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This article is cited in 3 scientific papers (total in 3 papers) Knot as a complete invariant of a Morse-Smale 3-diffeomorphism with four fixed points O. V. Pochinkaa, E. A. Talanovaab, D. D. Shubina a National Research University Higher School of Economics, Nizhny Novgorod, Russia b National Research Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 8, DOI: https://doi.org/10.4213/sm9814 
Bibliographic databases:
    § 1. Introduction and formulation of resultsRecall that a diffeomorphism $f\colon M^n\to M^n$ defined on an orientable connected closed smooth $n$-dimensional ($n\geqslant 1$) manifold $M^n$ is called a Morse-Smale diffeomorphism if:
If $\sigma_1$ and $\sigma_2$ are two different saddle periodic points of a Morse-Smale diffeomorphism and $W^{\mathrm{s}}_{\sigma_{1}}\cap W^{\mathrm{u}}_{\sigma_{2}}\neq\varnothing$, then the intersection $W^{\mathrm{s}}_{\sigma_{1}}\cap W^{\mathrm{u}}_{\sigma_{2}}$ is called a heteroclinic intersection, and its connected components of dimension $1$ are called heteroclinic curves. In [1] a complete topological classification of Morse-Smale diffeomorphisms on closed 3-manifolds was given; however, its significant part is the description of the invariant since it is designed for a wider class of diffeomorphisms. In some cases there are other, more natural invariants, which can be found without considering them as special case of the general invariant. Thus, in this paper, we establish that a complete invariant for a wide class of 3-diffeomorphisms whose nonwandering set consists of four points is the equivalence class of a Hopf knot in $\mathbb S^2\times\mathbb S^1$. Recall that a knot in manifold $M^n$ is a smooth embedding $\gamma\colon \mathbb S^1\to M^n$ or the image $L=\gamma(\mathbb S^1)$ of such an embedding. Two knots $L$ and $L'$ are called equivalent if there exists a homeomorphism $h\colon M^n\to M^n$ such that $h(L)=L'$. Let $[L]$ denote the equivalence class of $L$. A knot $L\subset\mathbb S^2\times\mathbb S^1$ is a Hopf knot if the homomorphism $i_{L*}$ induced by the inclusion $i_{L}\colon L\to\mathbb S^2\times\mathbb S^1$ is an isomorphism of the groups $\pi_1(L)\cong\pi_1(\mathbb S^2\times\mathbb S^1)\cong\mathbb Z$. Note that any Hopf knot is smoothly homotopic to the standard Hopf knot $L_0=\{x\}\times\mathbb S^1$ (see, for example, [2] and [3]), but in general they not equivalent. Mazur constructed a Hopf knot $L_M$ which is not equivalent to $L_0$ (see Figure 1). In [3] a countable family of pairwise nonequivalent Hopf knots was constructed (see Figure 2). Three-dimensional Morse-Smale diffeomorphisms with exactly four nonwandering points can be divided into two classes: in the first class each diffeomorphism has one saddle point, and in the second class each diffeomorphism has two saddle points. For the first class Bonatti and Grines [5] showed that the topological conjugacy class of a diffeomorphism is fully determined by the equivalence class of the Hopf knot obtained as the projections of the one-dimensional saddle separatrix. According to [6] and [5] any Hopf knot can be realized by means of a diffeomorphism in the first class on the 3-sphere. In this paper we consider diffeomorphisms in the second class. Namely, we look at the class $G$ of orientation-preserving Morse-Smale diffeomorphisms with the following properties on a closed manifold $M^3$:
Investigations of such systems are of interest primarily from the fundamental point of view: one examines the relationships between topology and dynamics. On the other hand systems with heteroclinic curves on $\mathbb R^3$ and $\mathbb S^3$ arise in many applied problems, for example, in the Lotka-Volterra model, suggested for modelling cognitive and emotional functions of the brain in [8]. Let $f\in G$. We denote the unstable separatrices of the point $\sigma_f^{1}$ by $\ell_f^1$ and $\ell_f^2$. Then (see, for example, [9]) the closure $\operatorname{cl}(\ell_f^i)$, $i=1,2$, of the one-dimensional unstable separatrix of $\sigma_f^{1}$ is homeomorphic to a simple closed curve, and it consists of the separatrix and two points, the saddle $\sigma_f^{1}$ and the sink $\omega_f$ (see Figure 3). Let $\mathbf{x}=(x_1,x_2,x_3)\in\mathbb R^3$, $|| \mathbf{x} ||=\sqrt{x_1^2 + x_2^2 + x_3^2}$, and let $a\colon\mathbb R^3\to\mathbb R^3$ be a diffeomorphism given by $a(\mathbf x)=\mathbf x/{2}$. Let $p\colon\mathbb R^3\setminus O\to\mathbb S^{2}\times\mathbb S^1$ be the map defined by
$$
\begin{equation*}
p(\mathbf x)=\biggl(\frac{\mathbf x}{\|\mathbf x\|}, \log_2(\|\mathbf x\|)\pmod 1\biggr).
