all $i\in I$ we have that
$\underline{\omega}_{0}\in\Omega^{1}(u_{i})$ implies
$\underline{\omega}_{0}\in\Omega^{0}(u_{i})$. Now define

$\displaystyle\mathcal{L}_{\underline{\omega}_{0}}=\\{$
$\displaystyle(\underline{\theta},\underline{\theta}^{\prime},s,t)\in\tilde{\mathcal{L}}\times\mathbbm{C}:\,\exists(\tilde{}\underline{\theta},\tilde{}\underline{\theta}^{\prime},\tilde{s},\tilde{t})\in\mathcal{L}^{0}_{\underline{\omega}_{0}}$
$\displaystyle\text{with
}d(\underline{\theta},\tilde{}\underline{\theta})\leq\rho^{5/2},\,d(\underline{\theta}^{\prime},\tilde{}\underline{\theta}^{\prime})\leq\rho^{5/2},\,|s-\tilde{s}|\leq\rho/2,\,|t-\tilde{t}|\leq\rho/2\\}.$

We will prove that $\mathcal{L}_{\underline{\omega}_{0}}$ is a recurrent
compact set for $K^{\underline{\omega}_{0}},\,K^{\prime}$. First, notice that
$\mathcal{L}^{0}_{\underline{\omega}_{0}}\subset\mathcal{L}_{\underline{\omega}_{0}}\subset\mathcal{L}^{1}_{\underline{\omega}_{0}}$.
Then $\mathcal{L}_{\underline{\omega}_{0}}$ is not empty. Now, let
$u=(\underline{\theta},\underline{\theta}^{\prime},s,t)\in\mathcal{L}_{\underline{\omega}_{0}}$,
we have that there is
$(\tilde{}\underline{\theta},\tilde{}\underline{\theta}^{\prime},\tilde{s},\tilde{t})\in\mathcal{L}^{0}_{\underline{\omega}_{0}}$
with the properties in the definition of
$\mathcal{L}_{\underline{\omega}_{0}}$. Since $\tilde{t}\in
L^{0}_{\underline{\omega}_{0}}(\tilde{}\underline{\theta},\tilde{}\underline{\theta}^{\prime},\tilde{s})$
then $|\tilde{t}|\leq 3(1+e^{R})$ (this is clear from the definition of
$L^{0}$ in subsection 4.3 and equation (3)). Thus $|t|<4(1+e^{R})$ and
$u\in\Delta_{i}$ for some $i\in I$. For
$u_{i}=(\underline{\theta}_{i},\underline{\theta}^{\prime}_{i},s_{i},t_{i})$
we have that

$d(\underline{\theta},\underline{\theta}_{i})<\rho^{5/2},\,d(\underline{\theta}^{\prime},\underline{\theta}^{\prime}_{i})<\rho^{5/2},\,|s-s_{i}|<\rho^{3},\,|t-t_{i}|<\rho^{3}.$

Therefore $u_{i}\in\mathcal{L}^{1}_{\underline{\omega}_{0}}$ and
$\underline{\omega}_{0}\in\Omega^{1}(u_{i})$, this implies that
$\underline{\omega}_{0}\in\Omega^{0}(u_{i})$ and there is some pair
$(\underline{b},\underline{b}^{\prime})\in\Sigma(\rho)\times\Sigma^{\prime}(\rho)$
such that
$T^{\underline{\omega}_{0}}_{\underline{b}}T^{\prime}_{\underline{b}^{\prime}}(u_{i})=(\tilde{\underline{\theta}}_{i},\tilde{\underline{\theta}}^{\prime}_{i},\tilde{s}_{i},\tilde{t}_{i})$
satisfies the properties (i) and (ii) described above.

We will prove that
$T^{\underline{\omega}_{0}}_{\underline{b}}T^{\prime}_{\underline{b}^{\prime}}(u)$
is in the interior of $\mathcal{L}_{\underline{\omega}_{0}}$. Write
$T^{\underline{\omega}_{0}}_{\underline{b}}T^{\prime}_{\underline{b}^{\prime}}(u)=(\underline{\theta}\underline{b},\underline{\theta}^{\prime}\underline{b}^{\prime},\hat{s},\hat{t})$,
using equation (3) we have

$\tilde{t}_{i}=\left(DF^{\underline{\theta}^{\prime}_{i}}_{\underline{b}^{\prime}}\right)^{-1}\cdot(t_{i}+s_{i}c^{\underline{\theta}_{i},\underline{\omega}_{0}}_{\underline{b}}-c^{\underline{\theta}^{\prime}_{i}}_{\underline{b}^{\prime}}),\text{
and
}\hat{t}=\left(DF^{\underline{\theta}^{\prime}}_{\underline{b}^{\prime}}\right)^{-1}\cdot(t+sc^{\underline{\theta},\underline{\omega}_{0}}_{\underline{b}}-c^{\underline{\theta}^{\prime}}_{\underline{b}^{\prime}}).$

Therefore $|\tilde{t}_{i}-\hat{t}|\lesssim\rho^{3/2}$. Analogously one has

$\hat{s}=\frac{DF^{\underline{\theta},\underline{\omega}_{0}}_{\underline{b}}}{DF^{\underline{\theta}^{\prime}}_{\underline{b}^{\prime}}}\cdot
s,\text{ and
}\tilde{s}_{i}=\frac{DF^{\underline{\theta}_{i},\underline{\omega}_{0}}_{\underline{b}}}{DF^{\underline{\theta}^{\prime}_{i}}_{\underline{b}^{\prime}}}\cdot
s_{i}.$

In this case we get $|\tilde{s}_{i}-\hat{s}|\lesssim\rho^{5/2}$. One also has
$d(\underline{\theta}\underline{b},\tilde{}\underline{\theta}_{i})\lesssim\rho^{7/2}$
and
$d(\underline{\theta}^{\prime}\underline{b}^{\prime},\tilde{\underline{\theta}}^{\prime}_{i})\lesssim\rho^{7/2}$.

Thanks to property (ii), we know that
$(\tilde{\underline{\theta}}_{i},\tilde{\underline{\theta}}^{\prime}_{i},\tilde{s}_{i},\tilde{t}_{i})\in\mathcal{L}^{0}_{\underline{\omega}_{0}}$.
Moreover, for any $(\underline{\eta},\underline{\eta}^{\prime},r,x)$ such that

$d(\underline{\eta},\underline{\theta}\underline{b})<\rho^{7/2},\,d(\underline{\eta}^{\prime},\underline{\theta}^{\prime}\underline{b}^{\prime})<\rho^{7/2},\,|r-\hat{s}|<\rho/4,\,|x-\hat{t}|<\rho/4,$

we have

$d(\underline{\eta},\tilde{}\underline{\theta}_{i})\leq\rho^{5/2},\,d(\underline{\eta}^{\prime},\tilde{}\underline{\theta}^{\prime}_{i})\leq\rho^{5/2},\,|r-\tilde{s}_{i}|\leq\rho/2,\,|x-\tilde{t}_{i}|\leq\rho/2.$

To conclude that
$(\underline{\eta},\underline{\eta}^{\prime},r,x)\in\mathcal{L}_{\underline{\omega}_{0}}$
we only need to show that
$(\underline{\eta},\underline{\eta}^{\prime},r)\in\tilde{\mathcal{L}}$.
Property (i) above implies that
$(\tilde{}\underline{\theta}_{i},\tilde{}\underline{\theta}^{\prime}_{i},r)\in\tilde{\mathcal{L}}$,
by the definition of the set $\tilde{\mathcal{L}}$ this means that
$\tilde{}\underline{\theta}_{i}\in\Sigma^{-}_{nr}$ and $r\in
E^{*}(\underline{a},\underline{a}^{\prime})$ for a pair
$(\underline{a},\underline{a}^{\prime})$ in
$\Sigma(\rho^{1/2})\times\Sigma^{\prime}(\rho^{1/2})$ such that
$(\tilde{}\underline{\theta}_{i},\tilde{}\underline{\theta}^{\prime}_{i})$
ends in it. However, since
$d(\underline{\eta},\tilde{}\underline{\theta}_{i})\lesssim\rho^{7/2},\,d(\underline{\eta}^{\prime},\tilde{}\underline{\theta}^{\prime}_{i})\lesssim\rho^{7/2}$
then $\underline{\eta}\in\Sigma^{-}_{nr}$ and
$(\underline{\eta},\underline{\eta}^{\prime})$ also ends in
$(\underline{a},\underline{a}^{\prime})$. Therefore
$(\underline{\eta},\underline{\eta}^{\prime},r)\in\tilde{\mathcal{L}}$ and
$(\underline{\eta},\underline{\eta}^{\prime},r,x)\in\mathcal{L}_{\underline{\omega}_{0}}$,
which shows that
$T^{\underline{\omega}_{0}}_{\underline{b}}T^{\prime}_{\underline{b}^{\prime}}(u)$
is in the interior of $\mathcal{L}_{\underline{\omega}_{0}}$. From the fact
that the sets $E^{*}(\underline{a},\underline{a}^{\prime})$ are closed and the
definition of the sets $L^{0}_{\underline{\omega}}$, it is not difficult to
prove that $\mathcal{L}_{\underline{\omega}_{0}}$ is a compact set. Therefore,
$\mathcal{L}_{\underline{\omega}_{0}}$ is a recurrent compact set for the pair
$K^{\underline{\omega}_{0}},\,K^{\prime}$.

###  4.2. Set of good scales

In this subsection we will define the sets
$E(\underline{a},\underline{a}^{\prime})$ which we use to construct
$\tilde{\mathcal{L}}$. Let
$(\underline{\theta},\underline{\theta}^{\prime},s)$ in the space of relative
scales, and points $x\in K(\theta_{0})$, $x^{\prime}\in
K^{\prime}(\theta_{0}^{\prime})$. Consider

$\displaystyle\lambda$
$\displaystyle=\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(x,x^{\prime}):=k^{\underline{\theta}^{\prime}}(x^{\prime})-sk^{\underline{\theta}}(x).$

Then $(\underline{\theta},\underline{\theta}^{\prime},s,\lambda)$ is the
unique relative configuration above
$(\underline{\theta},\underline{\theta}^{\prime},s)$ such that

$A(k^{\underline{\theta}}(x))=A^{\prime}(k^{\underline{\theta}^{\prime}}(x^{\prime})),$

(where $(\underline{\theta},A),(\underline{\theta}^{\prime},A^{\prime})$
represents this relative configuration).

Remember that, for some previously fixed $R>0$ (given by the scale recurrence
lemma),

$J_{R}=\\{s\in J:e^{-R}\leq|s|\leq e^{R}\\}$

and $\mathcal{S}_{R}=\Sigma^{-}\times{\Sigma^{\prime}}^{-}\times J_{R}$.

Let $d$, $d^{\prime}$ be the Hausdorff dimension of $K$, $K^{\prime}$,
respectively. We equip each set $K(\theta_{0})$ (resp.
$K^{\prime}(\theta_{0}^{\prime}))$ with the $d$-dimensional (resp.
$d^{\prime}$-dimensional) Hausdorff measure $\mu_{d}$ (resp.
$\mu_{d^{\prime}}$).

Then, for $(\underline{\theta},\underline{\theta}^{\prime},s)\in\mathcal{S}$,
we denote by $\mu(\underline{\theta},\underline{\theta}^{\prime},s)$ the image
under $\pi_{\underline{\theta},\underline{\theta}^{\prime},s}$ of
$\mu_{d}\times\mu_{d^{\prime}}$ on $K(\theta_{0})\times
K^{\prime}(\theta_{0}^{\prime})$.

As in the theory of Cantor sets in the real line, there are constants
$c_{11}>c_{10}>0$ such that, for $\theta_{0}\in\mathbb{A}$,
$\theta_{0}^{\prime}\in\mathbb{A}^{\prime}$:

$c_{10}<\mu_{d}\times\mu_{d^{\prime}}(K(\theta_{0})\times
K^{\prime}(\theta_{0}^{\prime}))<c_{11}\,.$

This can be proven using the results appearing in Zamudio’s thesis ([8]).
Indeed, for a given conformal Cantor set $K$ of dimension $d$, using equation
(1), one can find a sequence of coverings with size converging to zero and
$d$-volume bounded by $Cc_{0}^{d}$, namely the covering by the pieces of
$\Sigma(\rho)$, showing that $\mu_{d}(K)<\infty$.

On the other hand, lemma 1.2.3 of [8] gives that there exist constants
$C_{1}>0$, $L>0$, $r_{0}>0$, independent of $\rho$, such that for all
$\underline{a}\in\Sigma(\rho)$

$C_{1}^{-1}\left(\frac{\rho}{r}\right)^{-d}\leq\\#\\{\underline{b}\in\Sigma(\rho):\frac{r}{l}\leq
d(G(\underline{a}),G(\underline{b}))<r\\}\leq
C_{1}\left(\frac{\rho}{r}\right)^{-d},$

for all $l\geq L$, $r<r_{0}$. Using this lemma, we conclude that given a
finite cover of $K$ by balls $U_{i}$ of radii $r_{i}>0$, $i=1,\dots,n$, each
$U_{i}$ intersects at most $C_{1}\rho^{-d}r_{i}^{d}$ pieces $G(\underline{a})$
of $\Sigma(\rho)$ if $\rho$ and $r_{i}$ are sufficiently small. Since $U_{i}$
is a cover and $\\#\Sigma(\rho)>C^{-1}\rho^{-d}$, summing for all $i$ yields:

$C_{1}\rho^{-d}\left(\sum r_{i}^{d}\right)\geq C^{-1}\rho^{-d}$

and so $\sum r_{i}^{d}$ (and $\mu_{d}(K)$) is always bounded from zero. To
obtain the statement just restrict the arguments to $K(\theta_{0})$ and
${K^{\prime}}(\theta^{\prime}_{0})$ and take their product.

Notice that the same lemma 1.2.3 implies that there is a constant $c>0$ such
that for $\mu:=\mu_{d}\times\mu_{d^{\prime}}$, the product measure in
$\mathbbm{C}^{2}$,

$\mu(B(x,r))<cr^{d+d^{\prime}}$

for any ball of radius $r>0$. If $d+d^{\prime}>2$ this condition implies that:

$I_{2}(\mu):=\int\int|u-v|^{-2}d\mu(u)d\mu(v)<\infty$

This way, the proof of Theorem 9.7 in Mattila’s book (a Marstrand-type
theorem) [11] can be adapted444all one needs to verify is that for any points
$u,v\in\mathbbm{C}^{2}$, $Leb(\\{s:s\in
J_{R},\,|\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(u)-\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(v)|<\delta\\})<c\delta^{2}|u-v|^{-2}$,
where $c>0$ is some constant depending only on $R$. Notice that
$\pi_{\underline{\theta},\underline{\theta}^{\prime},s}=\pi_{s}\circ
F_{\underline{\theta},\underline{\theta}^{\prime}}$, where
$\pi_{s}(u)=u_{2}-s\cdot u_{1}$ for $u=(u_{1},u_{2})$ and
$F_{\underline{\theta},\underline{\theta}^{\prime}}=(k^{\underline{\theta}},k^{\underline{\theta}^{\prime}})$
are diffeomorphisms that distort area in a uniformly bounded way. A simple
manipulation shows that the measure is bounded above by
$c\delta^{2}|u_{1}-v_{1}|^{-2}$. If $|u_{1}-v_{1}|\leq\delta$ then
$\delta^{2}|u-v|^{-2}>\tilde{c}$ for some constant $\tilde{c}$ and the desired
inequality follows choosing $c$ big enough. If $|u_{1}-v_{1}|>\delta$, using
that $s$ is in $J_{R}$ a bounded set, one sees that $|u_{1}-v_{1}|^{-1}$ is
bounded by $c_{R}|u-v|^{-1}$, for some constant $c_{R}$ depending on $R$.  to
our context to show that for fixed
$(\underline{\theta},\underline{\theta}^{\prime})$ the measure
$\mu(\underline{\theta},\underline{\theta}^{\prime},s)$ is absolutely
continuous with respect to the Lebesgue measure for Lebesgue almost every $s$,
with density $\chi_{\underline{\theta},\underline{\theta}^{\prime},s}$ in
$L^{2}$ satisfying

$\int_{J_{R}}\|\chi_{\underline{\theta},\underline{\theta}^{\prime},s}\|_{L^{2}}^{2}\,ds\leq
c_{12}(R),$

where $c_{12}(R)$ is independent of $\underline{\theta}$,
$\underline{\theta}^{\prime}$.

