Proof. The proof is essentially the same as in [Reference Zamudio20]. For $\underline {a}\in \Sigma ^{fin}$ , let $d_{\underline {a}}$ be the metric

$$ \begin{align*} d_{\underline{a}}(x,y) = \inf_{\alpha} l(\alpha), \end{align*} $$

where $\alpha $ runs through all smooth curves inside $G^*(\underline {a})$ that connect x to y and $l(\alpha )$ denotes the lengths of such curves. Because g sends $G^*(a_0,a_1,\ldots ,a_n)$ diffeomorphically onto $G^*(a_1,\ldots ,a_n)$ and $m(Dg)> \mu $ , then

$$ \begin{align*} d_{(a_1, \ldots, a_n)}(g(x),g(y)) \geq \mu \cdot d_{a_0, \ldots, a_n}(x,y) \end{align*} $$

for all $x,y \in G^* (a_0, \ldots a_n)$ . Therefore,

$$ \begin{align*} \text{diam}_{(a_0, \ldots a_n)}(G^* (a_0, \ldots a_n)) \leq \mu ^{-1} \cdot \text{diam}_{(a_1, \ldots a_n)}(G^* (a_1, \ldots a_n)), \end{align*} $$

where $\text {diam}_{\underline {a}}$ is the diameter with respect to $d_{\underline {a}}$ . We conclude that, by induction,

$$ \begin{align*} \text{diam}(G^*(\underline{a}))\leq \text{diam}_{\underline{a}}(G^*(\underline{a})) \leq \mu^{-n} \cdot \text{diam}_{{a}_n}(G^*({a}_n)). \end{align*} $$

Taking any C larger than $\max _{a \in \mathbb {A}} \text {diam}(G^*(a))$ yields the result.

Proof. Observe that, because $p \in \Lambda _G$ , every backwards iterate of the segment in $\mathcal {L}^u_G(p)$ that connects $q_1$ and $q_2$ stays on the domain of the foliation. So, for every $n \in \mathbb {N}$ , we can define small neighborhoods $ N_i^n \subset N_i$ of $q_i$ , $i=1,\,2$ , such that $G^{-n}(N^n_i)$ is also on the domain of the foliation. Furthermore, this restriction can be done in such manner that, because p is periodic and, by the inclination lemma, $G^{-n}(N^n_1)$ and $G^{-n}(N^n_2)$ are $\delta $ close to each other on the $C^1$ metric for every $n> n_\delta $ . Also, we can assume that their tangent directions at $q^n_i=G^{-n}(q_i)$ are bounded away from $T_{q^n_i}W^u_G$ , $i=1,\,2.$ Let $\Pi ^n_u : N^n_1 \to N^n_2$ be the projection along the unstable foliation.

There is a small neighborhood $\tilde {U}\subset \mathbb {C}^2$ of $q_1^n$ and $q_2^n$ , and a $C^{1+\varepsilon }$ map $f:\tilde {U} \to \mathbb {D}\times \mathbb {D}$ such that the unstable leaves are taken onto the horizontal levels $\mathbb {D} \times \{z\} \text {, } z \in \mathbb {D}$ , and $N^n_1$ and $N^n_2$ are taken onto graphs $(h_1(z), z)$ and $(h_2(z),z)$ of $C^{1+\varepsilon }$ embeddings $h_1 \text { and } h_2$ with the domain being a small disk $\mathbb {D}$ too. Under this identification, $\Pi ^n_u$ is a $C^{1+\varepsilon }$ map that carries $(h_1(z), z)$ to $(h_2(z),z)$ , and, because $G^{-n}(N^n_1)$ and $G^{-n}(N^n_2)$ are $\delta $ close to each other on the $C^1$ metric, has a derivative $\delta $ close to the identity.

Now, the projection along unstable foliations commute with G. Therefore, $\Pi _u = G^{n} \circ \Pi ^n_u \circ G^{-n}$ . Using the chain rule to calculate the derivative of $D\Pi _u$ at $q_1$ , we obtain an expression of the form

$$ \begin{align*}A_1 \cdot A_2 \cdots A_n \cdot D\Pi_u^n \cdot B_n \cdots B_1,\end{align*} $$

where $B_i$ represents the restriction of $(DG)^{-1}$ to $T_{q_1^{i-1}}N_1^{i-1}$ and $A_i$ the restriction of $DG$ to $T_{q_2^i}N_2^i$ , for i from $1$ to n ( $q_1^0=q_1$ and $N_1^0=N_1$ ). However, all of these tangent spaces are, by induction, complex lines in $\mathbb {C}^2$ , so all the $A_i$ and $B_i$ are conformal. This way, the derivative of $\Pi _u$ is at most $\delta $ distant from being conformal. Making $\delta \rightarrow 0$ (or equivalently, $n \rightarrow \infty $ ), we have the desired conformality.

Proof of Theorem 2.5. We need to show that we can express $K_G=\Pi (W^s(p_G; {\textrm {loc}})\cap \Lambda _G)$ as a dynamically defined conformal Cantor set through the maps $f_i : \Pi (V_i) \to S((0,0); 3) $ , where $f_i= \Pi \circ \Pi _i\circ G^{-1} \circ \Pi ^{-1}$ . Let us show that $K_G$ is the maximal invariant set of these maps. Take $x \in W^s(p_G; {\textrm {loc}}) \cap \Lambda _G$ . Thus, $G^{-1}(x) \in \Lambda _G \subset U$ , so there exists $i \in \{1,2,3, \ldots , 9\}$ such that $G^{-1}(x) \in W_i $ , which implies $x \in V_i$ . Likewise, $y=\Pi _i(G^{-1}(x)) \in \Lambda _G$ . To show this, we see that $G^{n}(y) \in W^s(p_G; c_0) \cap U, \text { for all } n\geq 0 $ , as this set is carried into itself by forward iteration of G. Additionally, $G^{-n}(y) \in U \text { for all } n> 0 $ because $y \in W^u(G(x))$ and backwards iterations of unstable leaves always remain inside U by construction. So, $y \in \bigcap _{n \in \mathbb {Z} }G^n(U)=\Lambda _G$ , and in particular $y \in W^s(p_G; c_0) \cap U$ . Hence, as we have already shown, $y \in V_i \text { for some } i \in \{1,2,3,\ldots ,9\}$ . Repeating this argument inductively, we obtain that the orbit by the maps $f_i$ of $\Pi (x)$ always remains on $\bigcup _{i=1}^9 V_i$ .

However, if $x \in W^s(p_G; c_0) \cap U$ is such that the forward orbit of $\Pi (x)$ by the maps $f_i$ is always in $ \bigcup _{i=1}^9 V_i$ , then, using that projections along the unstable leaves commute with the map G and denoting by $x_n$ the $n^{th}$ term of the orbit of x by the $f_i$ , we can show, inductively, that $G^{-n}(x)=\Pi _u\circ \Pi ^{-1}(x_n) \text {, } (n> 0)$ , $\Pi _u$ being a projection along unstable leaves between two components of $W^s(p_G) \cap U$ . This implies that $G^{-n}(x) \in U \text {, } \text { for all } n>0$ , and as $G^{n}(x) \in U, \text { for all } n \geq 0$ , then $x \in \Lambda _G$ .

It is clear that the manifolds $W^s_G(p_G,{\textrm {loc}})$ and $W_i$ satisfy the properties of the transversal sections on Lemma 2.6. It is then clear that the maps $f_i$ are $C^{1+\varepsilon }$ and conformal at K, because of the density of periodic points (notice that $\Pi $ is a parameterization of a complex line).

The general case of a complex horseshoe can be done using Markov neighborhoods, as described in [Reference Pixton17]. The improvement from the work of Bowen [Reference Bowen2] is that the boxes are compact sets of the ambient space filled with our stable and unstable foliations. Given $G \in \mathrm{Aut}(\mathbb {C}^2)$ , $\Lambda $ and p as in the statement of Theorem 2.5, let $R_j, \, j=1, \ldots ,m$ be the boxes of a Markov neighborhood of $\Lambda $ . We consider $W^u_{\textrm {part}}$ , a large compact part of the unstable manifold $W^u_G(p)$ such that its intersection with each one of the boxes $R_j$ , $j = 1, \ldots , m$ , is equal to exactly one connected component of the intersection between $W^{u}_G(p)$ and $R_j$ . We also assume $p \in W^u_{\textrm {part}} \cap R_j$ for some value of j. This compact part exists because the horseshoe is mixing.

Then, define the sets $G(i,j)$ as $G^{-1}(R_j) \cap R_i \cap W^u_{\textrm {part}}$ and the maps

$$ \begin{align*} g_{(i,j)} : G(i,j) & \to W^u_{\textrm{part}} \\ q & \mapsto \Pi^s_j (G(q)) \end{align*} $$

for all $i,\,j=1,\ldots ,n$ , where $\Pi ^s_j$ denotes the projection along the stable leaves inside $R_j$ into $W^u_{\textrm {part}}$ . Notice that, in this case, there is no need to extend the foliations, given the existence of the Markov partition.

