---
abstract: 'In this paper, we study conjugacy invariants for 2-dimensional diffeomorphisms with homoclinic cubic tangencies (two-sided tangencies of the lowest order) under certain open conditions. Ordinary arguments used in past studies of conjugacy invariants associated with one-sided tangencies do not work in the two-sided case. We present a new method which is applicable to the two-sided case.'
address: 'Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, JAPAN'
author:
- Shinobu Hashimoto
title:  Moduli of surface diffeomorphisms with cubic tangencies
---

Introduction {#Intro}
============

Let $M$ be a closed surface and $\mathrm{Diff}^r(M)$ the space of $C^r$-diffeomorphisms with $C^r$-topology. Suppose that $f_i$ $(i=0,1)$ are elements of $\mathrm{Diff}^2(M)$ with two saddle fixed points $p_i$, $q_i$. We consider the case where $W^u(p_i)$ and $W^s(q_i)$ have a quadratic heteroclinic tangency $r_i$ and there exists a homeomorphism $h:M\to M$ with $f_1=h\circ f_0\circ h^{-1}$, $h(p_0)=p_1$, $h(q_0)=q_1$ and $h(r_0)=r_1$. Then, under some moderate conditions, Palis [@pa] proved that $\dfrac{\log |\lambda_0|}{\log |\mu_0|}= \dfrac{\log |\lambda_1|}{\log |\mu_1|}$, where $\lambda_i$ is the contracting eigenvalue of $Df(p_i)$ and $\mu_i$ is the expanding eigenvalue of $Df(q_i)$. Such topological conjugacy invariants are often called *moduli*. Following his result, de Melo [@dm] studied moduli of stability of two-dimensional diffeomorphisms $f$, that is, a minimal set of moduli which parametrize topological conjugacy classes of a neighborhood of $f$ in $\mathrm{Diff}^r(M)$. He detected moduli of stability for some classes of two-dimensional diffeomorphisms. In [@dm], he also showed that the restriction of the conjugacy homeomorphism $h$ on $W^u(p_0)\setminus \{p_0\}$ is a local diffeomorphism if $\dfrac{\log |\lambda_0|}{\log |\mu_0|}$ is irrational. Subsequently, Posthumus [@po] proved that the homoclinic version of the results due to Palis and de Melo. In fact, he proved that, if $f_i$ $(i=0,1)$ has a saddle fixed point $p_i$ with a homoclinic quadratic tangency $r_i$, then $\dfrac{\log |\lambda_0|}{\log |\mu_0|}= \dfrac{\log |\lambda_1|}{\log |\mu_1|}$ holds, where $\lambda_i,\mu_i$ are the contracting and expanding eigenvalues of $Df(p_i)$. Moreover, if $\dfrac{\log |\lambda_0|}{\log |\mu_0|}$ is irrational, then the eigenvalues are also moduli, that is, $\lambda_0=\lambda_1$ and $\mu_0=\mu_1$. Various results related to moduli concerning eigenvalues are obtained by some authors; see for example [@dmp; @dmvs; @pot; @gpvs]. However, in all of these results, the assumption that the tangency is quadratic or one-sided is crucial. In fact, some of their arguments do not work in the case that $q$ is a two-sided tangency, see Remark \[r\_one\_sided\] for the reason.

Here we consider surface diffeomorphisms of $C^3$-class with cubic homoclinic tangencies, which are two-sided tangencies of lowest order. Bifurcations of such a tangency is one of typical subjects in 2-dimensional dynamics, see for example [@kky; @ks1]. Moreover, Kiriki and Soma [@ks2] presented infinitely many (original) Hénon maps which have homoclinic cubic tangencies. In this paper, we will prove some results for surface diffeomorphisms with cubic homoclinic tangencies corresponding to those for diffeomorphisms with one-sided homoclinic tangencies in [@pa; @dm; @pot] . As far as the author knows, this paper is the first attempt to study moduli with respect to two-sided homoclinic tangencies. Our argument works if the expanding eigenvalue of $Df(p)$ is sufficiently small, where $p$ is a saddle fixed point with a homoclinic cubic tangency.

\[thm\_A\] Suppose that $M$ is a closed surface with Riemannian metric. Let $f_i$ $(i=0,1)$ be elements of $\mathrm{Diff}^3(M)$ each of which has a saddle fixed point $p_i$ and a homoclinic cubic tangency $q_i$ associated with $p_i$ and satisfies the following conditions.

1.  \[A1\] For $i=0,1$, there exists a neighborhood $U(p_i)$ of $p_i$ in $M$ such that $f|_{U(p_i)}$ is linear.

2.  \[A2\] $f_0$ is topologically conjugate to $f_1$ by a homeomorphism $h:M\to M$ with $h(p_0)=p_1$ and $h(q_0)=q_1$.

3.  \[A3\] Each $f_i$ $(i=0,1)$ satisfy the small expanding condition and one of the adaptable conditions with respect to $(p_i,q_i)$ in Section \[S\_adaptable\].

Then and $\eqref{M2}$ hold, where $\lambda_i$, $\mu_i$ are the eigenvalues of $Df_0(p_i)$ with $0<|\lambda_i|<1<|\mu_i|$.

1.  \[M1\] $\dfrac{\log |\lambda_0|}{\log |\mu_0|}=\dfrac{\log |\lambda_1|}{\log |\mu_1|}$.

2.  \[M2\] Moreover, if $\dfrac{\log |\lambda_0|}{\log |\mu_0|}$ is irrational, then $\mu_0=\mu_1$ and $\lambda_0=\lambda_1$.

Here we say that $f_0$ satisfies the *small expanding condition* at $p_0$ if $|\mu_0|=
1+\varepsilon$ with $0<\varepsilon<\varepsilon_0$ for the constant $\varepsilon_0$ given in Lemma \[l\_slow3’\]. Note that this condition depends on local expressions of $f_0$ such as near $p_0$ and near $f_0^{m_0}(q_0)$. In Section \[S\_pre\], we present a codimension two submanifold $\mathcal{C}$ of $\mathrm{Diff}^3(M)$ such that any element of $\mathcal{C}$ sufficiently close to $f_0$ also satisfies . In the case that $f$ is of class $C^\infty$, we know from Sternberg [@st] and Takens [@ta] that is an open dense condition in $\mathrm{Diff}^\infty(M)$.

Though we only consider the case of cubic tangencies, we believe that our method still works in the case of two-sided tangencies of higher order. So we propose the following question.

Is it possible to generalize our theorems to the case that diffeomorphisms have two-sided homoclinic tangencies of higher order?

We will finish the introduction by outlining the proof of the main theorem.

Let $f_0$ be a diffeomorphism satisfying the conditions of Theorem \[thm\_A\]. We may assume that $q_0$ and $r_0=\varphi(q_0)$ are contained in $W_{\mathrm{loc}}^u(p_0)$ and $W_{\mathrm{loc}}^s(p_0)$ respectively, where $\varphi=f_0^{m_0}$ for some positive integer $m_0$. For the proof of Theorem \[thm\_A\], we need to find out a useful connection between the eigenvalues $\mu_i$ and $\lambda_i$ for $i=0,1$. By applying Inclination Lemma, we have a sequence $\{\alpha_n^u\}$ of arcs in $W^u(p_0)$ which meet $W_{\mathrm{loc}}^s(p_0)$ transversely at single points $z_0\lambda_0^n$ and $C^3$-converge to a sub-arc of $W_{\mathrm{loc}}^u(p_0)$, see Figure \[fig\_alpha\_n\]. Then $\varphi(\alpha_n^u)$ contains an S-shaped arc $\gamma_{0,n}'$ framed by the rectangle $S_n$ as illustrated in Figure \[fig\_outline\].

We note that such arcs $\gamma_{0,n}'$ are subtle and vanish eventually as $n\to \infty$, see Figures \[fig\_Sn\] and \[fig\_rn\]. Since $h$ is not supposed to be smooth, one can not expect that $h$ sends $\gamma_n'$ to an S-shaped curve in $W^u(p_1)$. However Intersection Lemma (Lemma \[l\_intersection\]) shows that it actually holds, which is a key lemma in our argument. For the proof, we send $\gamma_{0,n}'$ to a curve $\widehat\gamma_1$ in a small neighborhood of $q_0$ by $f_0^{u_0}$ for some $u_0\in \mathbb{N}$ and pull it back near $r_0$ by $\varphi$. Repeating this process many times, one can amplify $\widehat\gamma_1$ and finally have a compressed S-shaped curve $\widehat\gamma_k$ near $q$ the diameter of which is substantial so that it can be distinguished by $h$. From this fact, we know that $h(\widehat\gamma_k)$ intersects a compressed S-shaped curve $\widehat\gamma_k^*$ in $W^u(p_1)$. It follows that there exists a sequence $\{r_n\}$ with $r_n\in \gamma_{0,n}'$ as illustrated in Figure \[fig\_outline\] such that $\bar r_n=h(r_n)$ is contained in the corresponding S-shaped curve $\bar\gamma_{0,n}'$ in $W^u(p_1)$. We note that the images of $r_n$, $\bar r_n$ by the orthogonal projections to the first coordinates are represented as $az_0\lambda_0^n+o(\lambda_0^n)$, $\bar a\bar z_0\lambda_1^n+o(\lambda_1^n)$ respectively for some non-zero constants $a$, $\bar a$. One can take subsequences $\{n(k)\}$, $\{m(k)\}$ of $\mathbb{N}$ such that $f_0^{m(k)}(r_{n(k)})$ converges to a point $x_0\in W_{\mathrm{loc}}^u(p_0)$. Then $f_1^{m(k)}(\bar r_{n(k)})$ also converges to $h(x_0)\in W_{\mathrm{loc}}^u(p_1)$. By using this fact, we will show that $\lim_{k\to \infty}\dfrac{m(k)}{n(k)}=-\dfrac{\log \lambda_0}{\log \mu_0}$ and $\lim_{k\to \infty}\dfrac{m(k)}{n(k)}=-\dfrac{\log \lambda_1}{\log \mu_1}$. This proves the assertion . The assertion is proved by together with standard arguments in [@dm; @po].