When one controls
$\|\chi_{\underline{\theta},\underline{\theta}^{\prime},s}\|_{L^{2}}$ , this
gives, by Cauchy-Schwarz inequality, a lower bound for the Lebesgue measure of
$\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(X)$, $X$ being a
subset of $K\times K^{\prime}$ with positive $(d+d^{\prime})$-dimensional
Hausdorff measure; indeed we have:

$\displaystyle\mu_{d}\times\mu_{d^{\prime}}(X)$
$\displaystyle\leq\int_{\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(X)}\chi_{\underline{\theta},\underline{\theta}^{\prime},s}(t)\,dt$
$\displaystyle\leq
Leb(\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(X))^{1/2}\|\chi_{\underline{\theta},\underline{\theta}^{\prime},s}\|_{L^{2}}$

and therefore

(10)
$Leb(\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(X))\geq(\mu_{d}\times\mu_{d^{\prime}}(X))^{2}\|\chi_{\underline{\theta},\underline{\theta}^{\prime},s}\|_{L^{2}}^{-2}\,.$

Fix $(\underline{\theta},\underline{\theta}^{\prime})$ in
$\Sigma^{-}\times\Sigma^{\prime-}$. Let $\underline{a}\in\Sigma(\rho^{1/2k})$,
$\underline{a}^{\prime}\in\Sigma^{\prime}(\rho^{1/2k})$, with
$a_{0}=\theta_{0}$ , $a_{0}^{\prime}=\theta_{0}^{\prime}$ . One has

$T_{\underline{a}}T^{\prime}_{\underline{a}^{\prime}}(\underline{\theta},\underline{\theta}^{\prime},s)=\left(\underline{\theta}\underline{a},\underline{\theta}^{\prime}\underline{a}^{\prime},s\cdot
DF^{\underline{\theta}}_{\underline{a}}/DF^{\underline{\theta}^{\prime}}_{\underline{a}^{\prime}}\right)$

and

$c_{13}^{-1}\leq\frac{|DF^{\underline{\theta}}_{\underline{a}}|}{|DF^{\underline{\theta}^{\prime}}_{\underline{a}^{\prime}}|}\leq
c_{13}\,.$

We therefore have

$\int_{J_{R}}\lVert\chi_{{T_{\underline{a}}T^{\prime}_{\underline{a}^{\prime}}(\underline{\theta},\underline{\theta}^{\prime},s)}}\rVert_{L^{2}}^{2}\,ds\leq
c_{12}^{\prime}(R),$

with $c_{12}^{\prime}(R)$ independent of $\underline{\theta}$,
$\underline{\theta}^{\prime}$, $\underline{a}$, $\underline{a}^{\prime}$. On
the other hand, one has

$\displaystyle\\#\,\Sigma(\rho^{1/2k})\leq c_{14}\,\rho^{-d/2k}\,,$
$\displaystyle\\#\,\Sigma^{\prime}(\rho^{1/2k})\leq
c_{14}\,\rho^{-d^{\prime}/2k}\,.$

We conclude that

$\int_{J_{R}}\sum_{\underline{a},\underline{a}^{\prime}}\|\chi_{{T_{\underline{a}}T^{\prime}_{\underline{a}^{\prime}}(\underline{\theta},\underline{\theta}^{\prime},s)}}\|_{L^{2}}^{2}\,ds\leq
c_{14}^{2}\,\rho^{-\frac{d+d^{\prime}}{2k}}\,c_{12}^{\prime}(R).$

We now define, with $c_{15}>0$ conveniently large to be determined later:

$\displaystyle E(\underline{\theta},\underline{\theta}^{\prime})$
$\displaystyle=\bigg{\\{}s\in
J_{R}:\,\|\chi_{\underline{\theta},\underline{\theta}^{\prime},s}\|_{L^{2}}^{2}\leq
c_{15}$ $\displaystyle\quad\text{and
}\quad\sum_{\underline{a},\underline{a}^{\prime}}\|\chi_{{T_{\underline{a}}T^{\prime}_{\underline{a}^{\prime}}(\underline{\theta},\underline{\theta}^{\prime},s)}}\|_{L^{2}}^{2}\leq
c_{15}\,\rho^{-\frac{d+d^{\prime}}{2k}}\bigg{\\}}.$

For $\underline{c}\in\Sigma(\rho^{1/2})$,
$\underline{c}^{\prime}\in\Sigma^{\prime}(\rho^{1/2})$, we define
$E(\underline{c},\underline{c}^{\prime})$ as the set of $s\in J_{R}$ such that
there exists $\underline{\theta}$, $\underline{\theta}^{\prime}$ ending
respectively with $\underline{c}$, $\underline{c}^{\prime}$ such that $s\in
E(\underline{\theta},\underline{\theta}^{\prime})$.

One has, for any $\underline{\theta}\in\Sigma^{-}$,
$\underline{\theta}^{\prime}\in\Sigma^{\prime-}$:

$Leb(J_{R}\setminus E(\underline{\theta},\underline{\theta}^{\prime}))\leq
c_{15}^{-1}(c_{12}(R)+c_{14}^{2}\,c_{12}^{\prime}(R));$

therefore, provided that

$c_{15}>c_{1}^{-1}(c_{12}(R)+c_{14}^{2}\,c_{12}^{\prime}(R)),$

we will have

$Leb(J_{R}\setminus E(\underline{c},\underline{c}^{\prime}))\leq c_{1}$

for all $\underline{c}\in\Sigma(\rho^{1/2})$,
$\underline{c}^{\prime}\in\Sigma^{\prime}(\rho^{1/2})$. This means that we can
apply the Scale recurrence Lemma with the family
$E(\underline{c},\underline{c}^{\prime})$ of subsets of $J_{R}$. The sets
$E^{*}(\underline{c},\underline{c}^{\prime})$ are then defined using this
lemma (see section 2.7), we can assume they are closed, this is justified in
the remark after the lemma.

###  4.3. Construction of $L^{0}_{\underline{\omega}}$

We now consider the family of random perturbations $g^{\underline{\omega}}$
again and proceed to construct the sets
$L_{\underline{\omega}}^{0}(\underline{\theta},\underline{\theta}^{\prime},s)$,
for
$(\underline{\theta},\underline{\theta}^{\prime},s)\in\tilde{\mathcal{L}}$.
For $\underline{a}\in\Sigma(\rho^{1/2k})$, let $\Sigma^{-}(\underline{a})$ be
the open and closed subset of $\Sigma^{-}$ formed by the $\underline{\theta}$
ending with $\underline{a}$. Choose a subset $\Sigma_{2}^{-}$ of
$\Sigma(\rho^{1/2k})$ such that

$\Sigma^{-}=\bigcup_{\Sigma_{2}^{-}}\Sigma^{-}(\underline{a})$

is a partition of $\Sigma^{-}$.

For $\underline{a}\in\Sigma_{2}^{-}$, define a subset
$\Sigma_{1}(\underline{a})$ of the subset $\Sigma_{1}$ (recall
$\Sigma_{1}\subset\Sigma(\rho^{1/k})$), as the set of words in $\Sigma_{1}$
starting with $\underline{a}$. For
$\underline{\theta}\in\Sigma^{-}(\underline{a})$, we also define
$\Sigma_{1}(\underline{\theta})=\Sigma_{1}(\underline{a})$.

Let $\underline{\theta}\in\Sigma^{-}$. We write

$\displaystyle\Omega$
$\displaystyle=[-1,+1]^{\Sigma_{1}(\underline{\theta})}\times[-1,+1]^{\Sigma_{1}\setminus\Sigma_{1}(\underline{\theta})},$
$\displaystyle\underline{\omega}$
$\displaystyle=(\underline{\omega}^{\prime},\underline{\omega}^{\prime\prime})$
and for such an $\underline{\omega}$, we set
$\displaystyle\underline{\omega}^{*}$
$\displaystyle=(\underline{0},\underline{\omega}^{\prime\prime}).$

This depends on $\underline{\theta}$, but nearby
$\underline{\widehat{\theta}}$ (with
$d(\underline{\theta},\underline{\widehat{\theta}})<c_{0}^{-1}\,\rho^{1/2k}$)
will belong to the same $\Sigma^{-}(\underline{a})$ and give the same
projection $\underline{\omega}^{*}$ of $\underline{\omega}$.

For
$(\underline{\theta},\underline{\theta}^{\prime},s)\in\tilde{\mathcal{L}}$,
the set
$L_{\underline{\omega}}^{0}(\underline{\theta},\underline{\theta}^{\prime},s)$
will actually only depend (as far as $\underline{\omega}$ is concerned) on the
projection $\underline{\omega}^{*}$ of $\underline{\omega}$ associated to
$\underline{\theta}$.

We will say that two words
$\underline{b}^{0},\underline{b}^{1}\in\Sigma(\rho)$ are independent if there
is no word $\underline{b}\in\Sigma(\rho^{1/2k})$ such that both
$\underline{b}^{0}$ and $\underline{b}^{1}$ start with $\underline{b}$.

With $c_{16}>0$ conveniently small, to be chosen in the following, let

$N=\bigg{[}c_{16}^{2}\,\rho^{-\frac{1}{2k}(d+d^{\prime}-2)}\bigg{]}.$

Let $(\underline{\theta},\underline{\theta}^{\prime},s)\in\tilde{\mathcal{L}}$
and $\underline{\omega}\in\Omega$.

We define
$L_{\underline{\omega}}^{0}(\underline{\theta},\underline{\theta}^{\prime},s)$
to be the set of points $(\underline{\theta},\underline{\theta}^{\prime},s,t)$
in the fiber for which there exist pairs
$(\underline{b}^{1},\underline{b}^{\prime
1}),\dots,(\underline{b}^{N},\underline{b}^{\prime N})$ in
$\Sigma(\rho)\times\Sigma^{\prime}(\rho)$, with $b_{0}^{i}=\theta_{0}$ ,
$b_{0}^{\prime i}=\theta_{0}^{\prime}$ such that, if we set

$T_{\underline{b}^{i}}^{\underline{\omega}^{*}}\,T^{\prime}_{\underline{b}^{\prime
i}}(\underline{\theta},\underline{\theta}^{\prime},s,t)=(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i},t_{i}),$

the following hold:

  * (i)

the words $\underline{b}^{1},\dots,\underline{b}^{N}$ are pairwise
independent;

  * (ii)

for $1\leq i\leq N$, $\underline{\theta}^{i}\in\Sigma_{nr}^{-}$ ;

  * (iii)

for $1\leq i\leq N$, and $|\tilde{s}-s_{i}|\leq\dfrac{2}{3}c_{4}\rho^{1/2}$,
$(\underline{\theta}^{i},\underline{\theta}^{\prime
i},\tilde{s})\in\tilde{\mathcal{L}}$;

  * (iv)

for $1\leq i\leq N$, $|t_{i}|\leq 2(1+e^{R})$.

We will use also a slightly smaller set
$L_{\underline{\omega}}^{-1}(\underline{\theta},\underline{\theta}^{\prime},s)$;
it is defined in the same way than
$L_{\underline{\omega}}^{0}(\underline{\theta},\underline{\theta}^{\prime},s)$,
but with (iii), (iv) replaced by:

  * (iii)’

for $1\leq i\leq N$, and $|\tilde{s}-s_{i}|\leq\dfrac{3}{4}c_{4}\rho^{1/2}$,
$(\underline{\theta}^{i},\underline{\theta}^{\prime
i},\tilde{s})\in\tilde{\mathcal{L}}$

  * (iv)’

for $1\leq i\leq N$, $|t_{i}|\leq 1+e^{R}$.

In the next section, we will prove the following estimate.

######  Proposition 4.2.

If $c_{16}$ has been chosen sufficiently small, there exists $c_{17}>0$ such
that, for any
$(\underline{\theta},\underline{\theta}^{\prime},s)\in\tilde{\mathcal{L}}$ and
any $\underline{\omega}\in\Omega$, the Lebesgue measure of
$L_{\underline{\omega}}^{-1}(\underline{\theta},\underline{\theta}^{\prime},s)$
is $>c_{17}$ .

##  5\. Proof of proposition 4.2

In this section we will prove proposition 4.2. First we prove some lemmas that
are necessary for the proposition. We follow the same argument as [1] sections
4.8-4.12 with some modifications.

Fix
$(\underline{\theta},\underline{\theta}^{\prime},s)\in\tilde{\mathcal{L}}$, we
will work with this triple throughout this section and at the end we will
prove that
$Leb(L^{-1}_{\underline{\omega}}(\underline{\theta},\underline{\theta}^{\prime},s))>c_{17}$.

Choose a subfamily $\Sigma_{2}$ of $\Sigma(\rho^{1/2k})$ of words starting
with $\theta_{0}$ such that

$K(\theta_{0})=\bigcup_{\Sigma_{2}}K(\underline{a})$

is a partition of $K(\theta_{0})$. Similarly, choose a subfamily
$\Sigma_{2}^{\prime}$ of $\Sigma^{\prime}(\rho^{1/2k})$ of words starting with
$\theta_{0}^{\prime}$ such that

$K^{\prime}(\theta_{0}^{\prime})=\bigcup_{\Sigma_{2}^{\prime}}K^{\prime}(\underline{a}^{\prime}).$

There is a constant $c_{19}>0$ such that, for each
$(\underline{a},\underline{a}^{\prime})\in\Sigma_{2}\times\Sigma_{2}^{\prime}$,
we have

$c_{19}^{-1}\,\rho^{\frac{1}{2k}(d+d^{\prime})}\leq\mu_{d}\times\mu_{d^{\prime}}(K(\underline{a})\times
K^{\prime}(\underline{a}^{\prime}))\leq
c_{19}\,\rho^{\frac{1}{2k}(d+d^{\prime})}\,.$

Let
$J(\underline{a},\underline{a}^{\prime}):=\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(G(\underline{a})\times
G(\underline{a}^{\prime}))$ and
$c(\underline{a},\underline{a}^{\prime})\coloneqq\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(c_{\underline{a}},c_{\underline{a}^{\prime}})\in\mathbbm{C}$
for $\underline{a}\in\Sigma_{2}$ and
$\underline{a}^{\prime}\in\Sigma^{\prime}_{2}$. Since $s$ is bounded above and
below, we have

(11)
$B(c(\underline{a},\underline{a}^{\prime}),c_{19}^{-1}\,\rho^{\frac{1}{2k}})\subset
J(\underline{a},\underline{a}^{\prime})\subset
B(c(\underline{a},\underline{a}^{\prime}),c_{19}\,\rho^{\frac{1}{2k}})\,,$

if $c_{19}$ is sufficiently large. We assume that the previous relations
involving $c_{19}$ hold for any other triples in
$\Sigma^{-}\times\Sigma^{\prime-}\times J_{R}$ and for any value of
$\underline{\omega}$, choosing $c_{19}$ large enough this can be easily
guaranteed.