Verifying that this set of data defines a dynamically defined Cantor set (up to a parameterization) and that it is equal to $\Lambda \cap V$ for some neighborhood V of p follows from the arguments on the example above almost ‘ipsis literis’. If the boxes $R_j$ are sufficiently small, then one can extend the maps $g_{(i,j)}$ to an $C^{1+\varepsilon }$ expanding map g defined in a small neighborhood of the union of the pieces $G(i,j)$ , as the projections cannot contract distances by much. The mixing property comes from the mixing dynamics of the horseshoe. To make this conformal Cantor set to be contained in $W^{u}_{\varepsilon }(p)$ , all one needs to do is to apply $G^{-N(p)}$ a sufficient amount of times, where $N(p)$ is the period of p. This finishes the proof.

Proof. We will first prove the result for $1 \leq r < 2$ . Consider for each $n \geq 2$ (and $\underline {\theta } \in \Sigma ^-$ ), the functions $\Psi ^{\underline {\theta }}_n: {\textrm {Im}}(k^{\underline {\theta }}_{n-1}) \to \mathbb {C}$ :

$$ \begin{align*} \Psi^{\underline{\theta}}_n= \Phi_{\underline{\theta}_n} \circ f_ {(\theta_{-n}, \theta{-n+1})}\circ \Phi_{\underline{\theta}_{n-1}}^{-1}. \end{align*} $$

Notice that

(3.1) $$ \begin{align} k^{\underline{\theta}}_n=\Psi^{\underline{\theta}}_n \circ \Psi^{\underline{\theta}}_{n-1} \circ \cdots \circ \Psi^{\underline{\theta}}_2 \circ k^{\underline{\theta}}_1. \end{align} $$

We proceed by controlling the functions $\Psi ^{\underline {\theta }}_n$ and showing that they are exponentially close to the identity.

First, the domain of $f_{(\theta _{-n}, \theta _{-n+1})}$ in the definition of $\Psi ^{\underline {\theta }}_n$ is $G^*(\underline {\theta }_{n-1})$ . Denoting its diameter by $r_{\underline {\theta }_{n}}$ , we know that $r_{\underline {\theta }_{n}} \leq C \cdot \mu ^{-n}$ for some constant $C>0$ , as shown in Lemma 2.1. However, we can do better. In what follows, all the C terms (with subscript, superscript, or without them) will always denote a positive real constant that will, in some way, depend on the other constants previously appearing in this proof, but never on $n \in \mathbb {N}$ or $\underline {\theta }$ . If a map f is $C^{1+\varepsilon }$ on an open set $U \subset \mathbb {C}$ , then, for any point $z \in U$ and $h \in \mathbb {C}$ small enough so that the segment joining z and $z+h$ is in U:

$$ \begin{align*} |f(z+h)-(f(z)+Df(z)\cdot h)| < \tilde{C} |h|^{1+\varepsilon}, \end{align*} $$

for some constant $\tilde {C}> 0$ . Consequently, $ f_{(\theta _{-n}, \theta _{-n+1})}:G^*(\underline {\theta }_{n-1}) \to G^*(\underline {\theta }_n)$ is $C_f \cdot r_{\underline {\theta }_{n-1}}^{1+\varepsilon }$ close to the map $A_{\underline {\theta }_n} \in \textrm{Aff}(\mathbb {C})$ described by

$$ \begin{align*}A_{\underline{\theta}_n}(c_{\underline{\theta}_{n-1}})=c_{\underline{\theta}_n} \quad\text{and}\quad DA_{\underline{\theta}_n} = Df_{(\theta_{-n}, \theta_{-n+1})}(c_{\underline{\theta}_{n-1}}),\end{align*} $$

and thus, if n is large enough:

$$ \begin{align*} r_{\underline{\theta}_{n}} & \leq |Df_{(\theta_{-n}, \theta_{-n+1})}(c_{\underline{\theta}_{n-1}})|\cdot\, r_{\underline{\theta}_{n-1}} + C_f \cdot r_{\underline{\theta}_{n-1}}^{1+\varepsilon} \\ & \leq r_{\underline{\theta}_{n-1}} \cdot ( | Df_{(\theta_{-n}, \theta_{-n+1})}(c_{\underline{\theta}_{n-1}}) | + C_1 \cdot \mu^{(-n+1)\varepsilon}). \end{align*} $$

Arguing by induction, using that $\log {(x+y)} \le \log {x} +({y}/{x})$ , we obtain

$$ \begin{align*} &\log r_{\underline{\theta}_{n}} \leq \log |Df_{\underline{\theta}_n}(c_{\theta_0})| + \Vert Dg\Vert \cdot \sum_{j=0}^{n-1} ( C_1 \cdot \mu^{-j\varepsilon} ) \leq \log |Df_{\underline{\theta}_n}(c_{\theta_0})| + C_2 , \end{align*} $$

so

$$ \begin{align*} r_{\underline{\theta}_{n}} \leq C' \cdot | Df_{\underline{\theta}_n}(c_{\theta_0})| \leq C' \cdot \mu^{-n}, \end{align*} $$

because $|Df_{(\theta _{-n}, \theta _{-n+1})}(x)|^{-1} \le \lVert Dg \rVert \; \text { for all } x \in G^*(\theta _{-n+1})$ , where $\lVert Dg \rVert $ denotes the $C^0$ norm of this function over its domain. In a completely analogous way, we can show, maybe enlarging $C'$ , that

(3.2) $$ \begin{align} {C'}^{-1} \cdot |Df_{\underline{\theta}_n}(c_{\theta_0})| \leq r_{\underline{\theta}_{n}} \leq C' \cdot |Df_{\underline{\theta}_n}(c_{\theta_0})| \end{align} $$

and so the size of the $G^*(\underline {\theta }_n)$ is controlled. This implies that

$$ \begin{align*} \Vert f_{(\theta_{-n}, \theta_{-n+1})}-A_{\underline{\theta}_n}\Vert \leq C \cdot |Df_{\underline{\theta}_n}(c_{\theta_0})|^{1+\varepsilon} \end{align*} $$

for some constant C, for all $\underline {\theta } \in \Sigma ^-$ . However, by construction, $ \Phi _{\underline {\theta }_n} \circ A_{\underline {\theta }_n} \circ \Phi _{\underline {\theta }_{n-1}}^{-1} = {\textrm {Id}}$ and $ D\Phi _{\underline {\theta }_n} = (Df_{{\underline {\theta }_n}}(c_{\theta _0}))^{-1} $ ; therefore,

$$ \begin{align*} \Vert \Psi^{\underline{\theta}}_n -{\textrm{Id}}\Vert & = \Vert \Psi^{\underline{\theta}}_n -\Phi_{\underline{\theta}_n}\circ A_{\underline{\theta}_n} \circ \Phi_{\underline{\theta}_{n-1}}^{-1} \Vert \leq \Vert D\Phi_{\underline{\theta}_n}\Vert \cdot \Vert f_{(\theta_{-n}, \theta_{-n+1})}-A_{\underline{\theta}_n} \Vert \\ & \leq |Df_{{\underline{\theta}_n}}(c_{\theta_0})|^{-1} \cdot C \cdot |Df_{\underline{\theta}_{n-1}}(c_{\theta_0})|^{1+\varepsilon}\\ & \leq C_3 \cdot {(\mu^{-\varepsilon})}^n \end{align*} $$

as we wished to obtain. This is enough to show that $\{k^{\underline {\theta }}_n\}_{n\geq 0}$ is a Cauchy sequence, at least in $C^0$ metric. In fact, for $m,l \geq 1$ ,

$$ \begin{align*} &\Vert k^{\underline{\theta}}_{m+l}-k^{\underline{\theta}}_m\Vert = \Vert \Psi^{\underline{\theta}}_{m+l} \circ \cdots \circ \Psi^{\underline{\theta}}_2 \circ k^{\underline{\theta}}_1 - \Psi^{\underline{\theta}}_{m} \circ \cdots \circ \Psi^{\underline{\theta}}_2 \circ k^{\underline{\theta}}_1 \Vert \\ & \quad\leq\! \sum_{j=1}^l \Vert \Psi^{\underline{\theta}}_{m+j} \circ \cdots \circ \Psi^{\underline{\theta}}_2 \circ k^{\underline{\theta}}_1 -\! \Psi^{\underline{\theta}}_{m+j-1} \circ \cdots \circ \Psi^{\underline{\theta}}_2 \circ k^{\underline{\theta}}_1 \Vert \leq\! \sum_{j=1}^l \Vert \Psi^{\underline{\theta}}_{m+j} \!-{\textrm{Id}} \Vert \\ &\quad \leq \sum_{j=1}^l C_3 \cdot {(\mu^{-\varepsilon})}^{m+j} \leq \frac{C_3 \cdot (\mu^{-\varepsilon})^{m}}{1-\mu^{-\varepsilon}}, \end{align*} $$

which implies that $\Vert k^{\underline {\theta }}_{m+l}-k^{\underline {\theta }}_m\Vert \rightarrow 0$ as $m \rightarrow \infty $ . Further, for any point $z \in {\textrm {Im}}(k^{\underline {\theta }}_{n-1})$ , we can calculate $ D\Psi ^{\underline {\theta }}_n (z) = D\Phi _{\underline {\theta }_n} \cdot Df_{(\theta _{-n, \theta {-n+1}})} ( \Phi _{\underline {\theta }_{n-1}}^{-1} (z) ) \cdot D\Phi _{\underline {\theta }_{n-1}}^{-1}$ . However, by hypothesis, we have that $D\Psi ^{\underline {\theta }}_n (c_{\theta _0}) = {\textrm {Id}}$ and

$$ \begin{align*}d(\Phi_{\underline{\theta}_{n-1}}^{-1} (z),\Phi_{\underline{\theta}_{n-1}}^{-1} (c_{\theta_0})) \leq \text{diam} (G^* (\underline{\theta}_{n-1})). \end{align*} $$