Preliminaries {#S_pre}
=============

Let $\{a_n\}$, $\{b_n\}$ be sequences with non-zero entries. Then $a_n\approx b_n$ means that $\dfrac{a_n}{b_n}\to 1$ as $n\to \infty$, and $a_n\sim b_n$ means that there exist constants $C$ and $C'$ independent of $n$ with $0<C'<1<C$ and satisfying $C' \leq \dfrac{a_n}{b_n}\leq C$ for any $n$. Suppose next that $\{a_n\}$, $\{b_n\}$ are sequences with non-negative entries. If there exists a constant $C'>0$ independent of $n$ and satisfying $a_n\leq C' b_n$ for any $n$, then we denote the property by $a_n\precsim b_n$.

Throughout the remainder of this paper, we suppose that $M$ is a closed connected surface and $f:M\to M$ is a $C^3$-diffeomorphism with a saddle fixed point $p$. Let $\mu$, $\lambda$ be the eigenvalues of $Df(p)$ with $$\label{eqn_mulambda}
0<|\lambda|<1<|\mu|.$$ Suppose moreover that $f$ is $C^3$-linearizable in a neighborhood $U(p)$ of $p$ in $M$. Then there exists a $C^3$-coordinate $(x,y)$ on $U(p)$ satisfying the following condition: $$\label{eqn_linear}
f(x,y)=(\mu x,\lambda y)$$ for any $(x,y)\in U(p)$. In particular, this implies that $p=(0,0)$, $W_{\mathrm{loc}}^u(p):=\{(x,y)\in U(p);\,y=0\}\subset W^u(p)$ and $W_{\mathrm{loc}}^s(p):=\{(x,y)\in U(p);\,x=0\}\subset W^s(p)$.

A non-transverse intersection point $q$ of $W^u(p)$ and $W^s(p)$ is called a *homoclinic tangency* associated with $p$. We fix a Riemannian metric on $M$ and denote the distance on $M$ induced from the metric by $d$. The tangency is of *order* $n$ if the limit $$\lim_{\begin{subarray}{l} w\in W_{\mathrm{loc}}^s(p),\\[2pt] w\to q\end{subarray}}\frac{d(w,W^u(p))}{[d(w,q)]^n}$$ exists and has non-zero value. See [@po Section 2] for the definition. If $n=2$ (resp. $n=3$), then the tangency $q$ is called *quadratic* (resp. *cubic*). Let $\mathcal{C}$ be the subspace of $\mathrm{Diff}^3(M)$ consisting of elements $f\in \mathrm{Diff}^3(M)$ satisfying the following conditions –.

1.  \[C1\] $f$ has a saddle periodic point $p$.

2.  \[C2\] There exists a homoclinic cubic tangency $q$ associated with $p$.

3.  \[C3\] $f$ satisfies the adaptable conditions in the sense of Section \[S\_adaptable\] with respect to $p,q$.

Note that $\mathcal{C}$ is a codimension two submanifold of $\mathrm{Diff}^3(M)$.

Let $q$ be a cubic tangency of $W^u(p)$ and $W^s(p)$. We assume that $q$ is contained in $W_{\mathrm{loc}}^u(p)  \subset U(p)$ if necessary replacing $q$ by $f^{-n}(q)$ with sufficiently large $n\in\mathbb{N}$. For the point $q$, there exists $m_0\in \mathbb{N}$ such that $r:=f^{m_0}(q)\in W_{\mathrm{loc}}^s(p) \subset U(p)$. Then one can rearrange the linearizing coordinate on $U(p)$ so that $q=(1,0)$, $r=(0,1)$. Moreover, we may suppose that $$U(p)=[-2,2]\times [-2,2],\ W_{\mathrm{loc}}^u(p)=[-2,2]\times \{0\},\ W_{\mathrm{loc}}^s(p)=\{0\}\times [-2,2].$$ Let $U(q)$, $U(r)$ be sufficiently small neighborhoods of $q$, $r$ in $U(p)$ respectively. Then the component $L^s(q)$ of $W^s(p)\cap U(q)$ containing $q$ is represented as $$L^s(q)= \{(x+1, y)\in U(q);\, y = v(x)\},$$ where $v$ is a $C^3$-function satisfying $$\label{eqn_vvv}
v(0) = v'(0) =v''(0) = 0\quad\text{and}\quad v'''(0)\neq 0.$$ Similarly, the component $L^u(r)$ of $W^u(p)\cap U(r)$ containing $r$ is represented as $$L^u(r)= \{(x, y+1)\in U(r);\, x = w(y)\},$$ where $w$ is a $C^3$-function satisfying $$\label{eqn_www}
w(0) = w'(0) =w''(0) = 0\quad\text{and}\quad w'''(0)\neq 0,$$ see Figure \[fig\_vw0\]

Recall that $q=(1,0)$, $r=(0,1)$ are cubic tangencies between $W^s(p)$ and $W^u(p)$ and $f^{m_0}(q)=r$ for some $m_0\in\mathbb{N}$. We set $f^{m_0}=\varphi$ for short. By and , $\varphi$ is represented in $U(q)$ as follows for some constants $a,b,c,d,e$. $$\label{eqn_varphi}
\varphi(x+1,y)=(ay+bxy+cx^3+H_1(x+1,y), 1+dx+ey+H_2(x+1,y)).$$ where $H_1$, $H_2$ are $C^3$-functions satisfying the following conditions. $$\label{eqn_H1H2}
\begin{split}
H_1(1,0)&=\partial_x H_1(1,0)=\partial_y H_1(1,0)=\partial_{xx} H_1(1,0)=\partial_{xy} H_1(1,0)\\
&=\partial_{xxx} H_1(1,0)= 0,\\
H_2(1,0)&=\partial_x H_2(1,0)=\partial_y H_2(1,0)=0.
\end{split}$$ Since $\varphi$ is a diffeomorphism, $$a,d\neq 0.$$ The fact that $q$ is a cubic tangency implies $$c\neq 0.$$ Here we put the following extra open condition. $$\label{EX1}
b\neq 0.$$

By and , the Jacobian matrix of $\varphi$ at $(x+1,y)$ is given as follows. $$\label{eqn_dphi}
\begin{split}
d\varphi_{(x+1,y)}&=
\begin{bmatrix} 
\,by+3cx^2+\partial_x H_1(x+1,y)  & a+bx+\partial_y H_1(x+1,y)\,\\ 
d+\partial_x H_2(x+1,y) & e+\partial_y H_2(x+1,y)
\end{bmatrix}\\
&=
\begin{bmatrix} 
\,by+3cx^2+o(x^2)+o(y)+O(xy)& a+bx+o(x)+O(y)\,\\ 
d+O(x)+O(y) & e+O(x)+O(y)
\end{bmatrix}.
\end{split}$$ Here we only consider the case satisfying the following condition, which belongs to Case $\mathrm{II}_{++}$ in Section \[S\_adaptable\]. $$\label{eqn_adaptable_0}
0<\lambda<1, \mu>1,a>0,b<0,c>0,d<0.$$ See Figure \[fig\_case0\] for the situation of $W_{\mathrm{loc}}^u(p)$ and $W_{\mathrm{loc}}^s(p)$ in the case of . Note that implies the extra condition .

One can set $\mu=1+\varepsilon$ for some $\varepsilon>0$. We only consider the case that $\varepsilon$ is sufficiently small.

Consider the rectangle $R_\varepsilon=[1+\varepsilon,(1+\varepsilon)^3]\times [0,\varepsilon^3]$ in $U(q)$. By , $$\label{eqn_vp(1+e)}
\begin{split}
\varphi(1+\varepsilon,0)&= (c\varepsilon^3+o(\varepsilon^3),1+d\varepsilon+o(\varepsilon)),\\
\varphi(1+\varepsilon,\varepsilon^3)&= ((a+c)\varepsilon^3+o(\varepsilon^3),1+d\varepsilon+o(\varepsilon)),\\
\varphi((1+\varepsilon)^3,0)&=(27c\varepsilon^3+o(\varepsilon^3), 1+3d\varepsilon+o(\varepsilon)),\\
\varphi((1+\varepsilon)^3,\varepsilon^3)&=((a+27c)\varepsilon^3+o(\varepsilon^3), 1+3d\varepsilon+o(\varepsilon)).
\end{split}$$ Let ${\rm pr}_x : U(p) \rightarrow W^u_{\mathrm{loc}}(p)$ and ${\rm pr}_y : U(p) \rightarrow W^s_{\mathrm{loc}}(p)$ be the orthogonal projections with respect to the linearizing coordinate on $U(p)$. Then there exist constants $\tau_0$, $\tau_1$ with $0<\tau_0<\tau_1$ independent of $\varepsilon$ and satisfying $$\label{eqn_prxR}
\mathrm{pr}_x(\varphi(R_\varepsilon))\subset [\tau_0\varepsilon^3,\tau_1\varepsilon^3].$$ Since $d<0$ by , it follows from that $$\label{eqn_prxR}
\mathrm{pr}_y(\varphi(R_\varepsilon))\subset [1+3.5d\varepsilon,1+0.5d\varepsilon]\subset [1+4d\varepsilon,1].$$

For any $\boldsymbol{x}\in R_\varepsilon$, let $u_0=u_0(\boldsymbol{x})$ be a uniquely determined positive integer such that $f^i(\varphi(\boldsymbol{x}))\in U(p)$ for $i=1,\dots,u_0$ and $\mathrm{pr}_x(f^{u_0}(\varphi(\boldsymbol{x})))
\subset ((1+\varepsilon)^2,(1+\varepsilon)^3]$. Since $\mathrm{pr}_x(f^{u_0}(\varphi(\boldsymbol{x})))=\mu^{u_0}\mathrm{pr}_x(\varphi(\boldsymbol{x}))$, $$1<(1+\varepsilon)^2 < \mu^{u_0}\mathrm{pr}_x(\varphi(\boldsymbol{x})) < \tau_1\mu^{u_0}\varepsilon^3.$$ Since $\mathrm{pr}_y(\varphi(\boldsymbol{x})) < 1$ by , it follows that $$\mathrm{pr}_y(f^{u_0}(\varphi(\boldsymbol{x})))=\lambda^{u_0}\mathrm{pr}_y(\varphi(\boldsymbol{x})) < \lambda^{u_0}.$$ Consider the following conditions for $\varepsilon > 0$: $$\tau_1 < \varepsilon^{-1}\quad\text{and}\quad (1+\varepsilon)^{\frac32} = \mu^{\frac32} < \lambda^{-1}.
\label{eqn_slow1}$$ If these conditions are satisfied, then the following inequalities $$\label{eqn_muu0}
1<1+\varepsilon < \mu^{u_0}\mathrm{pr}_x(\varphi(\boldsymbol{x})) < \tau_1\mu^{u_0}\varepsilon^3 < \mu^{u_0}\varepsilon^2$$ hold. This implies that $$\mathrm{pr}_y(f^{u_0}(\varphi(\boldsymbol{x})))=\lambda^{u_0}\mathrm{pr}_y(\varphi(\boldsymbol{x})) < \lambda^{u_0} < \mu^{-\frac{3}{2}u_0} < \varepsilon^3.$$ Thus the positive integer $u_0(\boldsymbol{x})$ satisfies $$\label{eqn_Rve}
f^{u_0(\boldsymbol{x})}(\varphi(\boldsymbol{x})) \in R_\varepsilon$$ for all $\boldsymbol{x} \in R_\varepsilon$. See Figure \[fig\_Re\].

