Moduli of 3-dimensional diffeomorphisms with saddle-foci

Figure 3.1.  The images of the parallel straight segments $\gamma_k^\natural$ in $D_a(p)$ by $h$

Figure 1.1.  A saddle-focus $p$ and a homoclinic quadratic tangency $q$ in $D_a(p)$

Figure 1.2.  The front curve $\widetilde \gamma_0$ divides $\widetilde H_0$ into the two sheets $\widetilde H_0^+$ and $\widetilde H_0^-$. The folding curve $\gamma_0$ of $H_0$ is the orthogonal image of $\widetilde\gamma_0$

Figure 1.3.  The half disk $\widetilde H_{0;u}^-$ meets $W^s(p)$ transversely at two points near $q$, one of which is $\widehat z$

Figure 1.4.  Trip from $\widetilde H_0^-$ to $\widetilde H_m$: $f^{u+v}(\widetilde H_0^-)\supset D$, $f^{m_0}(D)\supset D_0$, $f^m(D_0)\supset D_m$ and $f^{N+n_0}(D_m)\supset \widetilde H_m$, where $N$, $n_0$ are the positive integers with $f^N(q) = \widetilde q$ and $f^{n_0}(\widetilde q) = q_0$. The dotted line passing through $q$ represents a straight segment tangent to $\widetilde \rho$ at $q$

Figure 2.1.  The image $h(\widetilde H_{(j)})$ is contained in $\widehat H_{(j)}'$, but $h(\widetilde H_{(j)}^\pm)$ is not necessarily contained in $\widehat H_{(j)}'^\pm$

Figure 2.2.  The case of $\boldsymbol{x}, \boldsymbol{y}\in \widetilde H_{(j)}^+$, $\boldsymbol{x}'\in \widetilde H_{(j)}'^+$ and $\boldsymbol{y}'\in \widetilde H_{(j)}'^-$

Figure 2.3.  The shaded region represents $\mathcal{N}_\varepsilon(\widetilde\gamma_{m_j, n_j}', \widetilde H_{(j)}')$

Figure 2.4.  The situation which does not actually occur. $d_1: = {\rm dist}(\boldsymbol{x}', \boldsymbol{y}')<\nu(\varepsilon)$, $d_2: = {\rm dist}_{\widetilde H_{(j)}}(\boldsymbol{x}, \boldsymbol{y})<\delta(\varepsilon)$ and $d_3: = {\rm dist}_{\widetilde H_{(j)}'}(\boldsymbol{x}', \boldsymbol{y}')<\varepsilon$

Figure 4.1.  Correspondence of straight segments via $h$

Figure 4.2.  Correspondence via $h$ with respect to the new coordinate on $D_{a'}(p')$