Then, using that $Df_{(\theta _{-n, \theta {-n+1}})}$ is $\varepsilon $ -Hölder, we conclude that

(3.3) $$ \begin{align} \Vert D\Psi^{\underline{\theta}}_n -{\textrm{Id}}\Vert \leq C_f \cdot |D\Phi_{\underline{\theta}_n}| \cdot |D\Phi_{\underline{\theta}_{n-1}}|^{-1} \cdot r_{\underline{\theta}_{n-1}}^{\varepsilon} \leq C_4 \mu^{-n \varepsilon} , \end{align} $$

because $|D\Phi _{\underline {\theta }_n}|$ and $|D\Phi _{\underline {\theta }_{n-1}}|$ are comparable (because $D\Phi _{\underline {\theta }_n}= Df_{\underline {\theta }_{n}}(c_{\theta _0})^{-1}$ and so $|D\Phi _{\underline {\theta }_n}| \cdot |D\Phi _{\underline {\theta }_{n-1}}|^{-1}$ is controlled by $\Vert Dg\Vert $ ).

Now we can show that $\{\Vert Dk^{\underline {\theta }}_n\Vert \}_{n \geq 1}$ is bounded. Indeed, $\Vert Dk^{\underline {\theta }}_n \Vert \leq \prod _{j\geq 2} ^ {n}\Vert D\Psi ^{\underline {\theta }}_j \Vert \cdot \Vert Dk^{\underline {\theta }}_1 \Vert $ implies that:

$$ \begin{align*} \log ( \Vert Dk^{\underline{\theta}}_n \Vert ) & \leq {\sum_{j=2}^{n} \log \Vert D\Psi^{\underline{\theta}}_j \Vert } + C_0 \\ & \leq {\sum_{j=2}^{n} \log |(\Vert {\textrm{Id}}\Vert +\Vert D\Psi^{\underline{\theta}}_j - {\textrm{Id}}\Vert ) } + C_0 \leq {\sum_{j=2}^{n} C_4 \mu^{-j \varepsilon} } + C_0 \\ & \leq \frac{C_4\mu^{-2\varepsilon}+C_0-C_0\mu^{\varepsilon}}{1-\mu^{\varepsilon}} = C_5. \end{align*} $$

The same argument can be used to show that $\Vert (Dk^{\underline {\theta }}_n)^{-1}\Vert $ is bounded. It also follows that:

$$ \begin{align*} \Vert Dk^{\underline{\theta}}_{m+l}-Dk^{\underline{\theta}}_{m}\Vert & \leq \sum_{j=0}^{l-1} \Vert Dk^{\underline{\theta}}_{m+j+1}-Dk^{\underline{\theta}}_{m+j}\Vert \leq \sum_{j=0}^{l-1} \Vert D\Psi^{\underline{\theta}}_{m+j+1}-{\textrm{Id}}\Vert \cdot \Vert Dk^{\underline{\theta}}_{m+j}\Vert \\ & \leq C_5 \cdot {\sum_{j=0}^{l-1} C_4 \mu^{-(m+j+1) \varepsilon} } \leq C_6 \cdot \mu^{-m}, \end{align*} $$

which shows that $\{k^{\underline {\theta }}_n\}_{n\geq 0}$ is a Cauchy sequence also in the $C^1$ metric, and so it converges to a $C^1$ map $ k^{\underline {\theta }}$ . Because $\Vert (Dk^{\underline {\theta }}_n)^{-1}\Vert $ is bounded, this also implies that the inverse maps $\{(k^{\underline {\theta }}_n)^{-1}_{n\geq 0}\}$ also converge in the $C^1$ metric to the inverse of $k^{\underline {\theta }}$ .

We need to show that $ k^{\underline {\theta }}$ is $C^{1+\varepsilon }$ . This is true for $k^{\underline {\theta }}_{n}$ for all $n \geq 0$ . Indeed, for a given $\underline {\theta } \in \Sigma ^-$ , we write for $n \geq 0,\, x, y \in G^*(\theta _0)$ ’:

$$ \begin{align*}I_{n}(x,y)=|Dk^{\underline{\theta}}_{n}(x)-Dk^{\underline{\theta}}_{n}(y)| < H_n \cdot |x-y|^{\varepsilon}, \end{align*} $$

for some constant $H_n>0$ . By equations (3.1), (3.3), and the fact that $Dk^{\underline {\theta }}_{n}$ are bounded, we have that:

$$ \begin{align*} I_{n} (x,y) & = | D(\Psi_n \circ k^{\underline{\theta}}_{n-1})(x)-D(\Psi_n \circ k^{\underline{\theta}}_{n-1})(y) |\\ & \leq | D\Psi_n (k^{\underline{\theta}}_{n-1}(x)) (Dk^{\underline{\theta}}_{n-1}(x)-Dk^{\underline{\theta}}_{n-1}(y) ) | \\ & \quad + | ( D\Psi_n(k^{\underline{\theta}}_{n-1}(x))- D\Psi_n (k^{\underline{\theta}}_{n-1} (y)))Dk^{\underline{\theta}}_{n-1}(y) |\\ & \leq (1+C_4\cdot\mu^{(-n+1)\varepsilon})\cdot I_{n-1}(x,y) \\ & \quad+ e^{ C_5} \cdot \Vert Dg\Vert \cdot | Df_{(\theta_{-n}, \theta_{-n+1})}(\Phi_{\underline{\theta}_{n-1}}^{-1}(x)) - Df_{(\theta_{-n}, \theta_{-n+1})}(\Phi_{\underline{\theta}_{n-1}}^{-1}(y))|\\ & \leq (1+C_4\cdot\mu^{(-n+1)\varepsilon})\cdot I_{n-1}(x,y) + e^{C_5}\cdot \Vert Dg\Vert \cdot C_f \cdot \mu^{(-n+1)\varepsilon} \cdot |x-y|^{\varepsilon}\\ & \leq ((1+C_4\cdot\mu^{(-n+1)\varepsilon})\cdot H_{n-1}+ e^{C_5} \cdot \Vert Dg\Vert \cdot C_f \cdot \mu^{(-n+1)\varepsilon} )\cdot |x-y|^{\varepsilon}, \end{align*} $$

which inductively shows that these functions have Hölder continuous derivatives. Additionally, we can choose the Hölder constants satisfying the relation:

(3.4) $$ \begin{align} H_n \leq (1+C_4\cdot\mu^{(-n+1)\varepsilon}) \cdot H_{n-1}+ C_7 \cdot \mu^{(-n+1)\varepsilon}, \end{align} $$

and then the sequence $\{ H_n \}_{n \geq 1}$ is bounded. Effectively, it is crescent and if $H_{n-1}> 1 $ , then $H_n \leq (1+C_4\cdot \mu ^{(-n+1)\varepsilon } + C_7 \cdot \mu ^{(-n+1)\varepsilon }) \cdot H_{n-1} \leq (1+C_8\cdot \mu ^{(-n+1)\varepsilon }) \cdot H_{n-1}$ and using the same strategy as above, we have

$$ \begin{align*} \log H_n & \leq \log H_{n-1} + \log (1+C_8\cdot\mu^{(-n+1)\varepsilon}) \\ & \leq \sum_{j=1}^{n-1} \log (1+C_8\cdot\mu^{-j\varepsilon}) \leq \sum_{j=1}^{n-1} C_8\cdot\mu^{-j\varepsilon} \leq H, \end{align*} $$

as stated.

Finally, for each pair $x,y \in G^*(\theta _0)$ , there is $n\geq 0$ such that $\Vert Dk^{\underline {\theta }}_{n}-Dk^{\underline {\theta }}\Vert $ is less than $|x-y|^{\varepsilon }$ so, by triangle inequality, we have $|Dk^{\underline {\theta }}(x)-Dk^{\underline {\theta }}(y)| < (H+2) \cdot |x-y|^{\varepsilon }$ . By maybe enlarging H a little, the same estimates are true for the inverses of $k^{\underline {\theta }_n}$ and $k^{\underline {\theta }}$ .