Say $(\underline{a},\underline{a}^{\prime})$ is good if there are no more than
$c_{16}^{-1}\,\rho^{-1/2k(d+d^{\prime}-2)}$ pairs
$(\underline{\tilde{a}},\underline{\tilde{a}}^{\prime})$ such that the
distance between the points $c(\underline{a},\underline{a}^{\prime})$ and
$c(\underline{\tilde{a}},\underline{\tilde{a}}^{\prime})$ is less than
$(2+1/10)c_{19}\,\rho^{1/2k}$. Otherwise, say it is _bad_.

######  Lemma 5.1.

The number of bad pairs $(\underline{a},\underline{a}^{\prime})$ is less than
$2^{2}\cdot
21^{2}\cdot\pi\,c_{19}^{4}\,c_{15}\,c_{16}\,\rho^{-1/2k(d+d^{\prime})}$.

Before begining the proof, we remember the Vitali covering lemma. Let
$B_{1},\,B_{2},\,\dots,\,B_{n}\subset\mathbbm{R}^{d}$ be a finite collection
of balls. For $i=1,\,\dots,n$, denote by $3B_{i}$ the ball with same center as
$B_{i}$ but having radius three times larger. The lemma states that there
exists a subcollection of balls $B_{j_{1}},\,B_{j_{2}},\,\dots,\,B_{j_{k}}$
with the Vitali property, this is

  * •

The balls $B_{j_{1}},\,B_{j_{2}},\,\dots,\,B_{j_{k}}$ are pairwise disjoint
and

  * •

The union $B_{1}\cup B_{2}\cup\dots\cup B_{n}$ is contained in $3B_{j_{1}}\cup
3B_{j_{2}}\cup\dots\cup 3B_{j_{k}}$.

In the case that the balls $B_{i}$ are subsets of the complex plane and have
the same radius $R$, one can see that every point $z\in\mathbbm{C}$ is covered
by no more than $16$ of the balls
$3B_{j_{1}},\,3B_{j_{2}},\,\dots,\,3B_{j_{k}}$. Indeed, consider the ball $B$
centered at $z$ with radius $4R$. It contains all the balls
$B_{i_{1}},\,\dots,\,B_{i_{l}}$ such that $3B_{i_{1}},\,\dots,\,3B_{i_{l}}$
cover $z$. However, the balls $B_{i_{1}},\,\dots,\,B_{i_{l}}$ are pairwise
disjoint, and so there are no more than $(4R)^{2}/R^{2}=16$ of them inside
$B$, otherwise they would overlap.

###### Proof.

By construction of $\widetilde{\mathcal{L}}$, there exists
$\widetilde{\underline{\theta}}$, $\widetilde{\underline{\theta}}^{\prime}$,
$\tilde{s}$ with $d(\underline{\theta},\widetilde{\underline{\theta}})\leq
c_{0}\rho^{1/2}$,
$d(\underline{\theta}^{\prime},\widetilde{\underline{\theta}}^{\prime})\leq
c_{0}\rho^{1/2}$, $|s-\tilde{s}|\leq c_{2}\,\rho^{1/2}$ such that

$\left\|\chi_{\widetilde{\underline{\theta}},\widetilde{\underline{\theta}}^{\prime},\tilde{s}}\right\|_{L^{2}}^{2}\leq
c_{15}\,.$

Because of remark 2.6, the distance between the points
$k^{\underline{\theta}}(c_{\underline{a}})$ and
$k^{\tilde{\underline{\theta}}}(c_{\underline{a}})$ is of order $\rho^{1/2}$
for every
$(\underline{a},\underline{a}^{\prime})\in\Sigma_{2}\times\Sigma^{\prime}_{2}$
and the same is true for their $K^{\prime}$ versions. Thus, if $c_{19}$ is
sufficiently large, for each bad pair $(\underline{a},\underline{a}^{\prime})$
there are more than $c_{16}^{-1}\,\rho^{-1/2k(d+d^{\prime}-2)}$ pairs
$(\tilde{\underline{a}},\underline{\tilde{a}}^{\prime})$ satisfying

$|\pi_{\tilde{\underline{\theta}},{\tilde{\underline{\theta}}}^{\prime},\tilde{s}}(c_{\underline{a}},c_{\underline{a}^{\prime}})-\pi_{\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s}}(c_{\underline{\tilde{a}}},c_{{\underline{\tilde{a}}^{\prime}}})|\leq\frac{5}{2}c_{19}\rho^{\frac{1}{2k}}.$

From now on, we denote
$\pi_{\tilde{\underline{\theta}},{\tilde{\underline{\theta}}}^{\prime},\tilde{s}}(c_{\underline{a}},c_{\underline{a}^{\prime}})$
as $\tilde{c}(\underline{a},\underline{a}^{\prime})$ for any
$(\underline{a},\underline{a}^{\prime})\in\Sigma_{2}\times\Sigma^{\prime}_{2}$.

For each bad pair $(\underline{a},\underline{a}^{\prime})$, consider the disk
$J^{*}(\underline{a},\underline{a}^{\prime})$ of radius
$\frac{7}{2}\,c_{19}\rho^{\frac{1}{2k}}$ and center at
$\tilde{c}(\underline{a},\underline{a}^{\prime})$. Then the corresponding
$c_{16}^{-1}\,\rho^{-1/2k(d+d^{\prime}-2)}$ sets
$\pi_{\tilde{\underline{\theta}},{\tilde{\underline{\theta}}}^{\prime},\tilde{s}}(G(\tilde{\underline{a}})\times
G(\tilde{\underline{a}}^{\prime}))$ are subsets of
$J^{*}(\underline{a},\underline{a}^{\prime})$. This way,

$\displaystyle\int_{J^{*}(\underline{a},\underline{a}^{\prime})}{\chi_{\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s}}ds}=(\mu_{d}\times\mu_{d^{\prime}})\left(\pi^{-1}_{\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s}}(J^{*}(\underline{a},\underline{a}^{\prime}))\right)\geq\sum_{(\tilde{\underline{a}},\tilde{\underline{a}}^{\prime})}{(\mu_{d}\times\mu_{d^{\prime}})(G(\tilde{\underline{a}})\times
G(\tilde{\underline{a}}^{\prime}))}$ $\displaystyle\geq
c_{16}^{-1}\,\rho^{-1/2k(d+d^{\prime}-2)}\cdot
c_{19}^{-1}\rho^{\frac{d+d^{\prime}}{2k}}=(c_{16}c_{19})^{-1}\rho^{1/k}.$

Let $J^{*}$ be the union of all the disks
$J^{*}(\underline{a},\underline{a}^{\prime})$ corresponding to bad pairs and
$B$ be the number of these pairs. Choose a subcover of $J^{*}$ as in the
Vitali lemma, indexed by the pairs $(\underline{a},\underline{a}^{\prime})$
belonging to a subset $V$ of the set of bad pairs. It follows that

$B\cdot
c_{19}^{-1}\rho^{\frac{(d+d^{\prime})}{2k}}\leq(\mu_{d}\times\mu_{d^{\prime}})(\pi^{-1}_{(\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s})}(J^{*}))=\int_{J^{*}}{\chi_{\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s}}ds}\leq\sum_{(\underline{a},\underline{a}^{\prime})\in
V}\int_{3\,J^{*}(\underline{a},\underline{a}^{\prime})}{\chi_{\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s}}ds}.$

On the other hand, by Cauchy-Schwartz theorem,

$\displaystyle\left(\int_{3\,J^{*}(\underline{a},\underline{a}^{\prime})}{\chi_{\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s}}ds}\right)\cdot(c_{16}c_{19})^{-1}\rho^{1/k}$
$\displaystyle\leq\left(\int_{3\,J^{*}(\underline{a},\underline{a}^{\prime})}{\chi_{\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s}}ds}\right)^{2}$
$\displaystyle\leq\text{Leb}(3\,J^{*}(\underline{a},\underline{a}^{\prime}))\cdot$
$\displaystyle\int_{3\,J^{*}(\underline{a},\underline{a}^{\prime})}{\chi^{2}_{\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s}}ds}=\frac{21^{2}}{2^{2}}\pi
c^{2}_{19}\rho^{1/k}\cdot\int_{3\,J^{*}(\underline{a},\underline{a}^{\prime})}{\chi^{2}_{\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s}}ds},$

for every bad pair $(\underline{a},\underline{a}^{\prime})$. But the Vitali
covering covers each point $z\in\mathbbm{C}$ at most $16$ times, so

$\sum_{(\underline{a},\underline{a}^{\prime})\in
V}\int_{3\,J^{*}(\underline{a},\underline{a}^{\prime})}{\chi_{\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s}}ds}\leq\frac{21^{2}}{2^{2}}\pi\,c^{3}_{19}\,c_{16}\sum_{(\underline{a},\underline{a}^{\prime})\in
V}\int_{3\,J^{*}(\underline{a},\underline{a}^{\prime})}{\chi^{2}_{\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s}}ds}\leq
2^{2}\cdot
21^{2}\cdot\pi\,c^{3}_{19}\,c_{16}\int_{\mathbbm{C}}{\chi^{2}_{\tilde{\underline{\theta}},\tilde{\underline{\theta}}^{\prime},\tilde{s}}ds}.$

It follows that $B\leq 2^{2}\cdot
21^{2}\cdot\pi\,c_{19}^{4}\,c_{16}\,c_{15}\,\rho^{-(d+d^{\prime})/2k}$,
concluding the proof. ∎

Now, we construct the pairs $(\underline{b},\underline{b}^{\prime})$ amongst
which the pairs $(\underline{b}^{i},\underline{b}^{\prime i})$ of 4.2 must be
looked for. We make the following observation:

######  Lemma 5.2.

Let $\underline{\theta}\in\Sigma_{nr}^{-}$. The number of words
$\underline{c}\in\Sigma(\rho^{1/2})$ with $c_{0}=\theta_{0}$, such that
$\underline{\theta}\underline{c}\notin\Sigma_{nr}^{-}$ is $o(\rho^{-d/2})$ as
$\rho\to 0$, uniformly in $\underline{\theta}$.

Remember that
$(\underline{\theta},\underline{\theta}^{\prime},s)\in\widetilde{\mathcal{L}}$.
It follows from conclusion (ii) of the Scale recurrence Lemma (lemma 2.26) and
the last observation that we can find at least
$\frac{1}{2}c_{3}\,\rho^{-1/2(d+d^{\prime})}$ pairs
$(\underline{c}^{i},\underline{c}^{\prime
i})\in\Sigma(\rho^{1/2})\times\Sigma^{\prime}(\rho^{1/2})$ such that, writing
$T_{\underline{c}^{i}}T^{\prime}_{\underline{c}^{\prime
i}}(\underline{\theta},\underline{\theta}^{\prime},s)=(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i})$, we have:

  * •

$\underline{\theta}^{i}\in\Sigma_{nr}^{-}$ ;

  * •

$B(s_{i},c_{4}\rho^{1/2})\subset E^{*}(\underline{c}^{i},\underline{c}^{\prime
i})$.

As $(\underline{\theta}^{i},\underline{\theta}^{\prime i},s_{i})$ again
belongs to $\widetilde{\mathcal{L}}$, we can for each $i$ find at least
$\frac{1}{2}c_{3}\,\rho^{-1/2(d+d^{\prime})}$ pairs
$(\underline{d}^{ij},\underline{d}^{\prime
ij})\in\Sigma(\rho^{1/2})\times\Sigma^{\prime}(\rho^{1/2})$ (with the first
letter of $\underline{d}^{ij}$, $\underline{d}^{\prime ij}$ being the last one
of $\underline{c}^{i}$, $\underline{c}^{\prime i}$ respectively), such that
writing $T_{\underline{d}^{ij}}T^{\prime}_{\underline{d}^{\prime
ij}}(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i})=(\underline{\theta}^{ij},\underline{\theta}^{\prime ij},s_{ij})$, we
have

  * •

$\underline{\theta}^{ij}\in\Sigma_{nr}^{-}$ ;

  * •

$B(s_{ij},c_{4}\rho^{1/2})\subset
E^{*}(\underline{d}^{ij},\underline{d}^{\prime ij})$.

Concatenation of the $\underline{c}^{i}$, $\underline{c}^{\prime i}$ and
$\underline{d}^{ij},\underline{d}^{\prime ij}$ gives a family of words
$(\underline{b}^{ij},\underline{b}^{\prime ij})$ in
$\Sigma(\rho)\times\Sigma^{\prime}(\rho)$ with at least
$\frac{1}{4}c_{3}^{2}\,\rho^{-(d+d^{\prime})}$ elements.

We now consider the perturbed operators. In this case
$T^{\underline{\omega}}_{\underline{c}^{i}}T^{\prime}_{\underline{c}^{\prime
i}}(\underline{\theta},\underline{\theta}^{\prime},s)=(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\underline{\omega}))$ and by lemma 3.2 the distance between
$s_{i}(\underline{\omega})$ and $s_{i}$ is of order $c_{5}\rho^{1-1/2k}$.
Similarly one has

$T^{\underline{\omega}}_{\underline{d}^{ij}}T^{\prime}_{\underline{d}^{\prime
ij}}(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\underline{\omega}))=(\underline{\theta}^{ij},\underline{\theta}^{\prime
ij},s_{ij}(\underline{\omega}))$

and again the distance between $s_{ij}(\underline{\omega})$ and $s_{ij}$ is of
order $c_{5}\rho^{1-1/2k}$.

Now we fix
$(\underline{\widetilde{\theta}},\underline{\widetilde{\theta}}^{\prime},\tilde{s})$
such that $d(\underline{\theta},\underline{\widetilde{\theta}})\leq
c_{0}\rho^{1/2}$,
$d(\underline{\theta}^{\prime},\underline{\widetilde{\theta}}^{\prime})\leq
c_{0}\rho^{1/2}$, $|s-\tilde{s}|\leq c_{2}\,\rho^{1/2}$ and $\tilde{s}\in
E(\widetilde{\underline{\theta}},\widetilde{\underline{\theta}}^{\prime})$.

######  Lemma 5.3.

If $c_{16}$ has been chosen sufficiently small, there are at least
$\frac{1}{6}c_{3}\,c_{19}^{-2}\,\rho^{-\frac{d+d^{\prime}}{2k}}$ pairs
$(\underline{a},\underline{a}^{\prime})\in\Sigma_{2}\times\Sigma_{2}^{\prime}$
which are good and satisfy

$\left\|\chi_{T_{\underline{a}}T^{\prime}_{\underline{a}^{\prime}}(\widetilde{\underline{\theta}},\widetilde{\underline{\theta}}^{\prime},\tilde{s})}\right\|_{L^{2}}^{2}\leq
c_{16}^{-1}$

and such that at least
$\frac{1}{6}c_{3}c_{19}^{-1}\,\rho^{-(d+d^{\prime})(\frac{1}{2}-\frac{1}{2k})}$
pairs $(\underline{c}^{i},\underline{c}^{\prime i})$ start with
$(\underline{a},\underline{a}^{\prime})$.

###### Proof.

The proof is the same as in [1], bearing in mind the different but similar
bound in the number of bad pairs. ∎

We call the pairs verifying the properties of the previous lemma _excellent
pairs_. The following general lemma will be used later to estimate the measure
of the union of the perturbed version of the sets
$J(\underline{b}^{ij},\underline{b}^{\prime ij})$.

######  Lemma 5.4.

Let $J_{\alpha},J_{\alpha}^{\prime},K_{\alpha}$, $\alpha\in A$, be families of
sets in $\mathbbm{C}$ such that for some
$\lambda,\varepsilon,\nu,\sigma\in\mathbbm{R}^{+}$, and
$c_{\alpha},c_{\alpha}^{\prime}\in\mathbbm{C}$:

  * •

$B(c_{\alpha},\varepsilon)\subset J_{\alpha}\subset
B(c_{\alpha},\lambda\varepsilon)$, $K_{\alpha}\subset J_{\alpha}$,
$B(c_{\alpha}^{\prime},\varepsilon)\subset J_{\alpha}^{\prime}$.