Sequence of Rectangles {#S_Sbox}
======================

Let $f : M \rightarrow M$ be a $C^3$-diffeomorphism given in Section \[S\_pre\]. In particular, $f$ satisfies the linearizing condition on $U(p)$. It is not hard to show that $W^u(p)$ and $W^s(p)$ have a transverse intersection point other than $p$. See, for example, Lemma 1.2 in [@ks1]. Let $\delta^u$ be a segment in $W_{\mathrm{loc}}^u(p)$ with $\mathrm{Int}\delta^u \supset \{p,q\}$. Then, by Inclination Lemma, there exists a sequence $\{\alpha_n^u\}_{n=0}^\infty$ of arcs in $W^u(p)$ $C^3$-converging to $\delta^u$ and satisfying the following conditions:

-   $\alpha_0^u$ meets $W_{\mathrm{loc}}^s(p)$ transversely in a single point $\boldsymbol{z}_0=(0,z_0)$.

-   Each $\alpha_n^u$ contains $f^n(\boldsymbol{z}_0)=(0,z_0\lambda^n)$, and the intersection $\tilde\alpha_n^u=\alpha_n^u\cap U(q)$ is an arc meeting $L^s(q)$ transversely in a single point $c_n$ for any sufficiently large $n>0$.

See Figure \[fig\_alpha\_n\].

Note that $\alpha_0^u$ is represented by the graph of a $C^3$-function $y_0:\delta^u\to \mathbb{R}_+$, that is, $\alpha_0^u=\{(x,y_0(x))\,;\, x\in \delta^u\}$. Then each $\alpha_n^u$ is represented by the graph of the function $y_n:\delta^u\to \mathbb{R}_+$ with $$\label{eqn_psi}
y_n(x)=\lambda^n y_0(\mu^{-n}x)\quad\text{for}\quad x\in \delta^u.$$ We parametrise $\tilde\alpha^u_n$ in $[(1+\varepsilon)^{-3},(1+\varepsilon)^3]$ by $\alpha_n(t)=(t+1,\tilde y_n(t))$ with $
(1+\varepsilon)^{-3}-1\leq t\leq 
(1+\varepsilon)^3-1$, where $\tilde y_n(t)=y_n(t+1)$. By and , $$\begin{aligned}
\label{eqn_va}\varphi ( \alpha_n(t))&=
(a\tilde y_n(t)+bt\tilde y_n(t)+ct^3+\mathrm{h.o.t.},1+dt+e\tilde y_n(t)+\mathrm{h.o.t.})\\
\label{eqn_va'}d\varphi_{\alpha_n(t)}(\alpha_n'(t))&=
(a\tilde y_n'(t)+b\tilde y_n(t)+bt\tilde y_n'(t)+3ct^2+\mathrm{h.o.t.},\\
&\hspace{150pt} d+e\tilde y_n'(t)+\mathrm{h.o.t.}),\nonumber\end{aligned}$$ where the primes represent the derivative on $t$ and ‘h.o.t.’ denotes the sum of the higher order terms on $t$.

By , $$|\tilde y_n'(t)|=|y_n'(t+1)| = \lambda^n\mu^{-n}|y_0^{\prime}(\mu^{-n}(t+1))|.$$ Suppose that $\sigma$ is the maximum of $|y_0^{\prime}(x)|$ on $\delta^u$. Then $$|\tilde y_n'(t)|=|y_n'(t+1)| = \lambda^n\mu^{-n}|y_0^{\prime}(\mu^{-n}(t+1))| \le \lambda^n\mu^{-n}\sigma.$$ for any $n \in \mathbb{N}$. This implies that $$\label{eqn_y_n'}
|\tilde y_n'(t)| \precsim \lambda^n\mu^{-n}.$$ Suppose that $d\varphi_{\alpha_n(t)} (\alpha_n'(t))$ is vertical at $t=t_n$. Then $\lim_{n\to \infty}t_n=0$ and, by , $$b\tilde y_n(t_n)+(a+bt_n)\tilde y_n'(t_n)\approx -3ct_n^2.$$ Since $\tilde y_n(t)\approx \lambda^nz_0$ and $|\tilde y_n'(t)| \precsim \lambda^n\mu^{-n}$, this condition is equivalent to $$\label{eqn_3ctn}
3ct_n^2\approx -b\tilde y_n(t_n)\approx -b\lambda^nz_0.$$ It follows that, for all sufficiently large $n$, $d\varphi_{\alpha_n(t)} (\alpha_n'(t))$ is vertical at two points $t_{n,\pm}$ with $$\label{eqn_tn+-}
t_{n,\pm}\approx \pm\sqrt{\dfrac{-bz_0}{3c}}\lambda^{\frac{n}2}.$$ Let $\tilde t_{n,\pm}$ be the elements of $[(1+\varepsilon)^{-3}-1, (1+\varepsilon)^3-1]$ with $\tilde t_{n,-} < t_{n,-}$, $ t_{n,+} < \tilde t_{n,+}$ such that $\varphi(\alpha_n(\tilde t_{n,\pm}))$ is the intersection point of $\varphi(\alpha_n(t))$ and the vertical line $L_{n,\pm}$ tangent to $\varphi(\alpha_n(t))$ at $\varphi(\alpha_n(t_{n,\mp}))$. Let $S_n$ be the smallest orthogonal rectangle in $U(r)$ containing the four points $\varphi(\alpha_n(\tilde t_{n,-}))$, $\varphi(\alpha_n(t_{n,-}))$, $\varphi(\alpha_n(t_{n,+}))$, $\varphi(\alpha_n(\tilde t_{n,+}))$, see Figure \[fig\_Sn\].

Now we will estimate the size of $S_n$. Let $D_n$ be the distance between $S_n$ and $W_{\mathrm{loc}}^s(p)$. Then $$\label{eqn_SD}
\begin{split}
D_n&\approx 
a\tilde y_n(t_{n,+})+bt_{n,+}\tilde y_n(t_{n,+})+ct_{n,+}^3\\
&\approx
az_0\lambda^n+bz_0\sqrt{\frac{-bz_0}{3c}}\lambda^{\frac32 n}-\frac{bz_0}{3}\sqrt{\frac{-bz_0}{3c}}\lambda^{\frac{3}{2}n} \sim \lambda^n.
\end{split}$$ By , the width $W_{0,n}$ of $S_n$ is represented as $$\begin{aligned}
W_{0,n}&\approx (a\tilde y_n(t_{n,-})+bt_{n,-}\tilde y_n(t_{n,-})+ct_{n,-}^3)-(a\tilde y_n(t_{n,+})+bt_{n,+}\tilde y_n(t_{n,+})+ct_{n,+}^3)\\
&=a(\tilde y_n(t_{n,-})-\tilde y_n(t_{n,+}))+b(t_{n,-}\tilde y_n(t_{n,-})-t_{n,+}\tilde y_n(t_{n,+}))+c(t_{n,-}^3-t_{n,+}^3).\end{aligned}$$ It follows from Mean Value Theorem together with that $$|\tilde y_n(t_{n,-})-\tilde y_n(t_{n,+})|\precsim \lambda^n\mu^{-n}|t_{n,-} -t_{n,+}|\sim \lambda^{\frac{3}{2}n}\mu^{-n}.$$ Moreover, by , we have $$\begin{aligned}
c(t_{n,-}^3-t_{n,+}^3)&\approx  
c\left(t_{n,-}\left(\frac{-b\tilde y_n(t_{n,-})}{3c}\right)-t_{n,+}\left(\frac{-b\tilde y_n(t_{n,+})}{3c}\right)\right)\\
&=-\frac{b}3(t_{n,-}\tilde y_n(t_{n,-})-t_{n,+}\tilde y_n(t_{n,+})).\end{aligned}$$ Since $$\begin{aligned}
t_{n,-}\tilde y_n(t_{n,-})-t_{n,+}\tilde y_n(t_{n,+})&=(t_{n,-}-t_{n,+})\tilde y_n(t_{n,-})+t_{n,+}(\tilde y_n(t_{n,-})-y_n(t_{n,+}))\\
&\approx -\sqrt{\frac{-bz_0}{3c}}\lambda^{\frac{n}2}\cdot z_0\lambda^n
+O\left(\lambda^{\frac{n}2}\cdot \lambda^{\frac{3}{2}n}\mu^{-n}\right) \sim -\lambda^{\frac{3}{2}n},\end{aligned}$$ we have $$\label{eqn_SW}
\begin{split}
W_{0,n}\approx O(\lambda^{\frac{3}{2}n} \mu^{-n})+\frac{2b}{3} (t_{n,-}\tilde y_n(t_{n,-})-t_{n,+}\tilde y_n(t_{n,+})) \sim \lambda^{\frac{3}{2}n}.
\end{split}$$