Now, because the maps $k^{\underline {\theta }}$ are $C^{1+\varepsilon }$ , to prove that the sequence $k^{\underline {\theta }}_n$ converges to $k^{\underline {\theta }}$ in the $C^{1+\varepsilon }$ topology, it is sufficient to show that

(3.5) $$ \begin{align} \lim_{n \rightarrow \infty} \mathop{\mathrm{sup}}\limits_{x, y \in k^{\underline{\theta}}(G^*(\theta_0))} \frac{ \lvert {D( k^{\underline{\theta}}_n \circ (k^{\underline{\theta}})^{-1})(x)} - {D ( k^{\underline{\theta}}_n \circ (k^{\underline{\theta}})^{-1})}(y) \rvert } {\lvert x - y \rvert^\varepsilon } = 0. \end{align} $$

For a fixed value of $n> 0 $ , let $\underline {\theta }^n \in \Sigma ^-$ be the infinite word such that $\underline {\theta } = \underline {\theta }^n\underline {\theta }_n$ . Consequently, for $m> n$ ,

$$ \begin{align*} k^{\underline{\theta}}_n\circ (k^{\underline{\theta}}_m)^{-1} & = \Phi_{\underline{\theta}_n} \circ f_{\underline{\theta}_n} \circ (f_{\underline{\theta}_{m}})^{-1} \circ (\Phi_{\underline{\theta}_m})^{-1} \\ & = \Phi_{\underline{\theta}_n} \circ (f_{\underline{\theta}^n_{m - n}})^{-1} \circ (\Phi_{\underline{\theta}_m})^{-1} \\ & = \Phi_{\underline{\theta}_n} \circ (f_{\underline{\theta}^n_{m - n}})^{-1} \circ (\Phi_{\underline{\theta}^{n}_{m-n}})^{-1} \circ \Phi_{\underline{\theta}^{n}_{m-n}} \circ (\Phi_{\underline{\theta}_m})^{-1} \\ & = \Phi_{\underline{\theta}_n} \circ (k^{\underline{\theta}^n_{m - n}})^{-1} \circ \Phi_{\underline{\theta}^{n}_{m-n}} \circ (\Phi_{\underline{\theta}_m})^{-1}. \end{align*} $$

Remember that $ (D\Phi _{\underline {\theta }^n_{m-n}})^{-1} = Df_{\underline {\theta }^n_{m - n}}(c_{\theta _{-n}})$ has norm comparable to $r_{\underline {\theta }^n_{m-n}}$ . Now, because $f_{\underline {\theta }_n}(c_{\theta _0})$ could also be chosen as the base point for the piece $G(\theta _{-n})$ , equation (3.2) implies that $\lvert Df_{\underline {\theta }^n_{m - n}}(f_{\underline {\theta }_n}(c_{\theta _0})) \rvert $ is also comparable to $r_{\underline {\theta }^n_{m-n}}$ , and so $\lvert D ( \Phi _{\underline {\theta }^{n}_{m-n}} \circ (\Phi _{\underline {\theta }_m})^{-1} ) \rvert $ is comparable to $\lvert \Phi _{\underline {\theta }_n}^{-1} \rvert $ . Therefore, for $x, y \in k^{\underline {\theta }_m}(G^*(\theta _0))$ ,

$$ \begin{align*} &\lvert D(k^{\underline{\theta}}_n\circ (k^{\underline{\theta}}_m)^{-1} )(x) - D(k^{\underline{\theta}}_n\circ (k^{\underline{\theta}}_m)^{-1})(y) \rvert \\ &\quad\le C_9 \lvert D(k^{\underline{\theta}^n_{m - n}})^{-1} (\Phi_{\underline{\theta}^{n}_{m-n}} \circ (\Phi_{\underline{\theta}_m})^{-1}(x)) - D(k^{\underline{\theta}^n_{m - n}})^{-1} (\Phi_{\underline{\theta}^{n}_{m-n}} \circ (\Phi_{\underline{\theta}_m})^{-1} ( y) )\rvert \\ &\quad\le C_{9} (H+2) \lvert \Phi_{\underline{\theta}^{n}_{m-n}} \circ (\Phi_{\underline{\theta}_m})^{-1}(x) - \Phi_{\underline{\theta}^{n}_{m-n}} \circ (\Phi_{\underline{\theta}_m})^{-1}(y) \rvert^{\varepsilon} \\ &\quad\le C_{10} r_{\underline{\theta}_n}^{\varepsilon} \lvert x - y \rvert^{\varepsilon} \le C'\,C_{10} \mu^{-n\varepsilon} \lvert x - y \rvert^\varepsilon. \end{align*} $$

Finally, making $m \rightarrow \infty $ and then $n \rightarrow \infty $ , we prove the limit (3.5).

All the constants appearing in the estimates above depend continuously (actually, they are simple functions) on the $C^1$ norm of g as well as the Hölder constant and exponent of $Dg$ , and so, for any $g'$ sufficiently close to g, all of those estimates would be the same except with a minor pre-fixed error. This implies that the convergence we just showed is uniform not only over $\Sigma ^-$ but also on a small neighborhood of g in the Hölder topology.

The Hölder continuity of the association $\underline {\theta } \mapsto k^{\underline {\theta }}$ comes from the fact that, for some constant $C_{12}> 0$ ,

(3.6) $$ \begin{align} \lVert k^{\underline{\theta}} - k^{\underline{\theta}}_n \rVert_{C^{1+\varepsilon}} \le C_{12}r_{\underline{\theta}_n}^{\varepsilon}, \end{align} $$

from which, for any $\underline {\theta }^1, \, \underline {\theta }^2 \in \Sigma ^-$ ,

$$ \begin{align*} \lVert k^{\underline{\theta}^1} - k^{\underline{\theta}^2} \rVert_{C^{1+\varepsilon}} \le 2C_{12}r_{\underline{\theta}^1 \wedge \underline{\theta}^2}^{\varepsilon} = 2C_{12} d(\underline{\theta}^1, \underline{\theta}^2)^{\varepsilon}. \end{align*} $$

Equation (3.6) is a consequence of the fact that for $x, y\in k^{\underline {\theta }}(G^*(\theta _0))$ ,

$$ \begin{align*} \lvert D(k^{\underline{\theta}}_n\circ (k^{\underline{\theta}})^{-1})(x) - D(k^{\underline{\theta}}_n\circ (k^{\underline{\theta}})^{-1})(y)\rvert \le C_{10} r_{\underline{\theta}_n}^{\varepsilon} \lvert x - y \rvert^{\varepsilon} \end{align*} $$

and with just a small refinement of the estimates made above for

$$ \begin{align*} \lVert k^{\underline{\theta}}_{m+l} - k^{\underline{\theta}}_{m}\rVert \quad \text{and} \quad \lVert Dk^{\underline{\theta}}_{m+l}-Dk^{\underline{\theta}}_{m} \rVert \end{align*} $$

for $m, l> 0$ . Indeed, the terms in the series

$$ \begin{align*} \sum_{j=0}^{l-1} \lVert \Psi^{\underline{\theta}}_{m+j+1}-{\textrm{Id}} \rVert \quad \text{and} \quad \sum_{j=0}^{l-1} \lVert D\Psi^{\underline{\theta}}_{m+j+1}-{\textrm{Id}} \rVert \end{align*} $$

decay exponentially; therefore, these series can be controlled by $\lVert \Psi ^{\underline {\theta }}_{m}-{\textrm {Id}} \rVert $ and $\lVert D \Psi ^{\underline {\theta }}_{m}-{\textrm {Id}} \rVert $ , respectively.

If the map g defining the Cantor set is $C^r, r \geq 2$ , then the convergence also happens in the $C^r$ metric. This happens because the composition with affine maps on the definition of $\Psi ^{\underline {\theta }}_n$ ‘flattens’ the derivatives of $f_{(\theta _{-n}, \theta _{-n+1})}$ . As we have seen above, the first-order derivatives of the maps $\Psi ^{\underline {\theta }}_n$ are close to the identity, or close (in norm) to $1=r_{\underline {\theta }_n}^0$ . Analogously, the derivatives of order $r \in \mathbb {N}$ have norm less than $r_{\underline {\theta }_n}^{r-1}$ . A formula for the derivatives of higher order of a composition of two maps can be found in [Reference Constantine and Savits5]. It allow us to inductively bound the $C^r$ norm of $ k^{\underline {\theta }}_{n}$ , if r is an integer, and the convergence $k^{\underline {\theta }_n} \rightarrow k^{\underline {\theta }}$ is proved in this metric following the same type of argument in this proof. Moreover, following the same strategy as above, one can also show that if $r \notin \mathbb {N}$ is greater than $2$ , the maps $D^{\lfloor r \rfloor }k^{\underline {\theta }}$ are $r-\lfloor r \rfloor $ Hölder and that the convergence also happens in the $C^r$ metric, which completes the proof. The key is to analyze, in the expression of $D^{\lfloor r \rfloor }(f \circ g)$ , the terms involving derivatives of order $\lfloor r \rfloor $ of f and g. In the end, we have expressions similar to those in the proof, only involving many more terms.