  * •

$d(c_{\alpha},c_{\alpha}^{\prime})\leq\nu\varepsilon$,
$Leb(K_{\alpha})\geq\sigma^{-1}Leb(J_{\alpha})$.

Then

$Leb\left(\bigcup_{\alpha\in
A}J_{\alpha}^{\prime}\right)\geq\frac{1}{9(\lambda+\nu)^{2}}Leb\left(\bigcup_{\alpha\in
A}J_{\alpha}\right),$

and

$Leb\left(\bigcup_{\alpha\in
A}K_{\alpha}\right)\geq\frac{\sigma^{-1}}{9\lambda^{2}}Leb\left(\bigcup_{\alpha\in
A}J_{\alpha}\right).$

###### Proof.

Notice that $J_{\alpha}\subset
B(c_{\alpha}^{\prime},(\lambda+\nu)\varepsilon)$. Let $\tilde{A}$ be a subset
of $A$ such that the balls
$\\{B(c_{\alpha}^{\prime},(\lambda+\nu)\varepsilon)\\}_{\alpha\in\tilde{A}}$
have the Vitali property. Thus

$\displaystyle Leb\left(\bigcup_{\alpha\in A}J_{\alpha}\right)$
$\displaystyle\leq Leb\left(\bigcup_{\alpha\in
A}B(c_{\alpha}^{\prime},(\lambda+\nu)\varepsilon)\right)$ $\displaystyle\leq
Leb\left(\bigcup_{\alpha\in\tilde{A}}B(c_{\alpha}^{\prime},3(\lambda+\nu)\varepsilon)\right)$
$\displaystyle\leq[3(\lambda+\nu)]^{2}Leb\left(\bigcup_{\alpha\in\tilde{A}}B(c_{\alpha}^{\prime},\varepsilon)\right)$
$\displaystyle\leq 9(\lambda+\nu)^{2}Leb\left(\bigcup_{\alpha\in
A}B(c_{\alpha}^{\prime},\varepsilon)\right)\leq
9(\lambda+\nu)^{2}Leb\left(\bigcup_{\alpha\in A}J_{\alpha}^{\prime}\right),$

where in the passage from the second to the third line we use the fact that
the sets $B(c_{\alpha}^{\prime},\varepsilon)$, $\alpha\in\tilde{A}$, are
disjoint. This proves the first inequality. For the second, use again Vitali
to find a subset $A^{\prime}$ of $A$ such that the balls
$\\{B(c_{\alpha},\lambda\varepsilon)\\}_{\alpha\in A^{\prime}}$ have the
Vitali property. Then

$\displaystyle Leb\left(\bigcup_{\alpha\in A}J_{\alpha}\right)$
$\displaystyle\leq Leb\left(\bigcup_{\alpha\in
A^{\prime}}B(c_{\alpha},3\lambda\varepsilon)\right)$ $\displaystyle\leq
9\lambda^{2}\sum_{\alpha\in
A^{\prime}}Leb\left(B(c_{\alpha},\varepsilon)\right)$ $\displaystyle\leq
9\lambda^{2}\sigma\sum_{\alpha\in
A^{\prime}}Leb\left(K_{\alpha}\right)=9\lambda^{2}\sigma\cdot
Leb\left(\bigcup_{\alpha\in A^{\prime}}K_{\alpha}\right)\leq
9\lambda^{2}\sigma\cdot Leb\left(\bigcup_{\alpha\in A}K_{\alpha}\right).$

∎

######  Lemma 5.5.

Let $(\underline{a},\underline{a}^{\prime})$ be an excellent pair. Consider
all the pairs $(\underline{c}^{i},\underline{c}^{\prime i})$ described above
which begin with $(\underline{a},\underline{a}^{\prime})$, and for each pair
$(\underline{c}^{i},\underline{c}^{\prime i})$ consider the corresponding
pairs $(\underline{b}^{ij},\underline{b}^{\prime ij})$. Define the sets

$\displaystyle J^{\underline{\omega}}(\underline{b}^{ij},\underline{b}^{\prime
ij})$
$\displaystyle=\pi^{\underline{\omega}}_{\underline{\theta},\underline{\theta}^{\prime},s}(G^{\underline{\omega}}(\underline{b}^{ij})\times
G(\underline{b}^{\prime ij})),$ $\displaystyle
J_{2}^{\underline{\omega}}(\underline{a},\underline{a}^{\prime})$
$\displaystyle=\bigcup_{(\underline{c}^{i},\underline{c}^{\prime
i})}\bigcup_{(\underline{b}^{ij},\underline{b}^{\prime
ij})}J^{\underline{\omega}}(\underline{b}^{ij},\underline{b}^{\prime ij}).$

Then

$Leb\left(J_{2}^{\underline{\omega}}(\underline{a},\underline{a}^{\prime})\right)\gtrsim
c_{16}\rho^{1/k},$

and the constant can be chosen to be independent of $\underline{\omega}$.

###### Proof.

First we make the following observation:

(12)
$\pi_{\underline{\eta},\underline{\eta}^{\prime},w}\circ(f_{\underline{d}},f_{\underline{d}^{\prime}})=A\circ\pi_{T_{\underline{d}}T^{\prime}_{\underline{d}^{\prime}}(\underline{\eta},\underline{\eta}^{\prime},w)},$

where $A$ is an affine function with $|DA|\approx diam(G(\underline{d}))$, and
this holds for any
$(\underline{d},\underline{d}^{\prime})\in\Sigma(\alpha)\times\Sigma^{\prime}(\alpha)$,
for some $\alpha\in\mathbbm{R}^{+}$, and any
$(\underline{\eta},\underline{\eta}^{\prime},w)$.

For an excellent pair $(\underline{a},\underline{a}^{\prime})$ we consider the
associated pairs $(\underline{c}^{i},\underline{c}^{\prime i})$ and the sets

$\displaystyle J^{\underline{\omega}}(\underline{c}^{i},\underline{c}^{\prime
i})$
$\displaystyle=\pi^{\underline{\omega}}_{\underline{\theta},\underline{\theta}^{\prime},s}(G^{\underline{\omega}}(\underline{c}^{i})\times
G(\underline{c}^{\prime i})),$ $\displaystyle
J(\underline{c}^{i},\underline{c}^{\prime i})$
$\displaystyle=\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(G(\underline{c}^{i})\times
G(\underline{c}^{\prime i})),$
$\displaystyle\tilde{J}(\underline{c}^{i},\underline{c}^{\prime i})$
$\displaystyle=\pi_{\tilde{}\underline{\theta},\tilde{}\underline{\theta}^{\prime},\tilde{s}}(G(\underline{c}^{i})\times
G(\underline{c}^{\prime i})),$ $\displaystyle
J_{1}^{\underline{\omega}}(\underline{a},\underline{a}^{\prime})$
$\displaystyle=\bigcup_{(\underline{c}^{i},\underline{c}^{\prime
i})}J^{\underline{\omega}}(\underline{c}^{i},\underline{c}^{\prime i}).$

We will prove that the measure of
$J^{\underline{\omega}}_{1}(\underline{a},\underline{a}^{\prime})$ is at least
of the order $c_{16}\rho^{1/k}$. To do this, we use lemma 5.4 to see that the
measures of the sets $\bigcup
J^{\underline{\omega}}(\underline{c}^{i},\underline{c}^{\prime i})$, $\bigcup
J(\underline{c}^{i},\underline{c}^{\prime i})$ and
$\bigcup\tilde{J}(\underline{c}^{i},\underline{c}^{\prime i})$ are of the same
order. It is clear from equation (11) that the sets
$J^{\underline{\omega}}(\underline{c}^{i},\underline{c}^{\prime i})$,
$J(\underline{c}^{i},\underline{c}^{\prime i})$ and
$\tilde{J}(\underline{c}^{i},\underline{c}^{\prime i})$ are all contained in,
and contain, balls with radius of order $\rho^{1/2}$, and centered at the
points
$\pi^{\underline{\omega}}_{\underline{\theta},\underline{\theta}^{\prime},s}(c^{\underline{\omega}}_{\underline{c}^{i}},c_{\underline{c}^{\prime
i}})$,
$\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(c_{\underline{c}^{i}},c_{\underline{c}^{\prime
i}})$ and
$\pi_{\tilde{}\underline{\theta},\tilde{}\underline{\theta}^{\prime},\tilde{s}}(c_{\underline{c}^{i}},c_{\underline{c}^{\prime
i}})$ respectively. We remark that

$\displaystyle|\pi^{\underline{\omega}}_{\underline{\theta},\underline{\theta}^{\prime},s}(c^{\underline{\omega}}_{\underline{c}^{i}},c_{\underline{c}^{\prime
i}})-\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(c_{\underline{c}^{i}},c_{\underline{c}^{\prime
i}})|$
$\displaystyle=|s|\cdot|k^{\underline{\theta}}(c_{\underline{c}^{i}})-k^{\underline{\theta},\underline{\omega}}(c^{\underline{\omega}}_{\underline{c}^{i}})|$
$\displaystyle\leq|s|\cdot\left[|k^{\underline{\theta}}(c_{\underline{c}^{i}})-k^{\underline{\theta},\underline{\omega}}(c_{\underline{c}^{i}})|+|k^{\underline{\theta},\underline{\omega}}(c_{\underline{c}^{i}})-k^{\underline{\theta},\underline{\omega}}(c^{\underline{\omega}}_{\underline{c}^{i}})|\right]$
$\displaystyle\lesssim c_{5}\rho^{1-1/2k}\lesssim\rho^{1/2},$

we also have

$\displaystyle|\pi_{\tilde{}\underline{\theta},\tilde{}\underline{\theta}^{\prime},\tilde{s}}(c_{\underline{c}^{i}},c_{\underline{c}^{\prime
i}})-\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(c_{\underline{c}^{i}},c_{\underline{c}^{\prime
i}})|$
$\displaystyle\leq|k^{\tilde{}\underline{\theta}^{\prime}}(c_{\underline{c}^{\prime
i}})-k^{\underline{\theta}^{\prime}}(c_{\underline{c}^{\prime
i}})|+|s-\tilde{s}|\cdot|k^{\underline{\theta}}(c_{\underline{c}^{i}})|+|\tilde{s}|\cdot|k^{\underline{\theta}}(c_{\underline{c}^{i}})-k^{\tilde{}\underline{\theta}}(c_{\underline{c}^{i}})|$
$\displaystyle\lesssim\rho^{1/2},$

given that $d(\underline{\theta},\tilde{}\underline{\theta})\leq
c_{0}\rho^{1/2}$,
$d(\underline{\theta}^{\prime},\tilde{}\underline{\theta}^{\prime})\leq
c_{0}\rho^{1/2}$ and $|s-\tilde{s}|\leq c_{2}\rho^{1/2}$. All this allows us
to conclude that we can use lemma 5.4.

Now we can estimate the measure of
$J^{\underline{\omega}}_{1}(\underline{a},\underline{a}^{\prime})$. In the
following lines of equations we will be using: lemma 5.4 for the first three
lines, observation in equation (12) for the fifth and sixth line, equation
(10) in the seventh line, in the last line we use that
$(\underline{a},\underline{a}^{\prime})$ is an excellent pair and the fact
that $diam(G(\underline{c}^{i}/\underline{a}))$ is of order
$\rho^{\frac{1}{2}-\frac{1}{2k}}$.

$\displaystyle
Leb\left(J^{\underline{\omega}}_{1}(\underline{a},\underline{a}^{\prime})\right)$
$\displaystyle=Leb\left(\bigcup
J^{\underline{\omega}}(\underline{c}^{i},\underline{c}^{\prime i})\right)$
$\displaystyle\gtrsim Leb\left(\bigcup
J(\underline{c}^{i},\underline{c}^{\prime i})\right)$ $\displaystyle\gtrsim
Leb\left(\bigcup\tilde{J}(\underline{c}^{i},\underline{c}^{\prime i})\right)$
$\displaystyle=Leb\left(\pi_{\tilde{}\underline{\theta},\tilde{}\underline{\theta}^{\prime},\tilde{s}}\left(\bigcup
G(\underline{c}^{i})\times G(\underline{c}^{\prime i})\right)\right)$
$\displaystyle=Leb\left(A\circ\pi_{T_{\underline{a}}T^{\prime}_{\underline{a}^{\prime}}(\tilde{}\underline{\theta},\tilde{}\underline{\theta}^{\prime},\tilde{s})}\left(\bigcup
G(\underline{c}^{i}/\underline{a})\times G(\underline{c}^{\prime
i}/\underline{a}^{\prime})\right)\right)$ $\displaystyle\approx
diam(G(\underline{a}))^{2}\cdot
Leb\left(\pi_{T_{\underline{a}}T^{\prime}_{\underline{a}^{\prime}}(\tilde{}\underline{\theta},\tilde{}\underline{\theta}^{\prime},\tilde{s})}\left(\bigcup
G(\underline{c}^{i}/\underline{a})\times G(\underline{c}^{\prime
i}/\underline{a}^{\prime})\right)\right)$ $\displaystyle\gtrsim
diam(G(\underline{a}))^{2}\cdot\mu_{d}\times\mu_{d^{\prime}}\left(\bigcup
G(\underline{c}^{i}/\underline{a})\times G(\underline{c}^{\prime
i}/\underline{a}^{\prime})\right)^{2}\cdot\left\|\chi_{T_{\underline{a}}T^{\prime}_{\underline{a}^{\prime}}(\tilde{}\underline{\theta},\tilde{}\underline{\theta}^{\prime},\tilde{s})}\right\|_{L^{2}}^{-2}$
$\displaystyle\gtrsim
c_{16}\rho^{1/k}\cdot\left(\rho^{\left(\frac{1}{2}-\frac{1}{2k}\right)(d+d^{\prime})}\cdot\rho^{-(d+d^{\prime})(\frac{1}{2}-\frac{1}{2k})}\right)^{2}\approx
c_{16}\rho^{1/k}$

We now consider the sets

$\displaystyle J^{\underline{\omega}}(\underline{b}^{ij},\underline{b}^{\prime
ij})$
$\displaystyle=\pi^{\underline{\omega}}_{\underline{\theta},\underline{\theta}^{\prime},s}(G^{\underline{\omega}}(\underline{b}^{ij})\times
G(\underline{b}^{\prime ij})),$ $\displaystyle
J_{1}^{\underline{\omega}}(\underline{c}^{i},\underline{c}^{\prime i})$
$\displaystyle=\bigcup_{(\underline{b}^{ij},\underline{b}^{\prime
ij})}J^{\underline{\omega}}(\underline{b}^{ij},\underline{b}^{\prime ij}),$
$\displaystyle
J_{2}^{\underline{\omega}}(\underline{a},\underline{a}^{\prime})$
$\displaystyle=\bigcup_{(\underline{c}^{i},\underline{c}^{\prime
i})}J^{\underline{\omega}}_{1}(\underline{c}^{i},\underline{c}^{\prime i}).$

We will estimate the measure of
$J_{2}^{\underline{\omega}}(\underline{a},\underline{a}^{\prime})$. Notice
that $J^{\underline{\omega}}_{1}(\underline{c}^{i},\underline{c}^{\prime
i})\subset J^{\underline{\omega}}(\underline{c}^{i},\underline{c}^{\prime i})$
and if we are able to prove that
$Leb\left(J^{\underline{\omega}}_{1}(\underline{c}^{i},\underline{c}^{\prime
i})\right)\geq\sigma^{-1}\cdot
Leb\left(J^{\underline{\omega}}(\underline{c}^{i},\underline{c}^{\prime
i})\right)$, for some constant $\sigma$, then we can use lemma 5.4 with
$K_{\alpha}$ being
$J^{\underline{\omega}}_{1}(\underline{c}^{i},\underline{c}^{\prime i})$ and
$J_{\alpha}$ being
$J^{\underline{\omega}}(\underline{c}^{i},\underline{c}^{\prime i})$ to
conclude that