Next we estimate the height $H_{0,n}$ of $S_n$. For that, we estimate $W_{0,n}$ again by using $\tilde t_{n,+}$ and $t_{n,+}$ instead of $t_{n,-}$ and $t_{n,+}$. Since $\tilde t_{n,+}>t_{n,+}$, one can set $\tilde t_{n,+}=t_{n,+}+\rho_n\lambda^{\frac{n}2}$ for some $\rho_n>0$. $$\begin{aligned}
W_{0,n}&\approx 
a(\tilde y_n(\tilde t_{n,+})-\tilde y_n(t_{n,+}))+
b(\tilde t_{n,+}\tilde y_n(\tilde t_{n,+})-t_{n,+}\tilde y_n(t_{n,+}))
+c(\tilde t_{n,+}^3-\tilde t_{n,-}^3)\\
&=(a+b\tilde t_{n,+})(\tilde y_n(\tilde t_{n,+})-\tilde y_n(t_{n,+}))
+b(\tilde t_{n,+}-t_{n,+})\tilde y_n(t_{n,+})+c(\tilde t_{n,+}^3-t_{n,-}^3).\end{aligned}$$ Again by Mean Value Theorem together with , $$|\tilde y_n(\tilde t_{n,+})-\tilde y_n(t_{n,+})| \precsim \lambda^n\mu^{-n}\cdot \rho_n\lambda^{\frac{n}2} 
=\rho_n\lambda^{\frac{3}{2}n}\mu^{-n}.$$ Moreover, we have $$\begin{aligned}
(\tilde t_{n,+}-t_{n,+})\tilde y_n(t_{n,+}) &\sim \rho_n\lambda^{\frac{n}2}\cdot \lambda^n =\rho_n\lambda^{\frac{3}{2}n}\end{aligned}$$ and $$\begin{aligned}
\tilde t_{n,+}^3-t_{n,+}^3&=3\rho_n^2\lambda^n t_{n,+}+3\rho_n\lambda^{\frac{n}2}t_{n,+}^2+\rho_n^3\lambda^{\frac{3}{2}n}\\
&\approx \left(3\rho_n\sqrt{\dfrac{-bz_0}{3c}}-\frac{3bz_0}{c}+\rho_n^2\right)\rho_n\lambda^{\frac{3}{2}n}.\end{aligned}$$ This shows that $$\begin{aligned}
W_{0,n}&\sim \left( a\mu^{-n} + bz_0 + 3\rho_n\sqrt{\dfrac{-bz_0}{3c}}-\frac{3bz_0}{c}+\rho_n^2\right)
\rho_n\lambda^{\frac{3}{2}n}.\end{aligned}$$ Since $W_{0,n} \sim \lambda^{\frac{3}{2}n}$, it follows that $\rho_n \sim 1$ and hence $\tilde t_{n,+}\sim \lambda^{\frac{n}2}$. Similarly $-\tilde t_{n,-}\sim \lambda^{\frac{n}2}$. This implies that $$|\tilde t_{n,\pm}|\sim \lambda^{\frac{n}2}.$$ Therefore we have $$\label{eqn_SH}
\begin{split}
H_{0,n}&=(1+d\tilde t_{n,-}+e\tilde y_n(\tilde t_{n,-})) - (1+d\tilde t_{n,+}+e\tilde y_n(\tilde t_{n,+}))\\
&=d(\tilde t_{n,-} - \tilde t_{n,+}) + e(\tilde y_n(\tilde t_{n,-})-\tilde y_n(\tilde t_{n,+})) \sim \lambda^{\frac{n}{2}}+O(\lambda^{\frac{3}{2}n}\mu^{-n}) \sim \lambda^{\frac{n}{2}}.
\end{split}$$ In particular, $\{S_n\}$ is a sequence of rectangles converging to the cubic tangency $r$.

Slope Lemma {#S_SLPL}
===========

Let $\boldsymbol{v}=\begin{bmatrix}u\\v\end{bmatrix}\in T_{\boldsymbol{x}}(M)$ be a tangent vector at $\boldsymbol{x}\in U(p)$ with $u\neq 0$. Then we say that $|vu^{-1}|$ is the (absolute) *slope* of $\boldsymbol{v}$ and denote it by $\mathrm{Slope}(\boldsymbol{v})$.

Consider any tangent vector $\boldsymbol{v}_0=\begin{bmatrix}1\\ \delta\end{bmatrix}\in T_{\boldsymbol{x}}(M)$ at $\boldsymbol{x}=(x+1,y)\in R_\varepsilon$ with $|\delta|\leq \varepsilon^{\frac{5}{2}}$. We set $\boldsymbol{v}_0'=d\varphi_{(x+1,y)}(\boldsymbol{v}_0)$ and $\boldsymbol{v}_1=df^{u_0}_{\varphi(x+1,y)} (\boldsymbol{v}_0')$. By , $$\mathrm{Slope}(\boldsymbol{v}_0')\approx \frac{|d+e\delta|}{|3cx^2+a\delta|}.$$ Since $\varepsilon \le x$ and $|\delta|\leq \varepsilon^{\frac{5}{2}}$, $$\begin{aligned}
\mathrm{Slope}(\boldsymbol{v}_0') &\approx \frac{|d+e\delta|}{|3cx^2+a\delta|} \leq \frac{|d|+|e\delta|}{|3cx^2|-|a\delta|} \leq \frac{|d|+|e\varepsilon^{\frac{5}{2}}|}{|3c\varepsilon^2|-|a\varepsilon^{\frac{5}{2}}|}\\
& = \frac{|d|+|e\varepsilon^{\frac{5}{2}}|}{|3c|-|a\varepsilon^{\frac{1}{2}}|}\varepsilon^{-2} = \frac{|d|+|e\varepsilon^{\frac{5}{2}}|}{|3c|-|a\varepsilon^{\frac{1}{2}}|}\varepsilon^{\frac{1}{2}}\cdot \varepsilon^{-\frac{5}{2}}.\end{aligned}$$ By taking $\varepsilon_1>0$ sufficiently small, for any $0 < \varepsilon \le \varepsilon_1$, we have $$\begin{aligned}
\mathrm{Slope}(\boldsymbol{v}_0') 
\le 2\frac{|d|+|e\varepsilon^{\frac{5}{2}}|}{|3c|-|a\varepsilon^{\frac{1}{2}}|}\varepsilon^{\frac{1}{2}}\cdot \varepsilon^{-\frac{5}{2}} 
\le \frac{|3d|}{|2c|}\varepsilon^{\frac{1}{2}}\cdot \varepsilon^{-\frac{5}{2}} 
\le 1 \cdot \varepsilon^{-\frac{5}{2}}.\end{aligned}$$ Then,by and , we have $$\begin{aligned}
\mathrm{Slope}(\boldsymbol{v}_1)&=\mathrm{Slope}(\boldsymbol{v}_0')\lambda^{u_0}\mu^{-u_0}
\leq \varepsilon^{-\frac52}\lambda^{u_0}\mu^{-u_0}
\leq \varepsilon^{-\frac52}\mu^{-\frac52 u_0}
\leq \varepsilon^{-\frac52}\varepsilon^5=\varepsilon^{\frac52}.\end{aligned}$$ Thus we get the following lemma. See Figure \[fig\_slope\].

\[l\_slope\] Suppose that $f$ satisfies the conditions . Then there exists a constant $\varepsilon_1>0$ such that, if $\varepsilon \in (0, \varepsilon_1]$, then $$\label{eqn_slow2}
\mathrm{Slope}(\boldsymbol{v}_0')\leq \varepsilon^{-\frac{5}{2}}\quad\text{and}\quad \mathrm{Slope}(\boldsymbol{v}_1)\leq \varepsilon^{\frac{5}{2}}$$ for any tangent vector $\boldsymbol{v}_0\in T_{\boldsymbol{x}}(M)$ at $\boldsymbol{x}=(x+1,y)\in R_\varepsilon$ with $\mathrm{Slope}(\boldsymbol{v}_0)\leq \varepsilon^{\frac{5}{2}}$.

Fix a sufficiently small $s>0$ and set $\mathrm{pr}_x(S_n)=[s^-_n,s^+_n]$ for $n\in \mathbb{N}$. If $n$ is sufficiently large, then $[s_n^-,s_n^+]\subset (0,s]$. Let $\beta_n^u(s)$ be the component of $\varphi(\alpha_n^u) \cap \mathrm{pr}_x^{-1}((0,s])$ containing $\varphi(\alpha_n([\tilde t_{n,-},\tilde t_{n,+}]))$. For any $\boldsymbol{x}\in \beta_n^u(s)$, let $j_n(\boldsymbol{x})$ be a positive integer such that $f^j(\boldsymbol{x})\in U(p)$ for $j=1,\dots,j_n(\boldsymbol{x})$ and $\mathrm{pr}_x(f^{j_n(\boldsymbol{x})}(\boldsymbol{x})) \in [1+\varepsilon,(1+\varepsilon)^3]$. For any $\varepsilon>0$, one can take $s$ so that $\mathrm{pr}_x(f^{j_n(\boldsymbol{x})}(\boldsymbol{x})) \in R_\varepsilon$ for any $\boldsymbol{x} \in \beta_n^u(s)$. Let $\boldsymbol{v}(\boldsymbol{x})$ be a unit vector tangent to $\beta_n^u(s)$ at $\boldsymbol{x}$.

The following result is applied to $f_1$ in the proof of Theorem \[thm\_A\].

\[l\_slopef1\] Let $\varepsilon_1$ be the constant given in Lemma \[l\_slope\]. For any $\varepsilon \in (0, \varepsilon_1]$, there exist $s>0$ and $n_0\in \mathbb{N}$ such that $$\mathrm{Slope}(df_{\boldsymbol{x}}^{j_n(\boldsymbol{x})}(\boldsymbol{v}(\boldsymbol{x})))<\varepsilon^{\frac{5}{2}}$$ if $n\geq n_0$ and $\boldsymbol{x}\in \beta_n^u(s)\setminus S_n$.