\end{equation*}
\notag
$$
Let $V_{\omega_f}=W^{\mathrm{s}}_{\omega_f}\setminus\omega_f$. By virtue of the hyperbolicity of the sink $\omega_f$, there exists a diffeomorphism $\psi_f\colon V_{\omega_f}\to\mathbb R^3\setminus O$ conjugating $f$ and $a$. Let $p_{\omega_f}=p\psi_f\colon V_{\omega_f}\to\mathbb S^{2}\times\mathbb S^1$ and $L_f^{i}=p_{\omega_f}(\ell_f^i)$, $i=1,2$. Lemma 1.1. For any diffeomorphism $f\in G$ the sets $L_f^1$ and $L_f^2$ are equivalent Hopf knots in $\mathbb S^{2}\times\mathbb S^1$. Let $\mathcal L_f=[L_f^1]=[L_f^2]$ denote the equivalence class of these knots. Theorem 1.1. The diffeomorphisms $f,f'\in G$ are topologically conjugate if and only if $\mathcal L_f=\mathcal L_{f'}$. Thus, the equivalence class of the Hopf knot is a complete topological invariant for diffeomorphisms in the Pixton class $G$. Moreover, the following theorem holds. Theorem 1.2. For any equivalence class $\mathcal L$ of Hopf knots there exists a diffeomorphism $f_{\mathcal L}\colon\mathbb S^3\to\mathbb S^3\in G$ such that $\mathcal L_{f_{\mathcal L}}=\mathcal L$. An immediate consequence of Theorems 1.1 and 1.2 is the fact that the ambient manifold of a diffeomorphism in $G$ is the 3-sphere $\mathbb S^3$ (for an independent proof of this fact, see [10]). § 2. Compatible foliated neighbourhoodsFor $t\in(0,1]$ set $\mathcal N_{1}^t=\{(x_1,x_2,x_3)\in\mathbb{R}^3\colon x_1^2(x_2^2+x_3^2)< t\}$ and $\mathcal N_{2}^t=\{(x_1,x_2,x_3)\in\mathbb{R}^3\colon (x_1^2+x_2^2)x_3^2< t\}$, and for $i\in\{1,2\}$ set $\mathcal N^1_{i}=\mathcal N_{i}$. In the neighbourhood $\mathcal N_{1}$ we define a pair of transversal foliations $\mathcal{F}^{\mathrm{u}}_1$ and $\mathcal{F}^{\mathrm{s}}_{1}$ by
$$
\begin{equation*}
\mathcal{F}^{\mathrm u}_1=\bigcup_{(c_{2},c_3)\in Ox_2x_3}\{(x_1,x_2,x_3)\in \mathcal N_{1}\colon (x_{2},x_3)=(c_{2},c_3)\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathcal{F}^{\mathrm s}_{1}=\bigcup_{c_1\in Ox_1}\{(x_1,x_2,x_3)\in \mathcal N_{1} \colon x_1=c_1\}.
\end{equation*}
\notag
$$
In the neighbourhood $\mathcal N_{2}$ we define a pair of transversal foliations $\mathcal{F}^{\mathrm{u}}_2$ and $\mathcal{F}^{\mathrm{s}}_{2}$ by
$$
\begin{equation*}
\mathcal{F}^{\mathrm u}_2=\bigcup_{c_3\in Ox_3}\{(x_1,x_2,x_3)\in \mathcal N_{3} \colon x_3=c_3\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathcal{F}^{\mathrm s}_{2}=\bigcup_{(c_{1},c_2)\in Ox_1x_2}\{(x_1,x_2,x_3)\in \mathcal N_{3} \colon (x_{1},x_2)=(c_{1},c_2)\}.