$Leb\left(J_{2}^{\underline{\omega}}(\underline{a},\underline{a}^{\prime})\right)=Leb\left(\bigcup
J^{\underline{\omega}}_{1}(\underline{c}^{i},\underline{c}^{\prime
i})\right)\gtrsim Leb\left(\bigcup
J^{\underline{\omega}}(\underline{c}^{i},\underline{c}^{\prime
i})\right)\gtrsim c_{16}\rho^{1/k}.$

To prove that there is such $\sigma$ we will proceed similarly to what we did
when estimating
$Leb\left(J^{\underline{\omega}}_{1}(\underline{a},\underline{a}^{\prime})\right)$.
Note that $T_{\underline{c}^{i}}T^{\prime}_{\underline{c}^{\prime
i}}(\underline{\theta},\underline{\theta}^{\prime},s)=(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i})\in\tilde{\mathcal{L}}$ and then there exists
$(\tilde{\underline{\theta}}^{i},\tilde{\underline{\theta}}^{\prime
i},\tilde{s}_{i})$ such that $|s_{i}-\tilde{s}_{i}|\leq c_{2}\rho^{1/2}$,
$d(\tilde{\underline{\theta}}^{i},\underline{\theta}^{i})\leq
c_{0}\rho^{1/2}$, $d(\tilde{\underline{\theta}}^{\prime
i},\underline{\theta}^{\prime i})\leq c_{0}\rho^{1/2}$ and

(13)
$\left\|\chi_{\tilde{\underline{\theta}}^{i},\tilde{\underline{\theta}}^{\prime
i},\tilde{s}_{i}}\right\|_{L^{2}}^{2}\leq c_{15}.$

The sets

$\pi^{\underline{\omega}}_{\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\underline{\omega})}\left(G^{\underline{\omega}}(\underline{d}^{ij})\times
G(\underline{d}^{\prime
ij})\right),\pi_{\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}}\left(G(\underline{d}^{ij})\times G(\underline{d}^{\prime
ij})\right)\text{ and
}\pi_{\tilde{\underline{\theta}}^{i},\tilde{\underline{\theta}}^{\prime
i},\tilde{s}_{i}}\left(G(\underline{d}^{ij})\times G(\underline{d}^{\prime
ij})\right)$

are all contained in, and contain, balls with radius of order $\rho^{1/2}$,
and centered at the points

$\pi^{\underline{\omega}}_{\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\underline{\omega})}(c_{\underline{d}^{ij}}^{\underline{\omega}},c_{\underline{d}^{\prime
ij}}),\pi_{\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}}(c_{\underline{d}^{ij}},c_{\underline{d}^{\prime ij}})\text{ and
}\pi_{\tilde{\underline{\theta}}^{i},\tilde{\underline{\theta}}^{\prime
i},\tilde{s}_{i}}(c_{\underline{d}^{ij}},c_{\underline{d}^{\prime ij}})$

respectively. By lemma 3.2 we know that

$\|k^{\underline{\theta}^{i},\underline{\omega}}-k^{\underline{\theta}^{i}}\|_{C^{0}}\lesssim
c_{5}\rho^{1-1/2k},\,\,|c_{\underline{d}^{ij}}^{\underline{\omega}}-c_{\underline{d}^{ij}}|\lesssim
c_{5}\rho^{1+1/2k},$

on the other hand we also have
$|s_{i}-s_{i}(\underline{\omega})|\lesssim\rho^{1/2}$,
$\|k^{\underline{\theta}^{i}}-k^{\tilde{\underline{\theta}}^{i}}\|_{C^{0}}\lesssim\rho^{1/2}$
and $\|k^{\underline{\theta}^{\prime i}}-k^{\tilde{\underline{\theta}}^{\prime
i}}\|_{C^{0}}\lesssim\rho^{1/2}$, thus we can conclude that the distance
between any two centers is of order less than $\rho^{1/2}$. Therefore, we can
apply lemma 5.4 taking $J_{\alpha}$ as one of the families

$\pi^{\underline{\omega}}_{\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\underline{\omega})}\left(G^{\underline{\omega}}(\underline{d}^{ij})\times
G(\underline{d}^{\prime
ij})\right),\,\,\pi_{\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}}\left(G(\underline{d}^{ij})\times G(\underline{d}^{\prime
ij})\right),\,\,\pi_{\tilde{\underline{\theta}}^{i},\tilde{\underline{\theta}}^{\prime
i},\tilde{s}_{i}}\left(G(\underline{d}^{ij})\times G(\underline{d}^{\prime
ij})\right)$

and $J_{\alpha}^{\prime}$ as another of these families.

Using the previous analysis together with equations (12), (10), (13) and the
fact that the number of $(\underline{d}^{ij},\underline{d}^{\prime ij})$ is a
positive proportion of $\Sigma(\rho^{1/2})\times\Sigma^{\prime}(\rho^{1/2})$
we obtain

$\displaystyle
Leb\left(J^{\underline{\omega}}_{1}(\underline{c}^{i},\underline{c}^{\prime
i})\right)$ $\displaystyle=Leb\left(\bigcup
J^{\underline{\omega}}(\underline{b}^{ij},\underline{b}^{\prime ij})\right)$
$\displaystyle=Leb\left(\pi^{\underline{\omega}}_{\underline{\theta},\underline{\theta}^{\prime},s}\left(\bigcup
G^{\underline{\omega}}(\underline{b}^{ij})\times G(\underline{b}^{\prime
ij})\right)\right)$
$\displaystyle=Leb\left(A\circ\pi^{\underline{\omega}}_{T^{\underline{\omega}}_{\underline{c}^{i}}T^{\prime}_{\underline{c}^{\prime
i}}(\underline{\theta},\underline{\theta}^{\prime},s)}\left(\bigcup
G^{\underline{\omega}}(\underline{d}^{ij})\times G(\underline{d}^{\prime
ij})\right)\right)$ $\displaystyle\approx
Leb\left(J(\underline{c}^{i},\underline{c}^{\prime i})\right)\cdot
Leb\left(\bigcup\pi^{\underline{\omega}}_{\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\underline{\omega})}\left(G^{\underline{\omega}}(\underline{d}^{ij})\times
G(\underline{d}^{\prime ij})\right)\right)$ $\displaystyle\approx
Leb\left(J(\underline{c}^{i},\underline{c}^{\prime i})\right)\cdot
Leb\left(\bigcup\pi_{\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}}\left(G(\underline{d}^{ij})\times G(\underline{d}^{\prime
ij})\right)\right)$ $\displaystyle\approx
Leb\left(J(\underline{c}^{i},\underline{c}^{\prime i})\right)\cdot
Leb\left(\bigcup\pi_{\tilde{\underline{\theta}}^{i},\tilde{\underline{\theta}}^{\prime
i},\tilde{s}_{i}}\left(G(\underline{d}^{ij})\times G(\underline{d}^{\prime
ij})\right)\right)$ $\displaystyle\gtrsim
Leb\left(J(\underline{c}^{i},\underline{c}^{\prime
i})\right)\cdot\mu_{d}\times\mu_{d^{\prime}}\left(\bigcup
G(\underline{d}^{ij})\times G(\underline{d}^{\prime
ij})\right)^{2}\cdot\left\|\chi_{\tilde{\underline{\theta}}^{i},\tilde{\underline{\theta}}^{\prime
i},\tilde{s}_{i}}\right\|_{L^{2}}^{-2}$ $\displaystyle\gtrsim
Leb\left(J(\underline{c}^{i},\underline{c}^{\prime
i})\right)\cdot\left(\rho^{\frac{1}{2}(d+d^{\prime})}\cdot\rho^{-\frac{1}{2}(d+d^{\prime})}\right)^{2}\approx
Leb\left(J(\underline{c}^{i},\underline{c}^{\prime i})\right).$

This guarantees the existence of the desired constant $\sigma$ and finishes
the proof of the lemma. ∎

Now we can prove proposition 4.2. Consider the function

$\varphi^{\underline{\omega}}_{2}=\sum_{(\underline{a},\underline{a}^{\prime})}1_{J^{\underline{\omega}}_{2}(\underline{a},\underline{a}^{\prime})},$

where $1_{B}$ means indicator function of the set $B$ and the sum is over all
excellent pairs. We want to estimate the measure of the set

$X^{\underline{\omega}}=\\{t\in\mathbbm{C}:\varphi^{\underline{\omega}}_{2}(t)\geq
c^{\prime\prime}c_{16}^{2}\rho^{-\frac{1}{2k}(d+d^{\prime}-2)}\\},$

where $c^{\prime\prime}$ is a constant defined in the following way. Suppose
that we have two excellent pairs with the same first coordinate
$(\underline{a},\underline{a}^{\prime})$,
$(\underline{a},\tilde{\underline{a}}^{\prime})$ and such that
$J^{\underline{\omega}}(\underline{a},\underline{a}^{\prime})\cap
J^{\underline{\omega}}(\underline{a},\tilde{\underline{a}}^{\prime})\neq\emptyset$.
Then

$k^{\underline{\theta}^{\prime}}(y)-sk^{\underline{\theta},\underline{\omega}}(x)=k^{\underline{\theta}^{\prime}}(\tilde{y})-sk^{\underline{\theta},\underline{\omega}}(\tilde{x}),$

for some $(x,y)\in G^{\underline{\omega}}(\underline{a})\times
G(\underline{a}^{\prime})$, $(\tilde{x},\tilde{y})\in
G^{\underline{\omega}}(\underline{a})\times
G(\tilde{\underline{a}}^{\prime})$. Thus

$|y-\tilde{y}|\approx|x-\tilde{x}|\lesssim\rho^{1/2k},$

which shows that

$d(G(\underline{a}^{\prime}),G(\tilde{\underline{a}}^{\prime}))\lesssim\rho^{1/2k}.$

This implies that if we fix $(\underline{a},\underline{a}^{\prime})$, then the
number of possible pairs $(\underline{a},\tilde{\underline{a}}^{\prime})$ such
that $J^{\underline{\omega}}(\underline{a},\underline{a}^{\prime})\cap
J^{\underline{\omega}}(\underline{a},\tilde{\underline{a}}^{\prime})\neq\emptyset$
is bounded by a uniform constant, independent of $\rho$ and
$(\underline{a},\underline{a}^{\prime})$, we denote this constant by
$c^{\prime\prime}$ (this last statement is a consequence of lemma 1.2.3 in
[8]).

Notice that since $s\in J_{R}$ then $\varphi^{\underline{\omega}}_{2}$ is
supported in a ball of radius proportional to $1+e^{R}$ centered at $0$, thus
there is a constant $c$ such that

$\int\varphi^{\underline{\omega}}_{2}dt\leq(\sup\varphi^{\underline{\omega}}_{2})\cdot
Leb(X^{\underline{\omega}})+cc^{\prime\prime}c_{16}^{2}\rho^{-\frac{1}{2k}(d+d^{\prime}-2)}.$

Now we estimate $\sup\varphi^{\underline{\omega}}_{2}$ from above and
$\int\varphi^{\underline{\omega}}_{2}dt$ from below. By lemmas 5.5 and 5.3
there is a constant $c^{\prime}$ such that

(14) $\int\varphi^{\underline{\omega}}_{2}dt\geq
c^{\prime}c_{16}\rho^{1/k}\cdot\rho^{-\frac{d+d^{\prime}}{2k}}=c^{\prime}c_{16}\rho^{-\frac{1}{2k}(d+d^{\prime}-2)}.$

Let $x\in\mathbbm{C}$ and excellent pairs
$(\underline{a},\underline{a}^{\prime})$,
$(\tilde{\underline{a}},\tilde{\underline{a}}^{\prime})$ such that $x\in
J^{\underline{\omega}}(\underline{a},\underline{a}^{\prime})\cap
J^{\underline{\omega}}(\tilde{\underline{a}},\tilde{\underline{a}}^{\prime})$.
Remember that

$|\pi^{\underline{\omega}}_{\underline{\theta},\underline{\theta}^{\prime},s}(c^{\underline{\omega}}_{\underline{a}},c_{\underline{a}^{\prime}})-\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(c_{\underline{a}},c_{\underline{a}^{\prime}})|\lesssim
c_{5}\rho^{1-1/2k}=o(\rho^{1/2k}),$

where the $o$ notation means that, once we have chosen $c_{5}$, we can choose
any $\varepsilon>0$ and $c_{5}\rho^{1-1/2k}\leq\varepsilon\rho^{1/2k}$ will
hold provided $\rho$ is small enough. With this in mind we obtain

$\displaystyle|\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(c_{\underline{a}},c_{\underline{a}^{\prime}})-\pi_{\underline{\theta},\underline{\theta}^{\prime},s}(c_{\tilde{\underline{a}}},c_{\tilde{\underline{a}}^{\prime}})|$
$\displaystyle\leq
o(\rho^{1/2k})+|\pi^{\underline{\omega}}_{\underline{\theta},\underline{\theta}^{\prime},s}(c^{\underline{\omega}}_{\underline{a}},c_{\underline{a}^{\prime}})-x|+|x-\pi^{\underline{\omega}}_{\underline{\theta},\underline{\theta}^{\prime},s}(c^{\underline{\omega}}_{\tilde{\underline{a}}},c_{\tilde{\underline{a}}^{\prime}})|$
$\displaystyle\leq o(\rho^{1/2k})+2c_{19}\rho^{1/2k}.$

Given that $(\underline{a},\underline{a}^{\prime})$,
$(\tilde{\underline{a}},\tilde{\underline{a}}^{\prime})$ are excellent, we
conclude that there can be no more than
$c_{16}^{-1}\rho^{-\frac{1}{2k}(d+d^{\prime}-2)}$ excellent pairs intersecting
at $x$. Since
$\varphi^{\underline{\omega}}_{2}=\sum_{(\underline{a},\underline{a}^{\prime})}1_{J^{\underline{\omega}}(\underline{a},\underline{a}^{\prime})}$
we get that

$\varphi^{\underline{\omega}}_{2}(x)\leq
c_{16}^{-1}\rho^{-\frac{1}{2k}(d+d^{\prime}-2)}.$

We are ready to bound $Leb(X^{\underline{\omega}})$, using equation (14) and
the previous estimates

$c_{16}^{-1}\rho^{-\frac{1}{2k}(d+d^{\prime}-2)}Leb(X^{\underline{\omega}})+cc^{\prime\prime}c_{16}^{2}\rho^{-\frac{1}{2k}(d+d^{\prime}-2)}\geq
c^{\prime}c_{16}\rho^{-\frac{1}{2k}(d+d^{\prime}-2)},$

and from this we get

$Leb(X^{\underline{\omega}})\geq
c_{16}^{2}(c^{\prime}-cc^{\prime\prime}c_{16}).$

We fix $c_{16}$ small enough such that
$c_{17}:=c_{16}^{2}(c^{\prime}-cc^{\prime\prime}c_{16})>0$.