Proof. Given any $\underline {\theta }$ whose ending coincides with the word $\underline {a}$ , by the estimates in the proof of Lemma 3.1, there is some constant $C_{13}> 0$ such that

$$ \begin{align*} \log{\lvert D (k^{\underline{\theta}}_n \circ (k^{\underline{\theta}})^{-1})(z)\rvert} \,\, \leq C_{13} \cdot r_{{\underline{\theta}}_n}^{\varepsilon} \quad \text{and} \quad \log{\lvert D (k^{\underline{\theta}} \circ (k^{\underline{\theta}}_n)^{-1})(z')\rvert} \,\, \leq C_{13} \cdot r_{{\underline{\theta}}_n}^{\varepsilon} \end{align*} $$

for $z \in k^{\underline {\theta }}(G^{*}(\theta _0)) $ and $z' \in k^{\underline {\theta }}_n(G^{*}(\theta _0)) $ . Making $z=k^{\underline {\theta }}(x)$ and $z'=k^{\underline {\theta }}_n(y)$ , it follows that

(3.7) $$ \begin{align} \log{\lvert Dk^{\underline{\theta}}_n(x)\cdot(Dk^{\underline{\theta}}(x))^{-1}\rvert + \log{\lvert Dk^{\underline{\theta}}(y)\cdot(Dk^{\underline{\theta}}_n(y))^{-1} \rvert}} \le 2C_9 \cdot r_{{\underline{\theta}}_n}^{\varepsilon}, \end{align} $$

and so, using that $ \lvert A \rvert \cdot m(B) \le \lvert A \cdot B \rvert $ and $m(A) \cdot \lvert B \rvert \le \lvert A \cdot B \rvert $ for any two square matrices,

$$ \begin{align*} \log{\lvert Dk^{\underline{\theta}}_n(x)\rvert} + \log{m((Dk^{\underline{\theta}}(x))^{-1})} + \log{m(Dk^{\underline{\theta}}(y))} +\log{\lvert (Dk^{\underline{\theta}}_n(y))^{-1}\rvert} \le 2C_9 \cdot r_{{\underline{\theta}}_n}^{\varepsilon}.\end{align*} $$

However, by the definition of $k^{\underline {\theta }}_n$ and the fact that $m(A^{-1})=\lvert A \rvert ^{-1}$ for any invertible matrix A,

(3.8) $$ \begin{align} \frac{|Df_{\underline{a}}(x)|}{m(Df_{\underline{a}}(y))}=\lvert Dk^{\underline{\theta}}_n(x)\rvert \cdot \lvert (Dk^{\underline{\theta}}_n(y))^{-1} \rvert \le \frac{\lvert Dk^{\underline{\theta}}(x) \rvert}{m(Dk^{\underline{\theta}}(y))} \exp{(2C_9 \cdot r_{{\underline{\theta}}_n}^{\varepsilon})}, \end{align} $$

for any $x,\, y \in G^*(\theta _0)$ . As the association $\underline {\theta } \mapsto k^{\underline {\theta }}$ is continuous and $\Sigma ^-$ is compact, there is $C>0$ that bounds the right-hand side of equation (3.8) for all $x,\,y \in G^{*}(\theta _0)$ and $n \in \mathbb {N}$ . The second part is obtained in an analogous way, only changing the way we proceed from equation (3.7). In the case of the operator norm, it yields

$$ \begin{align*} \log{\lvert Dk^{\underline{\theta}}_n(x)\rvert} + \log{m((Dk^{\underline{\theta}}(x))^{-1})} + \log{\lvert Dk^{\underline{\theta}}(y)\rvert} +\log{m( (Dk^{\underline{\theta}}_n(y))^{-1})} \le 2C_9 \cdot r_{{\underline{\theta}}_n}^{\varepsilon}, \end{align*} $$

from which

$$ \begin{align*} \frac{|Df_{\underline{a}}(x)|}{|Df_{\underline{a}}(y)|} \le \frac{\lvert Dk^{\underline{\theta}}(x) \rvert}{\lvert Dk^{\underline{\theta}}(y) \rvert } \exp{(2C_9 \cdot r_{{\underline{\theta}}_n}^{\varepsilon})}, \end{align*} $$

and the claim follows because of the continuity of $k^{\underline {\theta }}(x)$ on $\underline {\theta }$ and x.

Proof. First, we can consider $c_a \in \textrm{int} \, G(a)$ . This can be done because of the additional hypothesis on the sets $G(a)$ , described at the end of §2.1, just before Definition 2.2. By the definition of $U_{K,\delta '}$ , we can choose $\tilde {c}_a$ such that $H(c_a)= \tilde {H}(\tilde {c}_a)$ , where $H: K \to \Sigma $ is the homeomorphism defined just after Definition 2.1, which implies $|\tilde {c}_a-c_a|< \delta $ for every $\delta '$ sufficiently small. Let us analyze the limit of $\Vert \tilde {k}_n^{\underline {\theta }}-{k}_n^{\underline {\theta }}\Vert _{C^0}$ .

Define $x_n = \Vert \tilde {k}_{n}^{\underline {\theta }}-{k}_{n}^{\underline {\theta }}\Vert _{C^0}$ for $n \in \mathbb {N}$ . Given $\varepsilon _0>0$ and $N \in \mathbb {N}$ , if the distance between $k_1^{\underline {\theta }}$ and $\tilde {k}_1^{\underline {\theta }}$ is small enough and so is $\delta '$ , then, by continuity, $x_n = \Vert \tilde {k}_n^{\underline {\theta }}-{k}_n^{\underline {\theta }}\Vert _{C^0} < \varepsilon _0$ for all $n \leq N$ . We will prove by induction that if $n\geq N$ , then there are constants $\tilde {C}_1$ , $\tilde {C}_2$ and $0<\lambda <1$ such that

(3.9) $$ \begin{align} x_{n+1}\leq (1+\tilde{C}_1\lambda^n)x_n + \tilde{C}_2 \lambda^n, \end{align} $$

and, for all $n \geq N$ , we can apply $\Psi _{n+1}^{\underline {\theta }}$ to the image of $\tilde {k}_n^{\underline {\theta }}$ . It is not true that we can always apply $\Psi _{N+1}^{\underline {\theta }}$ to the image of $\tilde {k}_N^{\underline {\theta }}$ ; at least not considering the same domain ${\textrm {Im}}(k^{\underline {\theta }_N})$ used in the definition at the beginning of the proof of Lemma 3.1. However, this domain can be extended to a larger set $V_{\varepsilon _1}({\textrm {Im}}(k^{\underline {\theta }_N}))$ , for some $\varepsilon _1>0$ , simply because

$$ \begin{align*} \Psi^{\underline{\theta}}_n= \Phi_{\underline{\theta}_n} \circ f_ {(\theta_{-n}, \theta{-n+1})}\circ \Phi_{\underline{\theta}_{n-1}}^{-1} \end{align*} $$

is well defined on $\Phi _{\underline {\theta }_{n-1}} (G(\theta _{-n+1})) \supset \Phi _{\underline {\theta }_{n-1}} (G(\underline {\theta }_{-n+1})) = {\textrm {Im}}(k^{\underline {\theta _n}})$ . So by making $N=n$ and observing that we can make $x_N \leq \varepsilon _0$ , the first part of the basis of induction is proved. Now, if we can apply $\Psi _{n+1}^{\underline {\theta }}$ to the image of $\tilde {k}_n^{\underline {\theta }}$ for some $n \geq N$ , we have, following the notation in the proof of Lemma 3.1, that

(3.10) $$ \begin{align} |\tilde{k}_{n+1}^{\underline{\theta}}(x)-{k}_{n+1}^{\underline{\theta}}(x)| & = |(\tilde{\Psi}_{n+1}^{\underline{\theta}}\circ\tilde{k}_{n}^{\underline{\theta}})(x)-({\Psi}_{n+1}^{\underline{\theta}}\circ{k}_{n}^{\underline{\theta}})(x)|\nonumber\\ & \leq |\Psi_{n+1}^{\underline{\theta}}(\tilde{k}_{n}^{\underline{\theta}}(x))-{\Psi}_{n+1}^{\underline{\theta}}({k}_{n}^{\underline{\theta}}(x))| + |(\tilde{\Psi}_{n+1}^{\underline{\theta}}- {\Psi}_{n+1}^{\underline{\theta}})(\tilde{k}_{n}^{\underline{\theta}}(x))| \nonumber\\ & \leq \Vert D{\Psi}_{n+1}^{\underline{\theta}}\Vert \cdot|\tilde{k}_{n}^{\underline{\theta}}(x)-{k}_{n}^{\underline{\theta}}(x)| + \Vert \tilde{\Psi}_{n+1}^{\underline{\theta}}- {\Psi}_{n+1}^{\underline{\theta}} \Vert \nonumber\\ & \leq (1+C_4 \mu^{-n \varepsilon}) \cdot|\tilde{k}_{n}^{\underline{\theta}}(x)-{k}_{n}^{\underline{\theta}}(x)| + 2 C_3 \mu^{-n \varepsilon}, \end{align} $$

which yields an estimate of the type in equation (3.9), and so, the induction basis is complete. However, with basic real analysis, one can show that there are constants $\tilde {C}_3$ and $\tilde {C}_4$ independent of $m \in \mathbb {N}$ such that, if equation (3.9) is true for all n such that $N\leq n\leq m$ , then

$$ \begin{align*} x_m \leq e^{\tilde{C}_3\lambda^N} (x_N + \tilde{C}_4 \lambda^N), \end{align*} $$

and so $x_m, \, m \geq N,$ can be made as small as $x_N$ by choosing N sufficiently large (and consequently the distance between $k_1^{\underline {\theta }}$ and $\tilde {k}_1^{\underline {\theta }}$ sufficiently small). This is enough to show that we can apply $\Psi _{m+1}^{\underline {\theta }}$ to the image of $\tilde {k}_m^{\underline {\theta }}$ to do the induction step.