We only consider the case where $\boldsymbol{x}$ is an element of $\beta_n^u(s)\setminus S_n$ with $\mathrm{pr}_x(\boldsymbol{x})\geq s_n^+$. Then $t\geq \tilde t_{n,+}$ holds if $\varphi(\alpha_n(t))=\boldsymbol{x}$. The proof in the case of $\mathrm{pr}_x(\boldsymbol{x})\leq s_n^-$ is done similarly. Since $\rho_n\sim 1$ and $\tilde t_{n,+}=t_{n,+}+\rho_n\lambda^{\frac{n}{2}}$, $t-t_{n,+}\geq \tilde t_{n,+}-t_{n,+}\sim \lambda^{\frac{n}2}.$ This implies that $$\label{eqn_t-t}
t^2-t_{n,+}^2\sim t^2\succsim \lambda^n.$$ In fact, if $t-t_{n,+}\geq \frac{t}2$, then $t^2-t_{n,+}^2=(t-t_{n,+})(t+t_{n,+})>\frac{t^2}2$ and hence holds. On the other hand, if $t-t_{n,+}\leq \frac{t}2$, then $t\leq 2t_{n,+}$ and so $t\sim \lambda^{\frac{n}2}$. It follows that $t+t_{n,+}\sim \lambda^{\frac{n}2}$ and $t-t_{n,+}\sim \lambda^{\frac{n}2}$. Then $t^2-t_{n,+}^2\sim \lambda^n\sim t^2$. Thus holds.

We set $\xi_n(t)=\mathrm{pr}_x(\boldsymbol{x})=\mathrm{pr}_x(\varphi(\alpha_n(t)))$. By , $$\label{eqn_xixi}
\begin{split}
\xi_n(t)&=a\tilde y_n(t)+bt\tilde y_n(t)+ct^3+\mathrm{h.o.t.},\\
\xi_n'(t)&=
a\tilde y_n'(t)+b\tilde y_n(t)+bt\tilde y_n'(t)+3ct^2+\mathrm{h.o.t.}
\end{split}$$ From the definition of $j_n(\boldsymbol{x})$, $$\mu^{j_n(\boldsymbol{x})} \xi_n(t)= \mu^{j_n(\boldsymbol{x})} \mathrm{pr}_x(\boldsymbol{x})= \mathrm{pr}_x(f^{j_n(\boldsymbol{x})}(\boldsymbol{x})) \in [1+\varepsilon,(1+\varepsilon)^3].$$ This implies that $\mu^{j_n(\boldsymbol{x})}\xi_n(t)\sim 1$. We note that $\xi_n'(t_{n,+})=0$. By Mean Value Theorem, $\tilde y_n(t)-\tilde y_n(t_{n,+})=\tilde y_n'(c)(t-t_{n,+})$ for some $t_{n,+}<c<t$. From this fact together with , , and , we know that $$\xi_n'(t)=\xi_n'(t)-\xi_n'(t_{n,+})\sim t^2-t_{n,+}^2\sim t^2.$$ By , $\mathrm{Slope}(\boldsymbol{v}(\boldsymbol{x}))\sim t^{-2}$. Hence we have $$\label{eqn_slopedf}
\mathrm{Slope}(df_{\boldsymbol{x}}^{j_n(\boldsymbol{x})}(\boldsymbol{v}(\boldsymbol{x}))) 
=\mathrm{Slope}(\boldsymbol{v}(\boldsymbol{x})) \cdot \frac{\lambda^{j_n(\boldsymbol{x})}}{\mu^{j_n(\boldsymbol{x})}} 
\sim t^{-2}\lambda^{j_n(\boldsymbol{x})}\xi_n(t).$$ Now we need to consider the following two cases.

[*Case 1.*]{} $ct^3\leq a\tilde y_n(t)$. By , $\xi_n(t)\sim \lambda^n$. Since $t^{-2}\precsim \lambda^{-n}$ by $t\succsim \lambda^{\frac{n}2}$, it follows from that $$\mathrm{Slope}(df_{\boldsymbol{x}}^{j_n(\boldsymbol{x})}(\boldsymbol{v}(\boldsymbol{x}))) \precsim \lambda^{-n}\lambda^{j_n(\boldsymbol{x})}\lambda^n=
\lambda^{j_n(\boldsymbol{x})}.$$

[*Case 2.*]{} $ct^3\geq a\tilde y_n(t)$. Again by , we have $\xi_n(t)\sim t^3$. Then, by , $$\mathrm{Slope}(df_{\boldsymbol{x}}^{j_n(\boldsymbol{x})}(\boldsymbol{v}(\boldsymbol{x}))) 
\sim t^{-2}\lambda^{j_n(\boldsymbol{x})}t^3=t \lambda^{j_n(\boldsymbol{x})}\precsim \lambda^{j_n(\boldsymbol{x})}.$$

Let $n_0(s)$ be the minimum positive integer with $s_{n_0(s)}^+ < s$. Since $\lim_{s\to +0}n_0(s)=\infty$, one can take $s=s(\varepsilon)>0$ such that our desired inequality holds for any $\boldsymbol{x}\in \beta_n^u(s)\setminus S_n$.

Sequence of rectangle-like boxes {#S_Bbox}
================================

Now we will define a sequence $\{B_{k,n}\}_{k=1}^\infty$ of rectangle-like boxes and estimate the sizes of them.

Recall that $\mathrm{pr}_x(S_n) = [s_n^-,s_n^+]$. Let $i_n$ be the positive integer with $(1+\varepsilon)^2<\mu^{i_n}s_n^{+}\leq (1+\varepsilon)^3$. By , $f^{i_n}(S_n)$ is contained in $R_\varepsilon$ for any sufficiently large $n$. We set $f^{i_n}(S_n)=B_{1,n}=B_1$ for short. Since $s_n^{+} \sim \lambda^n$ by and , we have $$\label{eqn_innsim1}
\mu^{i_n}\lambda^n \sim 1.$$ We denote the width and height of $B_1$ and the distance between $B_1$ and $W^u_{\mathrm{loc}}(p)$ by $W_{1,n}=W_1$, $H_{1,n}=H_1$ and $L_{1,n}=L_1$ respectively. It follows from , and that $$\label{eqn_W_1}
W_{1,n}\sim \lambda^{\frac32 n}\mu^{i_n}\sim \lambda^{\frac{n}2},\quad
H_{1,n}\sim \lambda^{\frac{n}2+i_n},\quad L_{1,n}\sim \lambda^{i_n}.$$ Note that, for any sufficiently large $n$, $H_1\ll L_1\ll W_1$. Consider a closed interval $\delta_1$ in $W_{\mathrm{loc}}^u(p)$ which is a small neighborhood of $\mathrm{pr}_x(B_1)$.

Let $v_i^{(1)}$, $e_i^{(1)}$ ($i=0,1,2,3$) be the vertices and edges of $B_1$ as illustrated in Figure \[fig\_B1\](a).

We consider the image $\varphi(B_1)$. By Lemma \[l\_slope\], for $i=0,2$, $$\mathrm{diam}(\mathrm{pr}_x(\varphi(e_i^{(1)}))) \succsim \varepsilon^{\frac{5}{2}}W_1 \sim \varepsilon^{\frac{5}{2}}\lambda^{\frac{n}2}.$$ On the other hand, for $i=1,3$, $$\mathrm{diam}(\mathrm{pr}_x(\varphi(e_i^{(1)}))) \precsim H_1 \sim \lambda^{\frac{n}2+i_n}.$$ Since $\lambda^{i_n}\varepsilon^{-\frac{5}{2}}$ can be supposed to be arbitrarily small for all sufficiently large $n$, $$\label{eqn_x+x-}
\begin{split}
x_+^{(1)}-x_-^{(1)} &\succsim \varepsilon^{\frac{5}{2}}W_1-O(\lambda^{\frac{n}2+i_n}) \sim \varepsilon^{\frac{5}{2}}\lambda^{\frac{n}{2}}-O(\lambda^{\frac{n}{2}+i_n}) \\
&=\varepsilon^{\frac52}\lambda^{\frac{n}{2}}\left(1-\frac{O(\lambda^{i_n})}{\varepsilon^{\frac{5}{2}}} \right) \sim \varepsilon^{\frac{5}{2}}\lambda^{\frac{n}{2}}.
\end{split}$$ where $x_+^{(1)}=\mathrm{pr}_x(\varphi(v_1^{(1)}))$ and $x_-^{(1)}=\mathrm{pr}_x(\varphi(v_3^{(1)}))$, see Figure \[fig\_B1\](b). Let $B_{1}'$ be the intersection $\mathrm{pr}_x^{-1}\bigl([x_-^{(1)}, x_+^{(1)}]\bigr)\cap \varphi(B_{1})$. Any compact region in $U(p)$ like $B_{1}'$ is called a *parallelogram-like box*.

Let $u_1$ be the positive integer with $(1+\varepsilon)^2 < \mu^{u_1}x_+^{(1)} \leq (1+\varepsilon)^3$. By , $f^{u_1}(B_1')$ is contained in $R_\varepsilon$ for any sufficiently large $n$. We denote $f^{u_1}(B_1')$ by $B_2$. We call that any compact region in $U(p)$ like $B_{2}$ is a *rectangle-like box*.

Let $B$ be either a parallelogram-like or rectangle-like box. The *horizontal width* of $B$ is the diameter of the interval $\mathrm{pr}_x(B)$. The *vertical height* of $B$ is the maximum of the lengths of $\eta(x_0)$ with $x_0\in \mathrm{pr}_x(B)$, where $\eta(x_0)$ is the intersection of $B$ and the vertical line $x=x_0$. See Figure \[fig\_vertical\] in the case of $B=B_1'$.

Suppose that $B$ is a rectangle-like box and $\delta$ is an almost horizontal arc in $U(q)$ with $B\cap \delta=\emptyset$ and $\mathrm{pr}_x(B)\subset \mathrm{pr}_x(\delta)$. Then the *vertical distance* between $B$ and $\delta$ is the maximum of $\sigma(x_1)$ with $x_1\in \mathrm{pr}_x(B)$, where $\sigma(x_1)$ is the length of the shortest segment in the vertical line $x=x_1$ connecting $B$ with $\delta$.