To finish the proof of the proposition we will show that
$\\{(\underline{\theta},\underline{\theta}^{\prime},s,t):t\in
X^{\underline{\omega}^{*}}\\}\subset
L_{\underline{\omega}}^{-1}(\underline{\theta},\underline{\theta}^{\prime},s)$.
Let $t\in X^{\underline{\omega}^{*}}$, then there are at least
$\left[c^{\prime\prime}c_{16}^{2}\rho^{-\frac{1}{2k}(d+d^{\prime}-2)}\right]$
excellent pairs $(\underline{a},\underline{a}^{\prime})$, each one with an
associated pair $(\underline{b}^{ij},\underline{b}^{\prime ij})$ which starts
with $(\underline{a},\underline{a}^{\prime})$ and such that $t\in
J^{\underline{\omega}^{*}}(\underline{b}^{ij},\underline{b}^{\prime ij})$. By
the definition of $c^{\prime\prime}$, we can extract from this family of
excellent pairs a subfamily $(\underline{a}^{l},\underline{a}^{\prime l})$,
$l=1,...,\left[c_{16}^{2}\rho^{-\frac{1}{2k}(d+d^{\prime}-2)}\right]$, such
that all firsts coordinates are different. For
$(\underline{a}^{l},\underline{a}^{\prime l})$ we denote the associated pair
$(\underline{b}^{ij},\underline{b}^{\prime ij})$ by
$(\underline{b}^{l},\underline{b}^{\prime l})$. We will prove the pairs
$(\underline{b}^{1},\underline{b}^{\prime
1}),...,(\underline{b}^{N},\underline{b}^{\prime N})$ have the properties
necessary to guarantee that
$(\underline{\theta},\underline{\theta}^{\prime},s,t)\in
L_{\underline{\omega}}^{-1}(\underline{\theta},\underline{\theta}^{\prime},s)$.
Write

$T^{\underline{\omega}^{*}}_{\underline{b}^{l}}T^{\prime}_{\underline{b}^{\prime
l}}(\underline{\theta},\underline{\theta}^{\prime},s,t)=(\underline{\theta}^{l},\underline{\theta}^{\prime
l},s_{l},t_{l})$

then:

  * (i)

Since all firsts coordinates of the excellent pairs are different we conclude
that $\underline{b}^{1},...,\underline{b}^{N}$ are pairwise independent.

  * (ii)

By the way in which $\underline{d}^{ij}$ was defined we get that all
$\underline{\theta}^{l}\in\Sigma^{-}_{nr}$.

  * (iii)’

By the scale recurrence lemma we know that $B(s_{ij},c_{4}\rho^{1/2})\subset
E^{*}(d^{ij},d^{\prime ij})$. We also know that
$|s_{ij}-s_{ij}(\underline{\omega}^{*})|\lesssim
c_{5}\rho^{1-1/2k}=o(\rho^{1/2})$, then, if $\rho$ is small enough, we have

$\\{\tilde{s}:|\tilde{s}-s_{ij}(\underline{\omega}^{*})|\leq\frac{3}{4}c_{4}\rho^{1/2}\\}\subset
E^{*}(d^{ij},d^{\prime ij}).$

We conclude that $(\underline{\theta}^{l},\underline{\theta}^{\prime
l},\tilde{s})\in\tilde{\mathcal{L}}$ if
$|\tilde{s}-s_{l}|\leq\frac{3}{4}c_{4}\rho^{1/2}$ (remember that for every $l$
there is $i,j$ such that $s_{l}=s_{ij}(\underline{\omega}^{*})$).

  * (iv)’

Given that $t\in
J^{\underline{\omega}^{*}}(\underline{b}^{l},\underline{b}^{\prime l})$, there
is $(x,y)\in\mathbbm{C}^{2}$ such that

$\displaystyle t$
$\displaystyle=k^{\underline{\theta}^{\prime}}(f_{\underline{b}^{\prime
l}}^{\underline{\omega}^{*}}(y))-sk^{\underline{\theta},\underline{\omega}^{*}}(f_{\underline{b}^{l}}^{\underline{\omega}^{*}}(x))$
$\displaystyle F^{\underline{\theta}^{\prime}}_{\underline{b}^{\prime
l}}(k^{\underline{\theta}^{\prime}\underline{b}^{\prime l}}(y))$
$\displaystyle=t+s\cdot
F^{\underline{\theta},\underline{\omega}^{*}}_{\underline{b}^{l}}(k^{\underline{\theta}\underline{b}^{l},\underline{\omega}^{*}}(x))$
$\displaystyle k^{\underline{\theta}^{\prime}\underline{b}^{\prime l}}(y)$
$\displaystyle=\left(F^{\underline{\theta}^{\prime}}_{\underline{b}^{\prime
l}}\right)^{-1}\left(t+s\cdot
F^{\underline{\theta},\underline{\omega}^{*}}_{\underline{b}^{l}}(k^{\underline{\theta}\underline{b}^{l},\underline{\omega}^{*}}(x))\right)$
$\displaystyle k^{\underline{\theta}^{\prime}\underline{b}^{\prime l}}(y)$
$\displaystyle=t_{l}+s_{l}\cdot
k^{\underline{\theta}\underline{b}^{l},\underline{\omega}^{*}}(x).$

Since $s_{l}\in J_{R}$ we conclude that $|t_{l}|\leq 1+e^{R}$.

##  6\. Proof of Proposition 4.1

In this section we will prove proposition 4.1. Given
$u=(\underline{\theta},\underline{\theta}^{\prime},s,t)\in\tilde{\mathcal{L}}\times\mathbbm{C}$,
remember the decomposition
$\underline{\omega}=(\underline{\omega}^{\prime},\underline{\omega}^{\prime\prime})$
where
$\underline{\omega}^{\prime}\in\mathbb{D}^{\Sigma_{1}(\underline{\theta})}$
and
$\underline{\omega}^{\prime\prime}\in\mathbb{D}^{\Sigma_{1}\setminus\Sigma_{1}(\underline{\theta})}$.
Recall that the set $\Sigma_{1}(\underline{\theta})$ is given by the words in
$\Sigma_{1}$ starting with the same word, in $\Sigma(\rho^{1/2k})$, in which
$\underline{\theta}$ finishes. In the same way as in [1], one uses Fubini’s
theorem to reduce the proof of proposition 4.1 to proving

(15) $\mathbb{P}^{\prime}(\Omega^{\prime}\setminus\Omega^{\prime
0}(u))\leq\exp(-c_{7}\rho^{-\frac{1}{2k}(d+d^{\prime}-2)}),$

where we have fixed $\underline{\omega}^{\prime\prime}$ such that
$u\in\mathcal{L}^{1}_{(\underline{0},\underline{\omega}^{\prime\prime})}$ and
$\Omega^{\prime}=\mathbb{D}^{\Sigma_{1}(\underline{\theta})}$, $\Omega^{\prime
0}(u)=\\{\underline{\omega}^{\prime}:(\underline{\omega}^{\prime},\underline{\omega}^{\prime\prime})\in\Omega^{0}(u)\\}$,
$\mathbb{P}^{\prime}$ is normalized Lebesgue measure in $\Omega^{\prime}$.

The fact that
$u=(\underline{\theta},\underline{\theta}^{\prime},s,t)\in\mathcal{L}^{1}_{(\underline{0},\underline{\omega}^{\prime\prime})}$
means that
$(\underline{\theta},\underline{\theta}^{\prime},s)\in\tilde{\mathcal{L}}$ and
there is
$(\tilde{}\underline{\theta},\tilde{\underline{\theta}}^{\prime},\tilde{s},\tilde{t})$
for which

$d(\underline{\theta},\tilde{}\underline{\theta})<2\rho^{5/2},\,d(\underline{\theta}^{\prime},\tilde{\underline{\theta}}^{\prime})<2\rho^{5/2},\,|s-\tilde{s}|<\rho,\,|t-\tilde{t}|<\rho,$

$(\tilde{}\underline{\theta},\tilde{\underline{\theta}}^{\prime},\tilde{s})\in\tilde{\mathcal{L}}$
and $\tilde{t}\in
L_{(\underline{0},\underline{\omega}^{\prime\prime})}^{0}(\tilde{}\underline{\theta},\tilde{\underline{\theta}}^{\prime},\tilde{s})$.
Notice that
$\Sigma_{1}(\underline{\theta})=\Sigma_{1}(\tilde{}\underline{\theta})$,
moreover
$u\in\mathcal{L}^{1}_{(\underline{0},\underline{\omega}^{\prime\prime})}$ if
and only if
$u\in\mathcal{L}^{1}_{(\underline{\omega}^{\prime},\underline{\omega}^{\prime\prime})}$
for any $\underline{\omega}^{\prime}\in\Sigma_{1}(\underline{\theta})$.

Next, $\tilde{t}\in
L_{(\underline{0},\underline{\omega}^{\prime\prime})}^{0}(\tilde{}\underline{\theta},\tilde{\underline{\theta}}^{\prime},\tilde{s})$
means that there are pairs $(\underline{b}^{1},\underline{b}^{\prime
1}),...,\,(\underline{b}^{N},\underline{b}^{\prime N})$ in
$\Sigma(\rho)\times\Sigma^{\prime}(\rho)$ such that if we set

$T^{(\underline{0},\underline{\omega}^{\prime\prime})}_{\underline{b}^{i}}T^{\prime}_{\underline{b}^{\prime
i}}(\tilde{}\underline{\theta},\tilde{\underline{\theta}}^{\prime},\tilde{s},\tilde{t})=(\tilde{\underline{\theta}}^{i},\tilde{\underline{\theta}}^{\prime
i},\tilde{s}_{i},\tilde{t}_{i})$

then:

  * (i)

the words $\underline{b}^{1},...,\,\underline{b}^{N}$ are pairwise
independent;

  * (ii)

$\tilde{\underline{\theta}}^{i}\in\Sigma^{-}_{nr}$;

  * (iii)

$(\tilde{\underline{\theta}}^{i},\tilde{\underline{\theta}}^{\prime
i},\tilde{s}_{i}^{\prime})\in\tilde{\mathcal{L}}$ if
$|\tilde{s}_{i}-\tilde{s}_{i}^{\prime}|<\frac{2}{3}c_{4}\rho^{1/2}$;

  * (iv)

$|\tilde{t}_{i}|\leq 2(1+e^{R})$.

Let $\underline{a}$ be the word in $\Sigma(\rho^{1/2k})$ in which
$\underline{\theta}$ ends, for each $\underline{b}^{i}$ define
$\underline{a}^{i}$ in $\Sigma_{1}(\underline{\theta})$ given by the
concatenation of $\underline{a}$ and a word at the beginning of
$\underline{b}^{i}$, in such a way that
$\underline{a}^{i}\in\Sigma_{1}(\underline{\theta})$ (this can be done since
$\underline{\theta}\underline{b}^{i}\in\Sigma_{nr}^{-}$). Notice that the
independence of the words $\underline{b}^{i}$ imply that the words
$\underline{a}^{i}$ are all different.

Now we consider the decomposition of
$\underline{\omega}^{\prime}\in\Omega^{\prime}$ as
$\underline{\omega}^{\prime}=(\omega_{1},...,\omega_{N},\tilde{\underline{\omega}}^{\prime})$,
where
$\tilde{\underline{\omega}}^{\prime}\in\mathbb{D}^{\Sigma_{1}(\underline{\theta})\setminus\\{a_{1},...,a_{N}\\}}$
and $\omega_{i}$ is the component of $\underline{\omega}^{\prime}$
corresponding to $\underline{a}^{i}$. We use again Fubini’s theorem to reduce
the proof of equation (15) to a similar formula in a smaller space. For
$\tilde{\underline{\omega}}^{\prime}$ fixed, we will prove that the set of
$(\omega_{1},..,\omega_{N})$ such that
$\underline{\omega}^{\prime}\notin\Omega^{\prime 0}(u)$ has measure
$\leq\exp(-c_{7}\rho^{-\frac{1}{2k}(d+d^{\prime}-2)})$.

To prove the desired inequality, we will prove that for each $\omega_{i}$
there is a set with positive measure such that whenever $\omega_{i}$ is in
this set we have $\underline{\omega}^{\prime}\in\Omega^{\prime 0}(u)$ (no
matter the value of $\omega_{j}$, $j\neq i$). More precisely, we will prove
that if $\omega_{i}$ is in this set then $\underline{b}^{i}$,
$\underline{b}^{\prime i}$ verify that if we set

$T_{\underline{b}^{i}}^{\underline{\omega}}T^{\prime}_{\underline{b}^{\prime
i}}(u)=(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\underline{\omega}),t_{i}(\underline{\omega}))$

then:

  * (i)

$(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}^{\prime})\in\tilde{\mathcal{L}}$ if
$|s_{i}^{\prime}-s_{i}(\underline{\omega})|<\frac{1}{2}c_{4}\rho^{1/2}$;

  * (ii)

$t_{i}(\underline{\omega})\in
L^{0}_{\underline{\omega}}(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\underline{\omega}))$.

The first property can be easily obtained. In fact, we already know (from
(iii) above) that
$(\tilde{\underline{\theta}}^{i},\tilde{\underline{\theta}}^{\prime
i},\tilde{s}_{i}^{\prime})\in\tilde{\mathcal{L}}$ if
$|\tilde{s}_{i}-\tilde{s}_{i}^{\prime}|<\frac{2}{3}c_{4}\rho^{1/2}$. Notice
that since
$d(\underline{\theta}^{i},\tilde{\underline{\theta}}^{i})\lesssim\rho^{7/2}$,
$d(\underline{\theta}^{\prime i},\tilde{\underline{\theta}}^{\prime
i})\lesssim\rho^{7/2}$ then $\tilde{\underline{\theta}}^{i}\in\Sigma^{-}_{nr}$
and the fiber of $\tilde{\mathcal{L}}$ over
$(\underline{\theta}^{i},\underline{\theta}^{\prime i})$ is the same as the
one over $(\tilde{\underline{\theta}}^{i},\tilde{\underline{\theta}}^{\prime
i})$, thus we only need to estimate
$|\tilde{s}_{i}-s_{i}(\underline{\omega})|$. Using that $|s-\tilde{s}|<\rho$,
$d(\underline{\theta},\tilde{\underline{\theta}})<2\rho^{5/2}$,
$d(\underline{\theta}^{\prime},\tilde{\underline{\theta}}^{\prime})<2\rho^{5/2}$
and lemma 3.2 one gets that
$|\tilde{s}_{i}-s_{i}(\underline{\omega})|=o(\rho^{1/2})$, then choosing
$\rho$ sufficiently small gives the desired property (for any value of
$\omega_{i}$).

For the second property we choose $\overline{\underline{\theta}}\in\Sigma^{-}$
such that $d(\overline{\underline{\theta}},\underline{\theta})<\rho^{5/2}$
(then $\overline{\underline{\theta}}$ ends with $\underline{a}$) and such that
it does not contain $\underline{a}$ anywhere else (this is possible since
$\underline{\theta}\in\Sigma^{-}_{nr}$). Set

$T^{\underline{\omega}}_{\underline{b}^{i}}T^{\prime}_{\underline{b}^{\prime
i}}(\overline{\underline{\theta}},\underline{\theta}^{\prime},s,t)=(\overline{\underline{\theta}}^{i},\underline{\theta}^{\prime
i},\overline{s}_{i}(\underline{\omega}),\overline{t}_{i}(\underline{\omega})).$

We will prove the following lemmas:

######  Lemma 6.1.

Once $\underline{\omega}^{\prime\prime}$ has been fixed, the number
$\overline{t}_{i}(\underline{\omega})$ only depends on $\omega_{i}$, not in
$\tilde{\underline{\omega}}^{\prime}$ or $\omega_{j}$ for $j\neq i$. Moreover,
if $c_{5}$ is big enough then there is a constant $c_{7}^{\prime}>0$ such that

$Leb\left(\\{\omega_{i}:\,\overline{t}_{i}(\underline{\omega})\in
L^{-1}_{\hat{\underline{\omega}}}(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\hat{\underline{\omega}}))\\}\right)\geq c_{7}^{\prime},$

where
$\hat{\underline{\omega}}=(\underline{0},\tilde{\underline{\omega}}^{\prime},\underline{\omega}^{\prime\prime})$.