Passing to the limit we show that

$$ \begin{align*} \Vert \tilde{k}^{\underline{\theta}}-{k}^{\underline{\theta}}\Vert _{C^0}<\varepsilon \end{align*} $$

provided that the distance between $k_1^{\underline {\theta }}$ and $\tilde {k}_1^{\underline {\theta }}$ is small enough, but this is controlled by the difference between $f_{\theta _{-1},\theta _0}$ and $\tilde {f}_{\theta _{-1},\theta _0}$ , which can be made small enough by choosing $\delta '>0$ sufficiently small.

The argument for the $C^{1}$ norm is similar:

$$ \begin{align*} &| D\tilde{k}_{n+1}^{\underline{\theta}} (x) -D{k}_{n+1}^{\underline{\theta}}(x)| = |D(\tilde{\Psi}_{n+1}^{\underline{\theta}}\circ\tilde{k}_{n}^{\underline{\theta}})(x)-D({\Psi}_{n+1}^{\underline{\theta}}\circ{k}_{n}^{\underline{\theta}})(x)|\\ &\quad \leq |D(\tilde{\Psi}_{n+1}^{\underline{\theta}}\circ\tilde{k}_{n}^{\underline{\theta}})(x)-\!D(\tilde{\Psi}_{n+1}^{\underline{\theta}}\circ{k}_{n}^{\underline{\theta}})(x)| + |D(\tilde{\Psi}_{n+1}^{\underline{\theta}}\circ{k}_{n}^{\underline{\theta}})(x)-\!D({\Psi}_{n+1}^{\underline{\theta}}\circ{k}_{n}^{\underline{\theta}})(x)| \\ &\quad \leq \Vert D{\tilde{\Psi}}_{n+1}^{\underline{\theta}}\Vert \cdot|D\tilde{k}_{n}^{\underline{\theta}}(x)-D{k}_{n}^{\underline{\theta}}(x)| + \Vert D\tilde{\Psi}_{n+1}^{\underline{\theta}}- D{\Psi}_{n+1}^{\underline{\theta}} \Vert \cdot |Dk_n ^{\underline{\theta}}(x)| \\ &\quad \leq (1+C_4 \mu^{-n \varepsilon}) \cdot|D\tilde{k}_{n}^{\underline{\theta}}(x)-D{k}_{n}^{\underline{\theta}}(x)| + 2 {C_4} {e^{C_5}} \mu^{-n \varepsilon}, \end{align*} $$

where $\tilde {\Psi }^{\underline {\theta }}_n$ is defined on $k^{\underline {\theta }}_n(G(\theta _0))$ because, as shown previously, we can enlarge a bit the domain of this function and $k^{\underline {\theta }}_n$ is sufficiently close to $\tilde {k}^{\underline {\theta }}_n$ ; and we use the estimate $\lVert D\tilde {\Psi }_{n+1}^{\underline {\theta }}-{\textrm {Id}}\rVert \le C_4 \mu ^{-n \varepsilon }$ . This estimate was proved for the unperturbed map in Lemma 3.1 (equation (3.3)), and is also true for the perturbation, maybe by enlarging $C_4$ a little, because this constant depends continuously on the map g defining the Cantor sets. The proof is completed proceeding as above.

Proof. From Lemma 3.1, in which we established the existence of limit geometries, we have that

$$ \begin{align*} k^{\underline{\theta}\theta_1} \circ (k^{\underline{\theta}}\circ f_{\theta_0, \theta_1})^{-1} & = \lim_{n \to \infty} k^{\underline{\theta}\theta_1}_{n+1} \circ (k^{\underline{\theta}}_{n}\circ f_{\theta_0, \theta_1})^{-1} \\ & = \lim_{n \to \infty} \Phi_{{(\underline{\theta}\theta_1})_{n+1}} \circ f_{{(\underline{\theta}\theta_1)_{n+1}}} \circ (\Phi_{\underline{\theta}_n} \circ f_{{\underline{\theta}_n}}\circ f_{\theta_0, \theta_1})^{-1} \\ & = \lim_{n \to \infty} \Phi_{{(\underline{\theta}\theta_1})_{n+1}} \circ \Phi_{\underline{\theta}_n} ^{-1} \end{align*} $$

which implies that the last limit exists and in particular belongs to $\textrm{Aff}(\mathbb {C})$ because this is a closed subset of the space of configurations. So, for any $\underline {\theta } = ( \ldots , \theta _{-1}, \theta _0) \in \Sigma ^- $ and $(\theta _0, \theta _1) \in B$ , we define $(F^{\underline {\theta } \theta _1}) ^{-1} = \lim _{ n \to \infty } \Phi _{{(\underline {\theta }\theta _1})_{n+1}} \circ \Phi _{\underline {\theta }_n} ^{-1} $ and we have that $F^{\underline {\theta } \theta _1} \circ k^{\underline {\theta }\theta _1} = k^{\underline {\theta }} \circ f_{\theta _0,\theta _1}$ as we wanted to show.

Claim 3.9. Let $\gamma : U \subset \mathbb {R}^2 \to \mathbb {R}^2$ be a $C^{1+\varepsilon }$ map defined on an open set U and X be a subset of U such that its convex hull is contained inside U. There is some constant $C'>0$ such that $\lvert D\gamma (x) - D\gamma (y) \rvert < C' \lvert x - y \rvert ^{\varepsilon } $ . Again, it is a simple observation from real analysis that

$$ \begin{align*}|\gamma(z+\delta) -(\gamma(z)+D\gamma(z) \cdot \delta)|< C' \cdot |\delta|^{1+\varepsilon}; \, z, \quad z+\delta \in X.\end{align*} $$

Let $P, \, Q \in \mathrm{Aff}(\mathbb {R}^2)$ be two affine maps, p be a point in X, and $\Gamma \in \mathrm{Aff}(\mathbb {R}^2)$ be the affine map defined by $\Gamma (z) = D\gamma (p)\cdot (z-p)+\gamma (p)$ . Suppose that P and $Q^{-1}$ expand distances by a factor comparable to $\mathrm{diam}(X)$ . This means there is some constant $\tilde {C}> 0$ such that $\tilde {C}^{-1}\mathrm{diam}(X) \le m(DP) \le \tilde {C}\mathrm{diam}(X) $ , and the same inequality is true with $\lvert DP \rvert $ , $\lvert DQ \rvert $ , and $m(DQ)$ instead of $m(DP)$ . Then there is some constant $C> 0$ depending only on $\tilde {C}$ and $C'$ such that $P\circ \gamma \circ Q$ is $C \cdot \mathrm{diam}(X)^{\varepsilon }$ - $C^{1+\varepsilon }$ close to the affine map $P \circ \Gamma \circ Q$ on the domain $Q^{-1}(X)$ .

Proof of Lemma 3.8. The expansion term of $F^{\underline {\theta }\underline {a}_n}$ , $DF^{\underline {\theta }\underline {a}_n}$ , has norm equal to

$$ \begin{align*} \dfrac{\text{diam}(k^{\underline{\theta}}(G(\underline{a}_n)))} {\text{diam}(k^{\underline{\theta}\underline{a}_n}(G(a_n)))}, \end{align*} $$

because of the relation $F^{\underline {\theta }\underline {a}_n} \circ k^{\underline {\theta }\underline {a}_n}= k^{\underline {\theta }} \circ f_{\underline {a}_n}$ applied to the domain $G(a_n)$ and the fact that $f_{\underline {a}_n}(G(a_n)) = G(\underline {a}_n)$ . Because the maps $k^{\underline {\theta }}$ , $\underline {\theta }\in \Sigma ^-$ , have a uniformly bounded derivative, there is a constant $C> 0$ such that the expansion term of $F^{\underline {\theta }\underline {a}_n}$ is less than $C \cdot \textrm {diam} (G(\underline {a}_n))$ and more than $C^{-1} \cdot \textrm {diam} (G(\underline {a}_n))$ .