Let $\delta_2$ be a sub-arc of $f^{u_1}(\varphi(\delta_1))\subset W^u(p)$ such that $\mathrm{pr}_x(\delta_2)$ is a small neighborhood of $\mathrm{pr}_x(B_2)$ in $W_{\mathrm{loc}}^u(p)$. We denote the horizontal width and vertical height of $B_2$ and the vertical distance between $B_2$ and $\delta_2$ by $W_2$, $H_2$ and $L_2$ respectively. By , and , $$\label{eqn_W2}
W_2=(x_+^{(1)}-x_-^{(1)})\mu^{u_1} \succsim \varepsilon^{\frac{5}{2}}\lambda^{\frac{n}2}\mu^{u_1} \geq 
\varepsilon^{-\frac{1}{2}}\tau_1^{-1}\lambda^{\frac{n}2} \sim \varepsilon^{-\frac{1}{2}}W_1.$$

For any $x_0$ with $x_-^{(1)}\leq x_0\leq x_+^{(1)}$, $\eta(x_0)$ is a vertical segment connecting $\varphi(e_0^{(1)})$ with $\varphi(e_2^{(1)})$. By this fact together with , one can show the vertical height $H_1'$ of $B_1'$ satisfies $H_1' \precsim H_1\varepsilon^{-\frac{5}{2}}$. It follows from and that $$\label{eqn_H2}
H_2=\lambda^{u_1}H_1' \precsim \mu^{-\frac{3}{2}u_1} \varepsilon^{-\frac{5}{2}}H_1 < (\varepsilon^2)^{\frac{3}{2}}\varepsilon^{-\frac{5}{2}}H_1 =\varepsilon^{\frac12}H_1.$$ Let $L_2$ be the vertical distance between $B_2$ and $\delta_2$. By using an argument similar to that for the estimation , we have $$\label{eqn_L2}
L_2\precsim \varepsilon^{\frac12}L_1.$$

The following lemma is obtained immediately from , and .

\[l\_slow3\] Let $\varepsilon_1>0$ be the constant given in Lemma \[l\_slope\]. Then there exists a constant $\varepsilon_0 \in (0, \varepsilon_1]$ such that, for any $\varepsilon \in (0, \varepsilon_0)$, the inequalities $$W_2\geq 10W_1,\quad H_2\leq 10^{-1}H_1\quad\mbox{and}\quad L_2\leq 10^{-1}L_1$$ hold.

If $\mu=1+\varepsilon$ for an $\varepsilon \in (0, \varepsilon_0)$, then we say that $f$ satisfies the *small expanding conditions* at $p$.

We repeat the process as above. Let $B_2'$ be the subset of $\varphi(B_2)$ cobounded by the vertical lines $x=x_-^{(2)}$ and $x=x_+^{(2)}$ passing through two of the four vertices of $\varphi(B_2)$ and satisfying $[x_-^{(2)}, x_+^{(2)}] \subset \mathrm{Int}(\mathrm{pr}_x(\varphi(B_2)))$. Let $u_2$ be the positive integer with $(1+\varepsilon)^2 < \mu^{u_2}x_+^{(2)} \leq (1+\varepsilon)^3$ and $f^{u_2}(B_2') \subset R_\varepsilon$ for sufficient large $n \in \mathbb{N}$. Set $B_3=f^{u_2}(B_2')$. Let $\delta_3$ be a sub-arc of $f^{u_2}(\varphi(\delta_2))$ such that $\mathrm{pr}_x(\delta_3)$ is a small neighborhood of $\mathrm{pr}_x(B_3)$ in $W_{\mathrm{loc}}^u(p)$. We denote the horizontal width and vertical height of $B_3$ and the vertical distance between $B_3$ and $\delta_3$ by $W_3$, $H_3$ and $L_3$ respectively.

The objects $B_{k}'$, $u_{k}$, $B_{k+1}$, $\delta_{k}$, $W_{k+1}$, $H_{k+1}$, $L_{k+1}$ $(k=3,4,5,\dots)$ are defined inductively if $$\label{eqn_prB}
B_j \subset R_\varepsilon$$ for $j=1,2,\dots,k$.

The top and bottom sides of the rectangle $B_1$ are horizontal and $\gamma_1=B_1 \cap W^u(p)$ consists of three proper arcs in $B_1$. By Slope Lemma I (Lemma \[l\_slope\]), for $k=2,3,\cdots$, the top and bottom sides of the rectangle-like box $B_k$ are almost horizontal and $\gamma_k=B_k \cap W^u(p)$ consists of three proper arcs in $B_k$. See Figure \[fig\_3on1\]. Thus we have the following lemma.

\[l\_slow3’\] Let $\varepsilon_0 >0$ be the constant given in Lemma \[l\_slow3\]. For any $\varepsilon \in (0, \varepsilon_0]$, there exists the maximum integer $k_0=k_0(\varepsilon,n)$ satisfying . Moreover, $$\label{eqn_slow3}
W_{k+1}\geq 10W_k,\quad H_{k+1}\leq 10^{-1}H_k\quad\mbox{and}\quad L_{k+1}\leq 10^{-1}L_k$$ hold for any $k=1,2,\dots,k_0$.

See Figure \[fig\_k0\] for the situation of Lemma \[l\_slow3’\].

We note that, since $W_1=W_{1,n}\sim \lambda^{\frac{n}2}$ by , $\lim_{n\to\infty}k_0(\varepsilon,n)=\infty$ for a fixed $\varepsilon$ with $0<\varepsilon\leq \varepsilon_0$.

Intersection Lemma {#S_ISL}
==================

Recall that $\mathcal{C}$ is the codimension two submanifold of $\mathrm{Diff}^3(M)$ defined in Section \[S\_pre\]. Let $f_0$, $f_1$ be elements of $\mathcal{C}$ satisfying the conditions – in Theorem \[thm\_A\]. In particular, $\varepsilon>0$ is taken so that Slope Lemmas I and II (Lemmas \[l\_slope\] and \[l\_slopef1\]) hold. Moreover, we suppose that $f_0$, $f_1$ satisfy the condition , which is one of the adaptable cases given in Section \[S\_adaptable\].

From now on, we set $f_0=f$ and use the notations in Sections \[S\_pre\]–\[S\_Bbox\]. Here the subscription ‘0’ is omitted from the notations. For example, $\lambda_0=\lambda$, $\mu_0=\mu$, $p_0=p$, $q_0=q$ and so on. We also set $f_1=\overline f$ and represent the notations for $\overline f$ by adding bars to the corresponding notations for $f$, e.g.  $\overline \lambda$, $\overline \mu$, $\overline p$, $\overline q$, $\overline m_0$, $\overline S_n$, $\overline B_k$, $\overline W_k$ and so on.

Let $h:M\to M$ be a homeomorphism with $\overline f=h\circ f\circ h^{-1}$. Here we note that $h(r)$ is not necessarily equal to $\overline r$. In fact, $h(r)=\overline r$ if and only if $m_0=\overline m_0$ or equivalently $\overline \varphi=h\circ \varphi\circ h^{-1}$. We may assume that $m_0\leq \overline m_0$ if necessary replacing $f$ and $\overline f$. Then $h(f^{\overline m_0-m_0}(r))=\overline r$. Since the constants appeared in depend on the coordinate on $U(p)$, one can not replace the coordinates on $U(p)$ or $U(\overline p)$ so as to satisfy $h(r)=\overline r$.

For any $C^1$-arc $\alpha$ in $U(p)$, the union of the end points of $\alpha$ is denoted by $\partial \alpha$. When any vector tangent to $\alpha$ is not vertical, the maximum $\mathrm{Slope}(\alpha)$ of $\mathrm{Slope}(\boldsymbol{v}(\boldsymbol{x}))$ for vectors $\boldsymbol{v}(\boldsymbol{x})$ tangent to $\alpha$ at $\boldsymbol{x} \in \alpha$ is well defined.

If $s>0$ is small enough, then $\gamma_{0,n}'=\beta^u_n(s) \cap S_n$ is equal to $\alpha^u_n \cap S_n$ for any sufficiently large $n \in \mathbb{N}$.

The following is a key lemma for the proof of Theorem \[thm\_A\].

\[l\_intersection\] Let $\gamma'_{0,n}=\beta^u_n(s) \cap S_n$ and $\overline \gamma'_{0,n}=\overline\beta^u_n(\overline{s}) \cap \overline{S}_n$. Then there exists an $n_0\in \mathbb{N}$ such that, for any $n \geq n_0$, $$\label{eqn_hfgamma}
h(f^{\overline m_0-m_0}(\gamma_{0,n}')) \cap \overline \gamma_{0,n}' \neq \emptyset.$$

We suppose that, for any $n_0\in \mathbb{N}$, there would exist $n>n_0$ such that $$\gamma_{0,n}'^{\,*} \cap \overline\gamma_{0,n}' = \emptyset,$$ where $\gamma_{0,n}'^{\,*}=h \circ f^{\overline{m}_0-m_0}(\gamma_{0,n}')$, and introduce a contradiction.

Recall that $i_n\in \mathbb{N}$ satisfies $(1+\varepsilon)^2 < f^{i_n}(s_n^+) \leq (1+\varepsilon)^3$ and $f^{i_n}(S_n) \subset R_{\varepsilon}$. For short, we set $$\gamma_1=\gamma_{1,n}:=f^{i_n}(\gamma_{0,n}') \quad \text{and} \quad 
\gamma_1^*=\gamma_{1,n}^*:=h(\gamma_1).$$ Then $\gamma_1^*=\overline f^{\,i_n-(\overline m_0-m_0)}(\gamma_{0,n}^*)$. Since $h(q)=\overline q$ and $\overline f=h \circ f \circ h^{-1}$, we have $h(1)=1, \,\,h(1+\varepsilon)=1+\overline\varepsilon, \,\,h\bigl((1+\varepsilon)^2\bigr)=(1+\overline\varepsilon)^2, \,\,h\bigl((1+\varepsilon)^3\bigr)=(1+\overline\varepsilon)^3$ and $1+\overline\varepsilon < \mathrm{pr}_x(\gamma_1^*) \leq (1+\overline\varepsilon)^3$. Strictly, $\gamma_1^*$ may slightly exceed $R_{\overline\varepsilon}$. Then we may rearrange our argument so that Lemmas \[l\_slope\] and \[l\_slopef1\] for $\overline f$ still hold if $\gamma_1^*$ is contained in a sufficiently small neighborhood of $R_{\overline\varepsilon}$. Then, by applying Lemma \[l\_slopef1\] to $\overline{f}$, one can show that $\gamma_1^*$ is a sub-arc almost parallel to $\overline{\delta}_1 \subset W^u_{\mathrm{loc}}(\overline p)$ and $\mathrm{Slope}(\gamma_1^*) < \overline{\varepsilon}^{\frac{5}{2}}$ for any sufficiently large $n$.