######  Lemma 6.2.

If $\overline{t}_{i}(\omega_{i})\in
L^{-1}_{\hat{\underline{\omega}}}(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\hat{\underline{\omega}}))$ then $t_{i}(\underline{\omega})\in
L^{0}_{\underline{\omega}}(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\underline{\omega}))$.

These two lemmas imply that

$Leb(\\{(\omega_{1},...,\omega_{N}):\,\,(\omega_{1},...,\omega_{N},\tilde{}\underline{\omega}^{\prime},\underline{\omega}^{\prime\prime})\notin\Omega^{0}(u)\\})\leq(1-c_{7}^{\prime})^{N}=e^{N\log(1-c_{7}^{\prime})},$

this finishes the proof of proposition 4.1.

###### Proof of lemma 6.1:.

From equation (3) we know that

$\overline{t}_{i}(\underline{\omega})=\left(DF^{\underline{\theta}^{\prime}}_{\underline{b}^{\prime
i}}\right)^{-1}\cdot(t+sc^{\overline{\underline{\theta}},\underline{\omega}}_{\underline{b}^{i}}-c^{\underline{\theta}^{\prime}}_{\underline{b}^{\prime
i}}).$

The dependency on $\underline{\omega}$ is on the term
$c^{\overline{\underline{\theta}},\underline{\omega}}_{\underline{b}^{i}}$.
Let $\hat{\underline{\theta}}$ such that
$\overline{\underline{\theta}}=\hat{\underline{\theta}}\underline{a}$, note
that $\hat{\underline{\theta}}$ does not contain the word $\underline{a}$, let
$b$ be the last letter of $\underline{b}^{i}$, we have

(16)
$c^{\overline{\underline{\theta}},\underline{\omega}}_{\underline{b}^{i}}=k^{\overline{\underline{\theta}},\underline{\omega}}(c^{\underline{\omega}}_{\underline{b}^{i}})=k^{\hat{\underline{\theta}}\underline{a},\underline{\omega}}(f^{\underline{\omega}}_{\underline{b}^{i}}(c^{\underline{\omega}}_{b}))=\left(F^{\hat{\underline{\theta}},\underline{\omega}}_{\underline{a}}\right)^{-1}\circ
k^{\hat{\underline{\theta}},\underline{\omega}}(f^{\underline{\omega}}_{\underline{a}\underline{b}^{i}}(c^{\underline{\omega}}_{b})).$

We will prove that, assuming $\underline{\omega}^{\prime\prime}$ is fixed,
this expression only depends on $\omega_{i}$ and not in $\omega_{j}$ for
$j\neq i$ or $\tilde{\underline{\omega}}^{\prime}$, and it depends in a very
specific way. Remember, see remark 3.3, that the base points
$c^{\underline{\omega}}_{a}$ were chosen to be pre-periodic points and that in
fact they do not depend on $\underline{\omega}$, i.e.
$c^{\underline{\omega}}_{a}=c_{a}$. Notice that
$\overline{\underline{\theta}}^{i}=\overline{\underline{\theta}}\underline{b}^{i}$
and
$d(\overline{\underline{\theta}}^{i},\underline{\theta}^{i})\lesssim\rho^{7/2}$,
then $\overline{\underline{\theta}}^{i}\in\Sigma^{-}_{nr}$ and $\underline{a}$
appears only once in $\overline{\underline{\theta}}^{i}$. We will study the
dependency on $\omega_{i}$ for the different terms in equation (16):

  * •

Let $\underline{\alpha}$ be a finite word at the end of
$\overline{\underline{\theta}}^{i}$ strictly shorter than
$\underline{a}\underline{b}^{i}$. It is easy to prove by induction that
$f^{\underline{\omega}}_{\underline{\alpha}}(c^{\omega}_{b})$ does not depend
on $\underline{\omega}^{\prime}$. Indeed, suppose that $\underline{\alpha}$
and $\underline{\beta}=(c,d)\underline{\alpha}$ are two such consecutive
words. Assume $f^{\underline{\omega}}_{\underline{\alpha}}(c^{\omega}_{b})$
does not depend on $\underline{\omega}^{\prime}$, we have

$f^{\underline{\omega}}_{\underline{\beta}}(c^{\omega}_{b})=f^{\underline{\omega}}_{(c,d)}\left(f^{\underline{\omega}}_{\underline{\alpha}}(c^{\omega}_{b})\right).$

If the word $\underline{\beta}$ is shorter than some word in
$\Sigma(\rho^{1/2k})$, then
$f^{\underline{\omega}}_{\underline{\alpha}}(c^{\omega}_{b})\in
G^{\underline{\omega}}(\underline{\gamma})\subset
V_{\rho}(G(\underline{\gamma}))$ such that
$(c,d)\underline{\gamma}\in\Sigma(\rho^{1/k})$ and $\underline{\gamma}$
contains a word in $\Sigma(\rho^{1/3k})$ repeated (remember that
$c^{\omega}_{b}$ is pre-periodic, with a uniform bounded period). Thus
$(c,d)\underline{\gamma}$ cannot belong to $\Sigma_{1}$ and using equation (4)
one gets that $f^{\underline{\omega}}_{\underline{\beta}}(c^{\omega}_{b})$
does not depend on $\underline{\omega}^{\prime}$ (in fact, in this case it
does not depend on $\underline{\omega}$).

If the word $\underline{\beta}$ is longer than any word in
$\Sigma(\rho^{1/2k})$ then it begins with a word in $\Sigma(\rho^{1/2k})$
which can not be $\underline{a}$. This implies that
$f^{\underline{\omega}}_{\underline{\alpha}}(c^{\omega}_{b})\in
G^{\underline{\omega}}(\underline{\gamma})\subset
V_{\rho}(G(\underline{\gamma}))$ such that
$(c,d)\underline{\gamma}\in\Sigma(\rho^{1/k})$, and the word
$(c,d)\underline{\gamma}$ is not in $\Sigma_{1}(\underline{\theta})$. Thus
$f^{\underline{\omega}}_{\underline{\beta}}(c^{\omega}_{b})$ does not depend
on $\underline{\omega}^{\prime}$.

  * •

Let $(a_{0},a_{1})$ be the first two letters of $\underline{a}$ and define
$\underline{\alpha}_{0}$ by
$\underline{a}\underline{b}^{i}=(a_{0},a_{1})\underline{\alpha}_{0}$. Define
$x_{0}=f_{a_{0},a_{1}}(f^{\underline{\omega}}_{\underline{\alpha}_{0}}(c^{\underline{\omega}}_{b}))$,
we already proved that $x_{0}$ does not depend on
$\underline{\omega}^{\prime}$. We have

$f^{\underline{\omega}}_{\underline{a}\underline{b}^{i}}(c^{\underline{\omega}}_{b})=f^{\underline{\omega}}_{a_{0},a_{1}}(f^{\underline{\omega}}_{\underline{\alpha}_{0}}(c^{\underline{\omega}}_{b}))=x_{0}+c_{5}\rho^{1+1/2k}\omega_{i}.$

  * •

We now study $k^{\hat{\underline{\theta}},\underline{\omega}}$. By definition

$k^{\hat{\underline{\theta}},\underline{\omega}}(z)=\lim_{n\to\infty}Df^{\underline{\omega}}_{\hat{\underline{\theta}}_{n}}(c_{\hat{\theta}_{0}})^{-1}(f^{\underline{\omega}}_{\hat{\underline{\theta}}_{n}}(z)-f^{\underline{\omega}}_{\hat{\underline{\theta}}_{n}}(c_{\hat{\theta}_{0}})).$

Using the same arguments as before we see that, since
$\hat{\underline{\theta}}$ does not contain $\underline{a}$ and
$c_{\hat{\theta}_{0}}$ is pre-periodic,
$f^{\underline{\omega}}_{\hat{\underline{\theta}}_{n}}(c_{\hat{\theta}_{0}})$
does not depend on $\underline{\omega}^{\prime}$. This also happens for $z$ in
a neighborhood of $c_{\hat{\theta}_{0}}$, hence
$Df^{\underline{\omega}}_{\hat{\underline{\theta}}_{n}}(c_{\hat{\theta}_{0}})$
is independent of $\underline{\omega}^{\prime}$. Again, the same arguments
prove that if $z\in G^{\underline{\omega}}(\underline{a})$ then
$f^{\underline{\omega}}_{\hat{\underline{\theta}}_{n}}(z)=f^{(\underline{0},\underline{\omega}^{\prime\prime})}_{\hat{\underline{\theta}}_{n}}(z)$.
We conclude that

$k^{\hat{\underline{\theta}},\underline{\omega}}(z)=k^{\hat{\underline{\theta}},(\underline{0},\underline{\omega}^{\prime\prime})}(z),\text{
and
}Dk^{\hat{\underline{\theta}},\underline{\omega}}(z)=Dk^{\hat{\underline{\theta}},(\underline{0},\underline{\omega}^{\prime\prime})}(z),$

for all $z\in G^{\underline{\omega}}(\underline{a})$.

  * •

We now treat
$F^{\hat{\underline{\theta}},\underline{\omega}}_{\underline{a}}$. One has
that

$F^{\hat{\underline{\theta}},\underline{\omega}}_{\underline{a}}(0)=c^{\hat{\underline{\theta}},\underline{\omega}}_{\underline{a}}=k^{\hat{\underline{\theta}},\underline{\omega}}(c^{\underline{\omega}}_{\underline{a}}),\,\,DF^{\hat{\underline{\theta}},\underline{\omega}}_{\underline{a}}=Dk^{\hat{\underline{\theta}},\underline{\omega}}(c^{\underline{\omega}}_{\underline{a}})\cdot
Df^{\underline{\omega}}_{\underline{a}}(c^{\underline{\omega}}_{a}),$

where $a$ is the last letter in $\underline{a}$. Given that
$\underline{a}\in\Sigma(\rho^{1/2k})$ and $c^{\underline{\omega}}_{a}=c_{a}$
is pre-periodic we conclude that
$c^{\underline{\omega}}_{\underline{a}}=f^{\underline{\omega}}_{\underline{a}}(c^{\underline{\omega}}_{a})$
and $Df^{\underline{\omega}}_{\underline{a}}(c^{\underline{\omega}}_{a})$ are
independent of $\underline{\omega}$. On the other hand, given that
$c^{\underline{\omega}}_{\underline{a}}\in
G^{\underline{\omega}}(\underline{a})$ and $\hat{\underline{\theta}}$ does not
contain $\underline{a}$ we obtain that
$k^{\hat{\underline{\theta}},\underline{\omega}}(c^{\underline{\omega}}_{\underline{a}})$
does not depend on $\underline{\omega}^{\prime}$. We conclude that
$F^{\hat{\underline{\theta}},\underline{\omega}}_{\underline{a}}$ is
independent of $\underline{\omega}^{\prime}$.

From the previous analysis we get that $\overline{t}_{i}(\underline{\omega})$
only depends on $\omega_{i}$ and not in $\omega_{j}$ for $j\neq i$ or
$\underline{\omega}^{\prime}$. Moreover, we have

$c^{\overline{\underline{\theta}},\underline{\omega}}_{\underline{b}^{i}}=\left(F^{\hat{\underline{\theta}},(\underline{0},\underline{\omega}^{\prime\prime})}_{\underline{a}}\right)^{-1}\circ
k^{\hat{\underline{\theta}},(\underline{0},\underline{\omega}^{\prime\prime})}(x_{0}+c_{5}\rho^{1+1/2k}\omega_{i}),$

taking derivative respect to $\omega_{i}$ we get

$D_{\omega_{i}}c^{\overline{\underline{\theta}},\underline{\omega}}_{\underline{b}^{i}}=\left(DF^{\hat{\underline{\theta}},(\underline{0},\underline{\omega}^{\prime\prime})}_{\underline{a}}\right)^{-1}\cdot
Dk^{\hat{\underline{\theta}},(\underline{0},\underline{\omega}^{\prime\prime})}(x_{0}+c_{5}\rho^{1+1/2k}\omega_{i})\cdot
c_{5}\rho^{1+1/2k}.$

Observe that since
$x_{0}+c_{5}\rho^{1+1/2k}\omega_{i}=f^{\underline{\omega}}_{\underline{a}\underline{b}^{i}}(c^{\underline{\omega}}_{b})$
belongs to the Cantor set $K^{\underline{\omega}}$ then the matrix
$Dk^{\hat{\underline{\theta}},(\underline{0},\underline{\omega}^{\prime\prime})}(x_{0}+c_{5}\rho^{1+1/2k}\omega_{i})$
is conformal. Moreover

$D_{\omega_{i}}\overline{t}_{i}(\underline{\omega})=\left(DF^{\underline{\theta}^{\prime}}_{\underline{b}^{\prime
i}}\right)^{-1}\cdot
sD_{\omega_{i}}c^{\overline{\underline{\theta}},\underline{\omega}}_{\underline{b}^{i}},$

therefore $\omega_{i}\to\overline{t}_{i}(\underline{\omega})$ defines a
holomorphic function, which we will denote by $\overline{t}_{i}(\omega_{i})$.
From the previous formulas it is not difficult to see that
$\|D_{\omega_{i}}\overline{t}_{i}(\omega_{i})\|\approx c_{5}$.

Now we show that
$\overline{t}_{i}(0)=\overline{t}_{i}(\underline{0},\underline{\omega}^{\prime\prime})$
is uniformly bounded. We already know that $|\tilde{t}_{i}|\leq 2(1+e^{R})$,
using this inequality together with
$d(\overline{\underline{\theta}},\tilde{\underline{\theta}})<3\rho^{5/2}$,
$d(\underline{\theta}^{\prime},\tilde{\underline{\theta}}^{\prime})<2\rho^{5/2}$,
$|t-\tilde{t}|<\rho$, $|s-\tilde{s}|<\rho$ and equation (3) one gets that
$|\overline{t}_{i}(0)|\lesssim 1$. Therefore, choosing $c_{5}$ big enough one
guarantees that the image of $\overline{t}_{i}(\omega_{i})$ contains
$L^{-1}_{\hat{\underline{\omega}}}(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\hat{\underline{\omega}}))$ (see lemma 6.3). Having chosen $c_{5}$
this way, the fact that $\|D_{\omega_{i}}\overline{t}_{i}(\omega_{i})\|\approx
c_{5}$ and proposition 4.2 gives that there is a constant $c_{7}^{\prime}$
such that

$Leb\left(\\{\omega_{i}:\,\overline{t}_{i}(\underline{\omega})\in
L^{-1}_{\hat{\underline{\omega}}}(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\hat{\underline{\omega}}))\\}\right)\geq c_{7}^{\prime}.$

∎

In the previous proof we used the following lemma, it is proven using standard
arguments in analysis, for completeness we present the proof.

######  Lemma 6.3.

Let $f:B(0,1)\subset\mathbbm{R}^{n}\to\mathbbm{R}^{n}$ be $C^{1}$ on $B(0,1)$
and such that $m(Df(x))>c_{5}$ for all $x\in B(0,1)$. Given $r$, if $c_{5}$ is
big enough, depending on $r$, then $B(0,r)\subset f(B(0,1))$.