However, because $h_n = h \circ (k^{\underline {\theta }})^{-1} \circ k^{\underline {\theta }} \circ f_{\underline {a}_n} = h \circ (k^{\underline {\theta }})^{-1} \circ F^{\underline {\theta }\underline {a}_n} \circ k^{\underline {\theta }\underline {a}_n}$ ,

$$ \begin{align*} DA_{h_n}^{-1} = Dh((k^{\underline{\theta}})^{-1}\circ F^{\underline{\theta}\underline{a}_n}(0))\cdot D(k^{\underline{\theta}})^{-1}(F^{\underline{\theta}\underline{a}_n}(0))\cdot DF^{\underline{\theta}\underline{a}_n}, \end{align*} $$

and $m( DA_{h_n}^{-1} )$ and $|DA_{h_n}^{-1}|$ are also controlled by $\textrm {diam} (G(\underline {a}_n))$ in the same way. Now, by Remark 3.7, the domain of interest of $h^{\underline {\theta }}$ on the relation

$$ \begin{align*} h^{\underline{\theta}\underline{a}_n}_n= h_n \circ (k^{\underline{\theta a}_n})^{-1} = h \circ f_{\underline{a}_n} \circ (k^{\underline{\theta a}_n})^{-1} = h \circ (k^{\underline{\theta}})^{-1} \circ F^{\underline{\theta}\underline{a}_n} = h^{\underline{\theta}} \circ F^{\underline{\theta}\underline{a}_n} \end{align*} $$

is the set $F^{\underline {\theta }\underline {a}_n} \circ k^{\underline {\theta }\underline {a}_n}(G(a_n)) = k^{\underline {\theta }} (G(\underline {a}_n))$ , whose size is also controlled by $\textrm {diam} (G(\underline {a}_n))$ . Because the $(k^{\underline {\theta }})^{-1}$ vary continuously with $\underline {\theta }$ in a compact set, the maps $h^{\underline {\theta }}$ are $C^{1+\varepsilon }$ and, if we enlarge $C> 0 $ ,

$$ \begin{align*} \lvert Dh^{\underline{\theta}}(x)-Dh^{\underline{\theta}}(y) \rvert \le C \cdot |x-y|^{\varepsilon}, \end{align*} $$

for all $\underline {\theta } \in \Sigma ^{-}$ . Then, using Claim 3.9 with $\gamma = h^{\underline {\theta }}$ , $P = A_{h_n}$ , $Q = F^{\underline {\theta }\underline {a}_n}$ , and $X = k^{\underline {\theta }} (G(\underline {a}_n))$ and remembering that $A_{h_n} \circ {h_n}^{\underline {\theta }\underline {a}_n}(0) = 0$ and $D(A_{h_n} \circ {h_n}^{\underline {\theta }\underline {a}_n})(0) = {\textrm {Id}}$ , we have

$$ \begin{align*} \Vert A_{h_n} \circ {h_n}^{\underline{\theta}\underline{a}_n}-{\textrm{Id}}\Vert = \Vert A_{h_n} \circ {h}^{\underline{\theta}} \circ F^{\underline{\theta}\underline{a}_n}-{\textrm{Id}}\Vert < C \cdot \text{diam}(G(\underline{a}_n))^{\varepsilon}. \end{align*} $$

The exponential decay of ratio $\mu $ is a consequence of Corollary 3.4.

Proof. If $h_0(K)$ and $h^{\prime }_0(K')$ are intersecting at a point $q= h_0(p) = h^{\prime }_0 (p')$ , ( $p \in K $ and $p' \in K'$ ), consider the sequences $H(p) = (a_0, a_1, \ldots ) \in \Sigma $ and $H'(p)=(a^{\prime }_0,a^{\prime }_1, \ldots ) \in \Sigma '$ , where H and $H'$ are the homeomorphisms defined in §2.1. We can construct a sequence of pairs of configurations $(h_n, h^{\prime }_n)$ , obtained by successively renormalizing by the functions $f_{(a_{i},a_ {i+1})}$ and $f_{(a^{\prime }_{j},a^{\prime }_{j+1})}$ , $i,\, j \ge 0$ , chosen in a careful order such that the ratio of diameters of the sets $G(\underline {b}_n)$ and $G(\underline {b}^{\prime }_n)$ are bounded away from $0$ and $\infty $ , where $(h_n,h^{\prime }_n) = (h\circ f_{\underline {b}_n}, h' \circ f_{\underline {b}^{\prime }_n})$ , $\underline {b}_n=(a_0,\,\ldots ,\,a_{r_n})$ , and $\underline {b}^{\prime }_n=(a^{\prime }_0,\,\ldots ,\,a^{\prime }_{r^{\prime }_n})$ (notice that $r_n \le n$ and ${r^{\prime }_n \le n}$ ).

Indeed, if $\underline {a}_n = (a_0,\,\ldots ,a_n)$ , by equation (3.2) of Lemma 3.1 and Corollary 3.2, $\text {diam}(G(\underline {a}_n))$ is comparable to $\text {diam}(G(\underline {a}_{n-1}))$ . Therefore, if we choose carefully which coordinate to act trivially, we can keep the ratios bounded.

Finally, such pairs of configurations are always intersecting, because the point q belongs to both their images. Using Claim 3.9, with $\gamma = h_0 \circ (k^{\underline {\theta }})^{-1}$ , $X = G(\underline {b}_n)$ , $P = A_{h_n}$ , $Q = F^{\underline {\theta }\underline {b}_n}$ ; and with $\gamma = h^{\prime }_0\circ (k^{\underline {\theta }'})^{-1}$ , $X = G(\underline {b}^{\prime }_n)$ , $P = A_{h_n}$ , $Q = F^{\underline {\theta }'\underline {b}^{\prime }_n}$ , in a similar way as done in the proof of Lemma 3.8, it follows that the sequence $[h_n, h^{\prime }_n]$ is relatively compact in $\mathcal {Q}$ .

However, let $[h_n,h^{\prime }_n]$ be such a relatively compact sequence. We can consider $(h_n, h_n') = \mathcal {T}_{\underline {b}_n,\underline {b}^{\prime }_n} (h_0,h^{\prime }_0)$ . Then $(h_n,h^{\prime }_n)$ is linked, because if it was not, the distance between the images of their scaled versions, $(A_{h_n}\circ h_n, A_{h_n}\circ h^{\prime }_n )$ , would go to infinity as $n \rightarrow \infty $ . Hence, choosing points $p_n \in h_n(K) \subset h_0(K)$ and $p^{\prime }_n \in h^{\prime }_n(K') \subset h^{\prime }_0(K')$ , it follows that $\lim _{n \rightarrow \infty } p_n = p = \lim _{n \rightarrow \infty } p^{\prime }_n $ , because the diameter of the sets $h_n(K)$ and $h^{\prime }_n(K')$ converge exponentially to $0$ as $n \rightarrow \infty $ , as they are controlled by $\text {diam}(G(\underline {b}_n))$ and $\text {diam} (G'(\underline {b}^{\prime }_n))$ , respectively, and they are always linked. Because K and $K'$ are closed, $p \in h_0(K) \cap h^{\prime }_0(K') \neq \emptyset $ as we wanted to show.

Proof. (1) We remember that $\mathcal {A}$ is a metric space, so for every point v of the recurrent compact set $\mathcal {L}$ , there is a closed ball around v that is carried by a renormalization (given by a pair of words $\underline {a}^{v}, \, {\underline {a}'}^{v}$ ) into the interior of $\mathcal {L}$ . Because this set is compact and by continuity of the renormalization operator, there is a finite number N of compact sets (balls) $\mathcal {L}^{i}$ whose union covers $\mathcal {L}$ and associated pair of words $(\underline {a}^{i},{\underline {a}'}^{i})$ for $1 \leq i \leq N$ such that $\mathcal {L}^i$ is carried into the interior of $\mathcal {L}$ by the renormalization associated to the pair $(\underline {a}^{i},{\underline {a}'}^{i})$ . Now, consider for every such pair, all the pairs of words $(\underline {b}^{i,j},{\underline {b}'}^{i,j})$ such that $\underline {b}^{i,j}$ is a subword of $\underline {a}^{i}$ and ${\underline {b}'}^{i,j}$ is a subword of ${\underline {a}'}^{i}$ . We construct an immediately recurrent Cantor set $\mathcal {L}'$ in the following way.

First, we choose one of the two pairs $(\underline {b}^{i,j_1},{\underline {b}'}^{i,j_1})$ with total size $1$ sharing the same beginning with $(\underline {a}^i,{\underline {a}'}^i)$ and write $(\underline {b}^{i,\overline {j_1}},{\underline {b}'}^{i,\overline {j_1}})$ as the word pair that needs to be concatenated to $(\underline {b}^{i,j_1},{\underline {b}'}^{i,j_1})$ to result in $(\underline {a}^{i},{\underline {a}'}^{i})$ . By continuity, there is a compact set $\mathcal {L}^{i(1)}$ such that ${\textrm {int}}\, \mathcal {L}^{i(1)} \supset T_{(\underline {b}^{i,j_1},{\underline {b}'}^{i,j_1})} \mathcal {L}^i $ and $T_{(\underline {b}^{i,\overline {j_1}},{\underline {b}'}^{i,\overline {j_1}})} \mathcal {L}^{i(1)} \subset {\textrm {int}}\, \mathcal {L} $ . This can be done in the same manner the sets $\mathcal {L}^i$ were constructed just above.