\end{equation*}
\notag
$$
We define two diffeomorphisms $a_i\colon\mathbb R^3\to\mathbb R^3$ by the formula:
$$
\begin{equation*}
a_1(\mathbf x)=\biggl(2x_1,\frac{x_2}{2},\frac{x_3}{2}\biggr)\quad\text{and} \quad a_2=a_1^{-1}.
\end{equation*}
\notag
$$
Note, that for $i\in\{1,2\}$ the set $\mathcal N_{i}^t$ is invariant under the diffeomorphism $a_i$, which maps leaves of $\mathcal{F}^{\mathrm{u}}_i$ (of $\mathcal{F}^{\mathrm{s}}_{i}$) to leaves of $\mathcal{F}^{\mathrm{u}}_i$ (of $\mathcal{F}^{\mathrm{s}}_{i}$, respectively). By virtue of [1] a saddle point $\sigma_f^i$ of a diffeomorphism $f\in G$ has a linearizing neighbourhood $N_f^{i}$ which is equipped with a homeomorphism $\mu_i\colon N_i\to {\mathcal N}_i$ conjugating the diffeomorphism $f|_{N_f^i}$ with $a_i|_{{\mathcal N}_i}$ and diffeomorphic on $N_i\setminus(W^{\mathrm s}_{\sigma_i\cup W^{\mathrm u}_{\sigma_i}})$. The foliations $\mathcal{F}^{\mathrm{u}}_i$ and $\mathcal{F}^{\mathrm{s}}_i$ induce $f$-invariant foliations $F^{\mathrm{u}}_i$ and $F^{\mathrm{s}}_i$ on the linearizing neighbourhood $N_i$ by means of the homeomorphism $\mu_i^{-1}$. Let $F^{\mathrm{u}}_{i,x}$ (${F}^{\mathrm{s}}_{i,x}$) denote the unique leave of the foliation $F^{\mathrm{u}}_i$ (of $F^{\mathrm{s}}_i$, respectively) containing the point $x\in N_f^i$. For $f\in G$ the set $H_f$ consists of a single heteroclynic curve. Hence it possesses an $f$-invariant neighbourhood $N_{H_f}\subset M^3$ with an $f$-invariant $C^{1,1}$-foliation $F$ of two-dimensional discs transversal to $H_f$. For each point $x\in N_{H_f}$ let $F_x$ denote the unique leaf of $F$ through this point. The pair of linearizing neighbourhoods $N_f^1$, $N_f^2$ of saddle points of $f$ is called a compatible system of neighbourhoods if for any point $x\in(N_f^1\cap N_f^2\cap N_{H_f})$ and the leaf $F_x$ of the foliation $F$ the following conditions hold (see Figure 4):
$$
\begin{equation*}
{F}^{\mathrm s}_{1,x}\cap F_x={F}^{\mathrm s}_{2,x}\cap({N}_f^{1}\cap N_{H_f})\quad\text{and}\quad {F}^{\mathrm u}_{2,x}\cap F_x={F}^{\mathrm u}_{1,x}\cap ({N}_f^{2}\cap N_{H_f}).
\end{equation*}
\notag
$$
Proposition 2.1 (see [1], Theorem 1). For any diffeomorphism $f\in G$ there exists a compatible system of neighbourhoods. § 3. Equivalence of the knots $L^1_f$ and $L^2_f$In this section we prove Lemma 1.1: for any diffeomorphism $f\in G$ the sets $L_f^1$ and $L_f^2$ are equivalent Hopf knots. Proof of Lemma 1.1. By [3], Lemma 2, two Hopf knots are equivalent if and only if they are isotopic. So, to prove the lemma it is sufficient to construct an isotopy between $L_f^1$ and $L_f^2$. Let (see Figure 5).