The intersection $\gamma_1'=\varphi(\gamma_1)\cap B_1'$ consists of mutually disjoint three arcs connecting the vertical sides of $B_1'$, see Figure \[fig\_3arcs\].

We set $\gamma_2=f^{u_1}(\gamma_1')$ and $\gamma_2^*=h(\gamma_2)$. Note that $\gamma_2$ is a disjoint union of three proper arcs in $B_2$ connecting the vertical sides of $B_2$. Let $\widehat\gamma_2^*$ be the smallest arc in $W^u(\overline p)$ containing $\gamma_2^*$. By applying Lemma \[l\_slopef1\] to $\overline{f}$, we have $\mathrm{Slope}(\widehat\gamma_2^*) < \overline{\varepsilon}^{\frac{5}{2}}$. In particular, $\widehat\gamma_2^*$ is almost parallel to $\overline{\delta}_2$. Repeating the same argument, one can have sequences $\{ \gamma_k\}$ satisfying the following conditions.

-   Each $\gamma_k$ is a disjoint union of three proper arcs in $B_k$ connecting the vertical sides of $B_k$.

-   For each $\gamma_k^*=h(\gamma_k)$, the smallest arc $\widehat\gamma_k^*$ in $W^u(\overline p)$ containing $\gamma_k^*$ is almost parallel to $\overline{\delta}_k.$

See Figure \[fig\_3on1\].

Take $\boldsymbol{x} \in \gamma_k$ arbitrarily and set $\boldsymbol{x}^* = h(\boldsymbol{x}) \in \gamma_k^*$. Since $h$ is uniformly continuous on $R_{\varepsilon}$, for any $\overline{l}>0$, there exists $l>0$ independent of $\boldsymbol{x}$ such that $h \circ f^{\overline{m}_0-m_0}(N_{l}(\boldsymbol{x})) \subset N_{\overline{l}}(\boldsymbol{x}^*)$, where $N_{l}(\boldsymbol{x})$ is the $l$-neighborhood of $\boldsymbol{x}$ and $N_{\overline{l}}(\boldsymbol{x}^*)$ is the $\overline{l}$-neighborhood of $\boldsymbol{x}^*$ in $M$. If $n$ is sufficiently large, then $N_{l}(\boldsymbol{x})$ must intersect the three arcs of $\gamma_k$. However, $N_{\overline{l}}(\boldsymbol{x}^*)$ intersects only one arc of $\gamma_k^*$. This gives a contradiction. Thus holds for all sufficiently large $n$.

Proof of Theorem \[thm\_A\]
===========================

Now we are ready to prove Theorem \[thm\_A\]. The proof is done by using our Intersection Lemma (Lemma \[l\_intersection\]) together with arguments in [@pa; @dm; @po] and so on. We only consider the case where both $f$ and $\overline f$ satisfy the condition , which belongs to Case $\mathrm{II}_{++}$ in Section \[S\_adaptable\], and the small expanding condition at $p$ and $\overline p$ respectively. The proof of any other adaptable case is done similarly.

By Intersection Lemma (Lemma \[l\_intersection\]), one can take $\overline{r}_n \in \overline{\gamma}_{0,n}' \cap h \circ f^{\overline{m}_0-m_0}(\gamma_{0,n}')$. Since $\overline{r}_n$ converges to $\overline{r}$ as $n \to \infty$, $r_n=(h \circ f^{\overline{m}_0-m_0})^{-1}(\overline{r}_n) \in \gamma_{0,n}'$ converges to $r$ as $n \to \infty$. See Figure \[fig\_rn\].

Let $W_{\mathrm{loc},+}^u(p)$ be the component of $W_{\mathrm{loc}}^u(p)\setminus \{p\}$ containing $q$. Take a fundamental domain $D$ for $f$ in $W^u_{\mathrm{loc},+}(p)$. Then there exist subsequences $\{ r_{n(k)} \} \subset \{ r_n \}$, $\{ m(k) \}$ of $\mathbb{N}$ and $\boldsymbol{x}_0 \in D$ satisfying the following conditions.

-   $r_{n(k)}$ converges to $r$ as $k \to \infty$.

-   $\boldsymbol{x}_{n(k)}:=f^{m(k)}(r_{n(k)})$ converges to $\boldsymbol{x}_0=(x_0,0)$ as $k \to \infty$.

-   $q_{n(k)}:=\varphi^{-1}(r_{n(k)})$ converges to $q$ as $k \to \infty$.

Then $$\begin{aligned}
 x_0&=\lim_{k\to\infty}\mathrm{pr}_x(\boldsymbol{x}_{n(k)}) = \lim_{k\to\infty}\mathrm{pr}_x(r_{n(k)})\mu^{m(k)}
=\lim_{k\to\infty}az_0\left(\lambda^{n(k)}+O\bigl(\lambda^{\frac{3}{2}n(k)}\bigr)\right)\mu^{m(k)}\\
&= \lim_{k\to\infty}az_0\lambda^{n(k)}\mu^{m(k)}.\end{aligned}$$ It follows that $\lim_{k \to \infty} \lambda^{n(k)} \mu^{m(k)} = \dfrac{x_0}{az_0}$. Then there exist constants $C_0$ and $C_1$ with $0<C_0<C_1$ and such that $$C_0<\lambda^{n(k)} \mu^{m(k)}<C_1$$ for any $k$. Taking the logarithms of this inequalities, we have $$\dfrac{\log C_0}{n(k)\log \mu} <  \dfrac{\log \lambda}{\log \mu} +\dfrac{m(k)}{n(k)} 
< \dfrac{\log C_1}{n(k)\log \mu}.$$ This shows that $\displaystyle \lim_{k \to \infty} \dfrac{m(k)}{n(k)} = -\dfrac{\log \lambda}{\log \mu}.$ By applying a similar argument to $\overline f$, one can prove $$\displaystyle \lim_{k \to \infty} \bar{\lambda}^{n(k)} \bar{\mu}^{m(k)-(\overline{m}_0-m_0)} 
= \dfrac{h(x_0)}{\bar{a}\bar{z}_0}$$ and hence $\displaystyle \lim_{k \to \infty} \dfrac{m(k)}{n(k)} = 
\lim_{k \to \infty} \dfrac{m(k)-(\overline{m}_0-m_0)}{n(k)} =-\dfrac{\log \bar{\lambda}}{\log \bar{\mu}}.$ Consequently, $\dfrac{\log \lambda}{\log \mu} = \dfrac{\log \bar{\lambda}}{\log \bar{\mu}}$ holds.

\[l\_h\_C\^1\] If $\dfrac{\log \lambda}{\log \mu}$ is irrational, then the restriction $h|_{W_+^u(p)}$ is locally $C^1$-diffeomorphic, where $W_+^u(p)$ is the component of $W^u(p)\setminus \{p\}$ containing $q$.

Let $s_n$ be the real number with $\mathrm{pr}_x(r_n)=\mu^{-s_n}$. Since $$\mathrm{pr}_x(r_n) \approx az_0\left(\lambda^{n} + O \bigl(\lambda^{\frac{3}{2}n}\bigr)\right)$$ by and , we have $$1=\mathrm{pr}_x(r_n)\mu^{s_n} \approx az_0\left(\lambda^{n}+O\bigl(\lambda^{\frac{3}{2}n}\bigr)\right)\mu^{s_n} \approx az_0\lambda^{n}\mu^{s_n}.$$ Thus $c_n=az_0\lambda^{n}\mu^{s_n}$ satisfies $\lim_{n\to \infty}c_n=1$. Moreover, $$\label{eqn_s_n}
s_n=\frac{\log c_n}{\log \mu}-\frac{\log (az_0)}{\log \mu}
-n\frac{\log \lambda}{\log \mu}.$$ Since $-\dfrac{\log \lambda}{\log \mu}$ is irrational, the set $$\left\{-\frac{\log (az_0)}{\log \mu}-n\dfrac{\log \lambda}{\log \mu}\mod 1\,;\  n=1,2,\dots \right\}$$ is dense in the interval $[0,1]$. Since $\lim_{n\to \infty}\log c_n=0$, the set $S=\{s_n\mod 1\,;\,n=1,2,\dots\}$ is also dense in $[0,1]$.