Then we inductively construct a sequence of compact sets $\mathcal {L}^{i(k)}$ , $k=1, \ldots , m$ such that for each $\mathcal {L}^{i(k)}$ , there is a renormalization by a pair of words of total size one that carries it into ${\textrm {int}}\, \mathcal {L}^{i(k+1)}$ for $k= 1, \ldots , m-1$ and carries $\mathcal {L}^{i(m)}$ into ${\textrm {int}}\, \mathcal {L}$ , where m is the total size of $(\underline {a}^i,{\underline {a}'}^i)$ . Then, taking $\mathcal {L}' = \bigcup _{i(k)} \mathcal {L}^{i(k)}$ , we have an immediately recurrent compact set.

(2) Using the decomposition $\mathcal {L} = \cup ^{N}_{i=1} \mathcal {L}_i$ , described in the previous argument, it is enough to show that for any $\varepsilon>0$ , there is $\delta> 0$ such that, for any pair $(\tilde {g}, \tilde {g}')$ in $U_{K,\delta } \times U_{K',\delta } $ , the renormalization operators associated to the words $\underline {a}^{i} \text { and } {\underline {a}'}^{i}$ , $i=1, \ldots , N$ , denoted by $\tilde {\mathcal {T}}_{\underline {a}^{i},{\underline {a}'}^{i}}=(\tilde {T}_{\underline {a}^{i}},\tilde {T}_{{\underline {a}'}^{i}})$ , satisfy $ \Vert \tilde {\mathcal {T}}-\mathcal {T}\Vert < \varepsilon $ .

These operators are obtained by composition of a finite number of operators arising from pairs of words of total size one, and so we need only to show that for any $\varepsilon '>0$ , a $\delta> 0$ can be found such that $|T_{(a,b)}-\tilde {T}_{(a,b)}|<\varepsilon '$ and $|T_{(a',b')}-\tilde {T}_{(a',b')}|<\varepsilon '$ for all pairs $(a,b) \in B$ and $(a',b') \in B'$ , or precisely, by Lemma 3.6, that for any $\underline {\theta } \in \Sigma ^-$ and $\theta _1 \in \mathbb {A}$ , $|F^{\underline {\theta }\theta _1} -\tilde {F}^{\underline {\theta }\theta _1} | < \varepsilon '$ , and its analogous version for $K' \text { and } \tilde {K}'$ . However, as seen in Lemma 3.6, $F^{\underline {\theta }\theta _1}=k^{\underline {\theta }} \circ f_{\theta _0,\theta _1} \circ (k^{\underline {\theta }\theta _1})^{-1}$ , and we need only to show that the values of all the functions and their derivatives above at a fixed point x do not change much when considering its $ \tilde {K} $ version. This is was done in Lemma 3.5.

(3) Given a recurrent compact set $\mathcal {L}$ relative to a pair of Cantor sets K and $K'$ , its image under

$$ \begin{align*} I : \mathcal{C} & \to \mathcal{Q} \\ [(A,\underline{\theta}),(A',\underline{\theta}')] & \mapsto [A \circ k^{\underline{\theta}},A' \circ k^{\underline{\theta}'}] \end{align*} $$

is also a compact set, because this association is continuous. In what follows, whenever we work with the set $\mathcal {L}$ in the context of $\mathcal {Q}$ , we are referring to the set $I(\mathcal {L})$ .

In this sense, any pair $(A\circ k^{\underline {\theta }},{A'}\circ {k}^{\underline {\theta }'})$ representing a relative affine configuration of limit geometries v belonging to $\mathcal {L}$ is intersecting, because, considering it a part of an immediately recurrent compact set, one can construct a sequence as in the hypothesis of Lemma 3.11. In light of the previous item, this implies that $A \circ \tilde {k}^{\underline {\theta }}$ and $A' \circ \tilde {{k}}^{\underline {\theta }'}$ also represent intersecting configurations for a pair of Cantor sets $(\tilde {K},\tilde {K'})$ sufficiently close to $(K,K')$ .

Thus, it is enough to show that, for any pair of Cantor sets $(K,K')$ that has a recurrent compact set $\mathcal {L}$ and a configuration pair $v \in \mathcal {L}$ represented by $[(A,\underline {\theta }),(A',\underline {\theta }')]$ , if $h : {\textrm {Im}}(A \circ k^{\underline {\theta }}) \to \mathbb {C}$ and $h' : {\textrm {Im}}(A \circ k^{\underline {\theta }'}) \to \mathbb {C}$ are embeddings $C^{1+\varepsilon }$ close to the identity, then $h \circ A \circ k^{\underline {\theta }}$ and $h' \circ A' \circ k^{\underline {\theta }'}$ are also intersecting.

To accomplish this, we will construct a relatively compact sequence of relative configurations $[h_n,h^{\prime }_n] \in \mathcal {Q}$ , with $[h_0,h^{\prime }_0] = [h \circ A \circ k^{\underline {\theta }}, h' \circ A' \circ k^{\underline {\theta }'}]$ , obtained inductively by renormalization. If we have such a sequence, the result follows from Lemma 3.11 again. We are going to represent each equivalence class $[h_n, h^{\prime }_n]$ by its representative that is scaled on the first coordinate (remember Definition 3.3). We are also interested on how ‘far’ each term of this sequence is from being in $\mathcal {L}$ , thus we write

$$ \begin{align*}[h_n,h^{\prime}_n] = [ \eta_n \circ k^{\underline{\theta}(n)}, \eta^{\prime}_n \circ B_n \circ k^{\underline{\theta}'(n)} ]\end{align*} $$

for some limit geometries

$$ \begin{align*}\underline{\theta}(n) &= (\ldots,\, \theta_{-n},\,\ldots,\, \theta_0, \,\ldots,\,\theta_{r_n})\in \Sigma^- \quad \text{ and }\\ \underline{\theta}'(n) &= (\ldots,\, \theta^{\prime}_{-n},\,\ldots,\, \theta^{\prime}_0, \,\ldots, \,\theta^{\prime}_{r^{\prime}_n} ) \in {\Sigma'}^-\end{align*} $$

and in what follows, we will show that these choices can be done so that $\eta _n$ and $\eta ^{\prime }_n$ are maps $C^1$ close to the identity in their domains, $ k^{\underline {\theta }(n)}(G(\theta _{n_r})) $ and $ k^{\underline {\theta }'(n)}(G(\theta _{n^{\prime }_r})) $ , respectively, $B_n \in \textrm{Aff}(\mathbb {C})$ and $ [ k^{\underline {\theta }(n)}, B_n \circ k^{\underline {\theta }'(n)} ] \in \textrm{int}\, \mathcal {L}$ for all $n \ge 1$ . If this is the case for all $n \ge 1$ , then the sequence is relatively compact and the proof is complete.

To prove the estimates for $\eta _n$ and $\eta ^{\prime }_n$ for large values of n, we are going to use Claim 3.9 many times. For $n=0$ , we make $\underline {\theta }(0)=\underline {\theta }$ and $\underline {\theta }'(0)=\underline {\theta }'$ , and so $\eta _0$ is the perturbed part of the scaled version of $h \circ A \circ k^{\underline {\theta }}$ relative to $k^{\underline {\theta }(0)} = k^{\underline {\theta }}$ ; in other words, if $A(z)=\alpha \cdot z + \beta $ and we denote $\Gamma _0(z)=h(\beta )+Dh(\beta )\cdot (z-\beta )$ ,

$$ \begin{align*} \eta_0 = A^{-1} \circ \Gamma_0^{-1} \circ h \circ A. \end{align*} $$

Moreover, if we denote $A'(z)=\alpha ' \cdot z+ \beta '$ and $\Gamma ^{\prime }_0(z)=h'(\beta ')+Dh'(\beta ')\cdot (z-\beta ')$ , then

$$ \begin{align*} \eta^{\prime}_0 \circ B_0 = A^{-1} \circ {\Gamma^{\prime}_0}^{-1} \circ h' \circ A'. \end{align*} $$

It follows from continuity that if h and $h'$ are sufficiently close to the identity in their domains, then $\eta _0$ is close to the identity and $\eta ^{\prime }_0 \circ B_0$ is close to $A^{-1} \circ A'$ .

Therefore, for $n=0$ , we make $\underline {\theta }(0)=\underline {\theta }$ , $\underline {\theta }'(0)=\underline {\theta }'$ , $B_0=A^{-1} \circ A'$ , $\eta _0 = A^{-1} \circ \Gamma _0^{-1} \circ h \circ A$ , and $\eta ^{\prime }_0 = A^{-1} \circ {\Gamma ^{\prime }_0}^{-1} \circ h' \circ A$ ; and the base of our construction is complete.