###### Proof.

Redefining $f$ as $f-f(0)$ and taking $r$ as $r+f(0)$, one can assume
$f(0)=0$. Since $f$ is $C^{1}$ and $m(Df(x))>c_{5}>0$ then $f$ is a $C^{1}$
local diffeomorphism, hence its image $f(B(0,1))$ is open. Let $h$ be a point
in $\mathbbm{R}^{n}\setminus f(B(0,1/2))$ which is closest to $0$, thus
$\lambda h\in f(B(0,1/2))$ for all $0\leq\lambda<1$. We can cover
$\overline{B}(0,1/2)$ by a finite number of open sets such that in each one of
these sets $f$ is a diffeomorphism onto its image. We can use this cover to
lift curves in $\overline{f(B(0,1/2))}$ (In fact, one can prove that
$f|_{B(0,1/2)}:B(0,1/2)\to f(B(0,1/2))$ is a covering map). Consider
$\alpha:[0,1]\to\overline{f(B(0,1/2))}$ given by $\alpha(t)=th$. Denote by
$\beta$ a lifting of $\alpha$, i.e. $f\circ\beta=\alpha$, such that
$\beta(0)=0$. The curve $\beta$ is $C^{1}$ and $\beta(1)\notin B(0,1/2)$,
otherwise this would contradict the choice of $h$. Therefore

$|h|=length(\alpha)=\int_{0}^{1}\left|\frac{d}{dt}(f\circ\beta)\right|dt\geq\int_{0}^{1}m(Df(\beta(t)))\left|\frac{d}{dt}\beta\right|dt\geq
c_{5}\int_{0}^{1}\left|\frac{d}{dt}\beta\right|dt\geq c_{5}/2.$

If $c_{5}>2r$ then $|h|>r$ and the desired result follows. ∎

###### Proof of lemma 6.2:.

Let $\overline{t}_{i}(\omega_{i})\in
L^{-1}_{\hat{\underline{\omega}}}(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\hat{\underline{\omega}}))$, then there exists pairs
$(\underline{d}^{j},\underline{d}^{\prime j})$, $1\leq j\leq N$, such that if
we write

$T^{\hat{\underline{\omega}}(i)}_{\underline{d}^{j}}T^{\prime}_{\underline{d}^{\prime
j}}(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\hat{\underline{\omega}}),\overline{t}_{i}(\omega_{i}))=(\underline{\theta}^{i}\underline{d}^{j},\underline{\theta}^{\prime
i}\underline{d}^{\prime j},\hat{s}_{(j)},\hat{t}_{(j)}),$

where $\hat{\underline{\omega}}(i)$ is obtained from
$\hat{\underline{\omega}}$ by setting the value $0$ in the coordinates
belonging to $\Sigma_{1}(\underline{\theta}^{i})$, then

  * (i)

$\underline{d}^{1}$,…,$\underline{d}^{N}$ are pairwise disjoint.

  * (ii)

$\underline{\theta}^{i}\underline{d}^{j}\in\Sigma^{-}_{nr}$.

  * (iii)’

$|s^{*}-\hat{s}_{(j)}|<\frac{3}{4}c_{4}\rho^{1/2}$ implies
$(\underline{\theta}^{i}\underline{d}^{j},\underline{\theta}^{\prime
i}\underline{d}^{\prime j},s^{*})\in\tilde{\mathcal{L}}$.

  * (iv)’

$|\hat{t}_{(j)}|<1+e^{R}$.

We will prove that $t_{i}(\underline{\omega})\in
L^{0}_{\underline{\omega}}(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\underline{\omega}))$. To do this, we will prove that if write

$T^{\underline{\omega}(i)}_{\underline{d}^{j}}T^{\prime}_{\underline{d}^{\prime
j}}(\underline{\theta}^{i},\underline{\theta}^{\prime
i},s_{i}(\underline{\omega}),t_{i}(\underline{\omega}))=(\underline{\theta}^{i}\underline{d}^{j},\underline{\theta}^{\prime
i}\underline{d}^{\prime j},s_{(j)},t_{(j)}),$

where $\underline{\omega}(i)$ is obtained from $\underline{\omega}$ by setting
the value $0$ in the coordinates belonging to
$\Sigma_{1}(\underline{\theta}^{i})$, then

  * (iii)

$|s^{*}-s_{(j)}|<\frac{2}{3}c_{4}\rho^{1/2}$ implies
$(\underline{\theta}^{i}\underline{d}^{j},\underline{\theta}^{\prime
i}\underline{d}^{\prime j},s^{*})\in\tilde{\mathcal{L}}$.

  * (iv)

$|t_{(j)}|<2(1+e^{R})$.

First, notice that by lemma 3.2 we have that
$|s_{i}(\underline{\omega})-s_{i}(\hat{\underline{\omega}})|\lesssim\rho^{1-1/2k}$.
To obtain $s_{(j)}$ and $\hat{s}_{(j)}$ we applied the same renormalizations,
with the same limit geometries but with different values of the perturbation
parameter, therefore
$|s_{(j)}-\hat{s}_{(j)}|\lesssim\rho^{1-1/2k}=o(\rho^{1/2})$. We conclude
that, taking $\rho$ small enough, item (iii)’ implies item (iii).

Now, to prove that (iv)’ implies (iv) we need stronger estimates. We have

$\displaystyle\hat{t}_{(j)}$
$\displaystyle=\left(DF^{\underline{\theta}^{\prime i}}_{\underline{d}^{\prime
j}}\right)^{-1}\cdot(\bar{t}_{i}(\omega_{i})+s_{i}(\hat{\underline{\omega}})c^{\underline{\theta}^{i},\hat{\underline{\omega}}(i)}_{\underline{d}^{j}}-c^{\underline{\theta}^{\prime
i}}_{\underline{d}^{\prime j}}),$ $\displaystyle t_{(j)}$
$\displaystyle=\left(DF^{\underline{\theta}^{\prime i}}_{\underline{d}^{\prime
j}}\right)^{-1}\cdot(t_{i}(\underline{\omega})+s_{i}(\underline{\omega})c^{\underline{\theta}^{i},\underline{\omega}(i)}_{\underline{d}^{j}}-c^{\underline{\theta}^{\prime
i}}_{\underline{d}^{\prime j}}).$

We will compare the corresponding terms:

  * •

Using that $d(\underline{\theta},\overline{\underline{\theta}})<\rho^{5/2}$,
one has

$|\overline{t}_{i}(\omega_{i})-t_{i}(\underline{\omega})|=\left|\left(DF^{\underline{\theta}^{\prime}}_{\underline{b}^{\prime
i}}\right)^{-1}\cdot
s\left[c^{\overline{\underline{\theta}},\underline{\omega}}_{\underline{b}^{i}}-c^{\underline{\theta},\underline{\omega}}_{\underline{b}^{i}}\right]\right|\lesssim\rho^{3/2}.$

  * •

Notice that $\underline{\omega}(i)$ and $\hat{}\underline{\omega}(i)$ only
differ at their values in the coordinates
$(\underline{\omega}_{1},...,\underline{\omega}_{N})$. Using that
$\underline{\theta}^{i}\underline{d}^{j}\in\Sigma_{nr}^{-}$ one sees that
$\underline{a}$ is not contained in $\underline{d}^{j}$. Then, from the
arguments used in lemma 6.1, we see that
$c^{\underline{\omega}(i)}_{\underline{d}^{j}}=c^{\hat{}\underline{\omega}(i)}_{\underline{d}^{j}}$.
Moreover, since $\underline{\theta}^{i}$ ends in
$\underline{a}\underline{b}^{i}$ and
$\underline{\theta}^{i}\underline{d}^{j}\in\Sigma^{-}_{nr}$ then

$f^{\underline{\omega}(i)}_{\underline{\theta}^{i}_{n}}(c^{\underline{\omega}(i)}_{\underline{d}^{j}})=f^{\hat{}\underline{\omega}(i)}_{\underline{\theta}^{i}_{n}}(c^{\hat{}\underline{\omega}(i)}_{\underline{d}^{j}})$

for all $n$ such that $\underline{\theta}^{i}_{n}$ is strictly shorter that
$\underline{a}\underline{b}^{i}$. For the same reasons we also have

$f^{\underline{\omega}(i)}_{\underline{\theta}^{i}_{n}}(c_{\theta_{0}})=f^{\hat{}\underline{\omega}(i)}_{\underline{\theta}^{i}_{n}}(c_{\theta_{0}})$

for all $n$ such that $\underline{\theta}^{i}_{n}$ is strictly shorter than
$\underline{a}\underline{b}^{i}$. Both equalities still hold for $z$ in a
neighborhood of either of the points. Thus we can use remark 3.3 and lemma 3.2
to obtain that

$\displaystyle|c^{\underline{\theta}^{i},\underline{\omega}(i)}_{\underline{d}^{j}}-c^{\underline{\theta}^{i},\hat{}\underline{\omega}(i)}_{\underline{d}^{j}}|$
$\displaystyle=|k^{\underline{\theta}^{i},\underline{\omega}(i)}(c^{\underline{\omega}(i)}_{\underline{d}^{j}})-k^{\underline{\theta}^{i},\hat{}\underline{\omega}(i)}(c^{\hat{}\underline{\omega}(i)}_{\underline{d}^{j}})|$
$\displaystyle\lesssim
diam(G^{\underline{\omega}(i)}(\underline{a}\underline{b}^{i}))\lesssim
diam(G^{\underline{\omega}(i)}(\underline{a}))diam(G^{\underline{\omega}(i)}(\underline{b}^{i}))$
$\displaystyle\lesssim
diam(G(\underline{a}))diam(G(\underline{b}^{i}))\approx\rho^{1+1/2k}.$

  * •

Notice that $\underline{\omega}$ and $\hat{}\underline{\omega}$ only differ in
the values associated to the coordinates $(\omega_{1},...,\omega_{N})$. We
will use $\overline{\underline{\theta}}$ again. Consider

$T^{\underline{\omega}}_{\underline{b}^{i}}T^{\prime}_{\underline{b}^{\prime
i}}(\overline{}\underline{\theta},\underline{\theta}^{\prime},s)=(\overline{\underline{\theta}}\underline{b}^{i},\underline{\theta}^{\prime}\underline{b}^{\prime
i},\overline{s}_{i}(\underline{\omega}))$

and

$T^{\hat{}\underline{\omega}}_{\underline{b}^{i}}T^{\prime}_{\underline{b}^{\prime
i}}(\overline{}\underline{\theta},\underline{\theta}^{\prime},s)=(\overline{\underline{\theta}}\underline{b}^{i},\underline{\theta}^{\prime}\underline{b}^{\prime
i},\overline{s}_{i}(\hat{}\underline{\omega})).$

Since $d(\underline{\theta},\overline{}\underline{\theta})<\rho^{5/2}$ one
gets
$|s_{i}(\underline{\omega})-\overline{s}_{i}(\underline{\omega})|\lesssim\rho^{5/2}$
and
$|s_{i}(\hat{}\underline{\omega})-\overline{s}_{i}(\hat{}\underline{\omega})|\lesssim\rho^{5/2}$.
Thus we only need to estimate
$|\overline{s}_{i}(\underline{\omega})-\overline{s}_{i}(\hat{}\underline{\omega})|$.
Remember $\hat{\underline{\theta}}$ which verified
$\overline{}\underline{\theta}=\hat{}\underline{\theta}\underline{a}$, write

$k^{\overline{\underline{\theta}},\underline{\omega}}=(F^{\hat{}\underline{\theta},\underline{\omega}}_{\underline{a}})^{-1}\circ
k^{\hat{}\underline{\theta},\underline{\omega}}\circ
f^{\underline{\omega}}_{\underline{a}},$

then

$Dk^{\overline{\underline{\theta}},\underline{\omega}}=(DF^{\hat{}\underline{\theta},\underline{\omega}}_{\underline{a}})^{-1}\cdot(Dk^{\hat{}\underline{\theta},\underline{\omega}}\circ
f^{\underline{\omega}}_{\underline{a}})\cdot
Df^{\underline{\omega}}_{\underline{a}}.$

Using the analysis and notation from lemma 6.1 we see that:
$F^{\hat{}\underline{\theta},\underline{\omega}}_{\underline{a}}$ does not
depend on $\underline{\omega}^{\prime}$,
$Dk^{\hat{}\underline{\theta},\underline{\omega}}(z)=Dk^{\hat{}\underline{\theta},\hat{}\underline{\omega}}(z)$
for all $z\in G^{\underline{\omega}}(\underline{a})\cup
G^{\hat{}\underline{\omega}}(\underline{a})$ (in particular
$c^{\underline{\omega}}_{\underline{a}\underline{b}^{i}}$ and
$c^{\hat{}\underline{\omega}}_{\underline{a}\underline{b}^{i}}$). We also have

$c^{\hat{}\underline{\omega}}_{\underline{a}\underline{b}^{i}}=x_{0},\,\,c^{\underline{\omega}}_{\underline{a}\underline{b}^{i}}=x_{0}+c_{5}\rho^{1+1/2k}\omega_{i}$

and
$Df^{\underline{\omega}}_{\underline{a}}(c^{\underline{\omega}}_{\underline{b}^{i}})=Df^{\hat{}\underline{\omega}}_{\underline{a}}(c^{\hat{}\underline{\omega}}_{\underline{b}^{i}})$.
Therefore

$\displaystyle|Dk^{\overline{\underline{\theta}},\underline{\omega}}(c^{\underline{\omega}}_{\underline{b}^{i}})-Dk^{\overline{\underline{\theta}},\hat{}\underline{\omega}}(c^{\hat{}\underline{\omega}}_{\underline{b}^{i}})|$
$\displaystyle=\frac{|Df^{\underline{\omega}}_{\underline{a}}(c^{\underline{\omega}}_{\underline{b}^{i}})|}{|DF^{\hat{}\underline{\theta},\underline{\omega}}_{\underline{a}}|}\cdot|Dk^{\hat{}\underline{\theta},\underline{\omega}}(x_{0}+c_{5}\rho^{1+1/2k}\omega_{i})-Dk^{\hat{}\underline{\theta},\underline{\omega}}(x_{0})|$
$\displaystyle\lesssim\rho^{1+1/2k},$

Here we used that limit geometries are $C^{2}$ and the norm of
$D^{2}k^{\hat{\underline{\theta}},\underline{\omega}}$ can be uniformly
bounded. Using that
$DF^{\underline{\theta},\underline{\omega}}_{\underline{d}}=Dk^{\underline{\theta},\underline{\omega}}(c^{\underline{\omega}}_{\underline{d}})\cdot
Df^{\underline{\omega}}_{\underline{d}}(c_{d})$ for any word
$\underline{d}\in\Sigma^{fin}$ that ends with the letter $d$, one can conclude
that

$\displaystyle|DF^{\overline{}\underline{\theta},\underline{\omega}}_{\underline{b}^{i}}-DF^{\overline{}\underline{\theta},\hat{}\underline{\omega}}_{\underline{b}^{i}}|$
$\displaystyle=|Df^{\underline{\omega}}_{\underline{b}^{i}}(c_{b^{i}})|\cdot|Dk^{\overline{}\underline{\theta},\underline{\omega}}(c^{\underline{\omega}}_{\underline{b}^{i}})-Dk^{\overline{}\underline{\theta},\hat{}\underline{\omega}}(c^{\hat{}\underline{\omega}}_{\underline{b}^{i}})|\lesssim\rho^{2+1/2k}.$

And from this

$\left|\frac{DF^{\overline{}\underline{\theta},\underline{\omega}}_{\underline{b}^{i}}}{DF^{\underline{\theta}^{\prime}}_{\underline{b}^{\prime
i}}}-\frac{DF^{\overline{}\underline{\theta},\hat{}\underline{\omega}}_{\underline{b}^{i}}}{DF^{\underline{\theta}^{\prime}}_{\underline{b}^{\prime
i}}}\right|\lesssim\rho^{1+1/2k},$

therefore
$|\overline{s}_{i}(\underline{\omega})-\overline{s}_{i}(\hat{\underline{\omega}})|\lesssim\rho^{1+1/2k}$
and
$|s_{i}(\underline{\omega})-s_{i}(\hat{\underline{\omega}})|\lesssim\rho^{1+1/2k}$.

From the previous estimates we conclude that
$|\hat{t}_{(j)}-t_{(j)}|\lesssim\rho^{1/2k}$, then if $\rho$ is small enough
(iv)’ implies (iv). ∎

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