Take a point $x_0$ of $[\mu^{-1},1]$ arbitrarily, and let $\sigma \in [0,1]$ be the real number with $\mu^{-\sigma}=x_0$. Since $[\mu^{-1},1]$ is a fundamental domain for $f$ in $W_{\mathrm{loc},+}^u(p)$, it follows from the density of $S$ that there exist subsequences $\{n(k)\}$, $\{m(k)\}$ of $\mathbb{N}$ such that $\lim_{k\to \infty}(s_{n(k)}-m(k))=\sigma$. Then $$\label{eqn_wsm}
\begin{split}
x_0&=\mu^{-\sigma}=\lim_{k\to \infty}\mu^{-s_{n(k)}+m(k)} =\lim_{k\to \infty} \mathrm{pr}_x(r_{n(k)})\mu^{m(k)}\\
&=\lim_{k\to \infty} az_0 \left( \lambda^{n(k)}+O(\lambda^{\frac{3}{2}n(k)})\right)\mu^{m(k)} =\lim_{k\to \infty} az_0 \lambda^{n(k)}\mu^{m(k)}.
\end{split}$$ Thus we have $\displaystyle \lim_{k \to \infty} \lambda^{n(k)} \mu^{m(k)} = \dfrac{x_0}{az_0}.$

Since $\overline{f}$ is conjugate to $f$ via $h$, $\mathrm{pr}_x(\overline{r}_{n(k)})\bar{\mu}^{m(k)-(\overline{m}_0-m_0)}$ converges to $h(x_0)$. As above, we have $$\displaystyle \lim_{k \to \infty} \bar{\lambda}^{n(k)} \bar{\mu}^{m(k)-(\overline{m}_0-m_0)} = \dfrac{h(x_0)}{\bar{a}\bar{z}_0}.$$ If we set $\tau = \dfrac{\log \bar{\mu}}{\log \mu} =\dfrac{\log \bar{\lambda}}{\log \lambda}$, then $\bar{\mu}=\mu^\tau$ and $\bar{\lambda}=\lambda^\tau$. It follows that $$\dfrac{x^\tau_0}{a^\tau z_0^\tau} = \dfrac{h(x_0)}{\bar{a}\bar{z}_0}\bar\mu^{\overline{m}_0-m_0}.$$ Thus $h|_{W^u_{\mathrm{loc},+}(p)}$ is a $C^1$-diffeomorphism represented as $$h(x) = \dfrac{\bar{a}\bar{z}_0}{a^\tau z^\tau_0\,\bar\mu^{\overline{m}_0-m_0}}x^\tau,$$ where $W^u_{\mathrm{loc},+}(p)$ is the component of $W^u_{\mathrm{loc}}(p)\setminus \{p\}$ containing $q$. Since $W^u_{+}(p)=\bigcup_{n=0}^\infty f^n(W^u_{\mathrm{loc},+}(p))$ and both $f$ and $\overline f$ are $C^3$-diffeomorphisms, $h|_{W_+^u(p)}$ is locally $C^1$-diffeomorphic. This completes the proof.

Take a sequence $\{ q_j \}$ on $W^u_{\mathrm{loc},+}(p)$ converging to $q$ and set $t_j = \varphi(q_j)$. See Figure \[fig\_x1\_2\].

Let $t^{\prime}_j$ be the image of $t_j$ by the horizontal projection to $W^s_{{\rm loc}}(p)$. Obviously, both $t_j$ and $t^{\prime}_j$ converge to $r$ as $k \to \infty$. There exist subsequences $\{ t_{j(k)} \}$ of $\{ t_j \}$, $\{ l(k) \}$ of $\mathbb{N}$ and a point $x_1 $ of $W^u_{{\rm loc}}(p)$ with $\displaystyle \lim_{k \to \infty} f^{l(k)}(t_{j(k)}) = x_1$. Then the following approximations $$x_1 \sim {\rm pr}_x(t_{j(k)}) \mu^{l(k)} \sim [ d(t^{\prime}_{j(k)},r)]^3 \mu^{l(k)} 
\sim [ d(q_{j(k)},q)]^3 \mu^{l(k)}$$ hold. It follows that $\mu^{-l(k)} \sim [ d(q_{j(k)},q)]^3$. Similarly, we have $\bar{\mu}^{-l(k)+(\overline m_0-m_0)} \sim [ d(\bar{q}_{j(k)},\bar{q})]^3$, where $\bar{q}_{j(k)}=h(q_{j(k)})$. Since $h|_{W^u_{+}(p)}$ is locally $C^1$-diffeomorphic by Lemma \[l\_h\_C\^1\], $$d(\bar{q}_{j(k)},\bar{q}) \sim d(q_{j(k)},q).$$ Thus $$\biggl( \dfrac{\bar{\mu}}{\mu} \biggr)^{-l(k)} \sim 
\biggl( \dfrac{d(\bar{q}_{j(k)},\bar{q})}{d(q_{j(k)},q)} \biggr)^3\bar\mu^{-(\overline m_0-m_0)} 
\sim 1.$$ This implies that $\mu = \bar{\mu}$. By , we also have $\lambda = \bar{\lambda}$. This completes the proof of the part .

\[r\_one\_sided\] Some arguments used in the case that the tangency between $W^s(p)$ and $W^u(p)$ is one-sided (for example [@pa; @dm; @po]) can not be applicable to the two-sided case. Here we explain the reason.

Suppose that a homoclinic tangency $q_0$ is one-sided, say a quadratic tangency. Take an arc $\gamma$ in $U(q_0)$ meeting $W_{\mathrm{loc}}^u(p_0)$ orthogonally at $q_0$. Let $\{ w_i \}$ be a sequence in $\gamma$ converging to $q_0$ from above. Then $$\label{ineq_dist_1}
d(w_i,W^s(p_0)) \approx d(w_i,W^u_{\mathrm{loc}}(p_0))$$ holds. On the other hand, their images by the conjugacy homeomorphism $h$ satisfy $$\label{ineq_dist_2}
d(h(w_i),W^s(p_1)) \leq d(h(w_i),W^u_{\mathrm{loc}}(p_1)).$$ See Figure \[fig\_technique\](a).

By using and , one can shown that $\dfrac{\log \lambda_1}{\log \mu_1} \leq \dfrac{\log \lambda_0}{\log \mu_0}$. By applying the same argument to $h^{-1}$, we also have $\dfrac{\log \lambda_1}{\log \mu_1} \geq \dfrac{\log \lambda_0}{\log \mu_0}$, and hence $\dfrac{\log \lambda_1}{\log \mu_1} =\dfrac{\log \lambda_0}{\log \mu_0}$.

Now we consider the case of 2-sided tangencies, say cubic tangencies, and $\{ w_i \}$ is a sequence as above. Then the approximation still holds. However, the inequality would not hold as is suggested in Figure \[fig\_technique\](b). So it might be difficult to get the inequality $\dfrac{\log \lambda_1}{\log \mu_1} \leq \dfrac{\log \lambda_0}{\log \mu_0}$ only by arguments in [@pa; @dm; @po]. Thus we need another idea in the study of moduli associated with two-sided homoclinic tangencies.

Adaptable conditions {#S_adaptable}
====================

In this section, we will present conditions on the signs of $a$, $bc$, $\lambda$ and $\mu$ under which any arguments presented throughout the previous sections are valid.

Recall that we have set $$U(p)=[-2,2]\times [-2,2],\ W_{\mathrm{loc}}^u(p)=[-2,2]\times \{0\},\ W_{\mathrm{loc}}^s(p)=\{0\}\times [-2,2].$$ The union $W_{\mathrm{loc}}^u(p)\cup W_{\mathrm{loc}}^s(p)$ divides $U(p)$ to four components. The closures of these components containing $(1,1)$, $(-1,1)$, $(-1,-1)$ and $(1,-1)$ are called the first, second, third and fourth quadrants of $U(p)$ and denoted by $Q_1$, $Q_2$, $Q_3$ and $Q_4$, respectively. In our argument it is required that $\varphi(R_\varepsilon)$ or some substitution is in $Q_1$. If $R_\varepsilon$ lies in $Q_2$, then we may use $$R^-_\varepsilon = [(1+\varepsilon)^{-3}, (1+\varepsilon)^{-1}] \times [0, \varepsilon^3]$$ instead of $R_\varepsilon$. Then $\varphi(R^-_\varepsilon)$ is in $Q_1$. See Figure \[fig\_vpRe\].

Thus one can arrange the placement of $\varphi(R_\varepsilon)$ suitably under any conditions on the signs of $a$, $bc$, $\lambda$ and $\mu$.

\[def\_adaptable\] $f$ satisfies the *adaptable condition* with respect to $(p,q)$ if, for all sufficiently large positive integers $n$ (or positive even or odd integers), there exists a rectangle $S_n$ defined as in Section \[S\_Sbox\] and either $S_n$ or its image $f(S_n)$ lies in $Q_1$.

As was seen in Section \[S\_Sbox\], $S_n$ exists if and only if there exists $t_n$ satisfying the condition $$\tag{\ref{eqn_3ctn}'}\label{eqn_3ctn'}
3ct_n^2\approx -b\lambda^nz_0$$ which corresponding to . Here $z_0$ is the positive constant as illustrated in Figure \[fig\_alpha\_n\].

Now we will see that the existence of $S_n$ and the placements of $S_n$ and $f(S_n)$ are strictly determined by the signs of $a$, $bc$, $\lambda$ and $\mu$, which are classified to the sixteen cases as in Table \[table\_1\]

First we suppose that $\lambda>0$. Then there exists $t_n$ satisfying if and only if $bc<0$. Moreover, if $a>0$, then $S_n$ is in $Q_1$, which belongs to Case II$_+$. See Figure \[fig\_Cases\](1). If $a<0$, then $S_n$ is in $Q_2$. Hence $f(S_n)$ is $Q_1$ if $\mu<0$, which is in Case IV$_{+-}$. See Figure \[fig\_Cases\](2).

Next we suppose that $\lambda<0$. Then there exists $t_n$ satisfying if and only if either (i) $bc<0$ and $n$ is even or (ii) $bc>0$ and $n$ is odd. In the case (i), $S_n$ is in $Q_1$ if $a>0$, which belongs to Case II$_-$. See Figure \[fig\_Cases\](1). If $a<0$ and $\mu<0$, then $f(S_n)$ is in $Q_1$, which belongs to Case IV$_{--}$. See Figure \[fig\_Cases\](2). On the other hand, in the case (ii), $S_n$ is in $Q_1$ if $a<0$, which belongs to Case III$_-$. See Figure \[fig\_Cases\](3). If $a>0$ and $\mu<0$, then $f(S_n)$ is in $Q_1$, which belongs to Case I$_{--}$. See Figure \[fig\_Cases\](4).

Thus we have the following preposition.

\[p\_adaptable\] If one of Cases [I$_{--}$, II, III$_{-}$, IV$_{--}$]{} and [IV$_{+-}$]{} holds, then $f$ satisfies the adaptable condition with respect to $(p,q)$.

It follows from the proposition that $f$ satisfies the adaptable condition in nine of the sixteen cases in Table \[table\_1\].

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