Since the diffeomorphism $f|_{W^{\mathrm{u}}_{\sigma_f^{2}}}$ is topologically conjugate to a linear dilation, the orbit space $(W^{\mathrm{u}}_{\sigma_f^{2}}\setminus\sigma_f^{2})/f$ is homeomorphic to the two-dimensional torus. Since $W^{\mathrm{u}}_{\sigma_f^{2}}\cap V_{\omega_f}=W^{\mathrm{u}}_{\sigma_f^{2}}\setminus (H_f\cup \sigma_f^{2})$ and the orbit space $H_f/f$ is homeomorphic to the circle, the set $C_f$ is homeomorphic to a two-dimensional annulus. Moreover, the homomorphism $i_{C_f*}$ induced by the inclusion $i_{C_f}\colon C_f\to\mathbb S^2\times\mathbb S^1$ is an isomorphism of the groups $\pi_1(C_f)\cong\pi_1(\mathbb S^2\times\mathbb S^1)\cong\mathbb Z$. Let Since $N^1_f\cap V_{\omega_f}=N^1_f\setminus W^{\mathrm{s}}_{\sigma_f^{1}}$, the set $U^1_f$ is a disjoint union of solid tori: $U^1_f=U^{1,1}_f\sqcup U^{1,2}_f$, which are tubular neighbourhoods of the knots $L^1_f$ and $L^2_f$. Let
$$
\begin{equation*}
T^i_f=\partial U^{1,i}_f \quad\text{and}\quad S^i_f=T^i_f\cap C_f
\end{equation*}
\notag
$$
(see Figure 6). Since the set $W^{\mathrm{s}}_{\sigma_f^{1}}\cap{W^{\mathrm{u}}_{\sigma_f^{2}}}$ consists of a unique noncompact $f$-invariant curve, the set $\partial N^{1}_{f}\cap{W^{\mathrm{u}}_{\sigma_f^{2}}}$ consists of two noncompact $f$-invariant curves as well. Their projections onto $\mathbb S^2\times\mathbb S^1$ are the curves $S^1_f\sqcup S^2_f$ (see Figure 6). This implies that $S^1_f$ and $S^2_f$ are isotopic Hopf knots in $\mathbb S^2\times\mathbb S^1$. Thus, the knots $S^i_f$ and $L^i_f$ are generators of the solid torus $U^{1,i}_f$. Hence they bound a two-dimensional annulus in the solid torus and, consequently, are isotopic. The lemma is proved. § 4. Hopf knot equivalence class as a complete invariant of topological conjugacy in the class $G$In this section we prove Theorem 1.1: two diffeomorphisms $f,f'\in G$ are topologically conjugate if and only if $\mathcal L_f=\mathcal L_{f'}$. Proof of Theorem 1.1. Necessity. Let the diffeomorphisms $f,f'\in G$ be topologically conjugate by means of a homeomorphism $h\colon M^3\to M^3$. Since $h$ maps invariant manifolds of fixed points of $f$ to invariant manifolds of fixed points of $f'$ preserving their stability, we obtain $h(W^{\mathrm{s}}_{\omega_f}) = W^{\mathrm{s}}_{\omega_{f'}}$ and $h(\ell^i_f) = \ell^i_{f'}$. Since $hf=f'h$, $h$ defines a homeomorphism $\widehat h\colon\mathbb S^2\times\mathbb S^1\to\mathbb S^2\times\mathbb S^1$ by This implies that $\widehat h(L^i_f)=L^i_{f'}$ and, consequently, the Hopf knots $L^i_f$ and $L^i_{f'}$ are equivalent.
Sufficiency. Let $\mathcal L_f=\mathcal L_{f'}$. Thus there exists a homeomorphism $\widehat h_0\colon \mathbb S^2\times\mathbb S^1\to\mathbb S^2\times\mathbb S^1$ such that $\widehat h_0(L^1_f)=L^1_{f'}$. We modify $\widehat h_0$ step by step to construct a homeomorphism $h\colon M^3\to M^3$ conjugating $f$ with $f'$. In doing this we use the notation of Lemma 1.1 and § 2, putting primes for the diffeomorphism $f'$. Let
$$
\begin{equation*}
U^2_f=p_{\omega_f}(N^2_f) \quad\text{and}\quad U_f=U^1_f\cup U^2_f
\end{equation*}
\notag
$$
(see Figure 7).
Step 1. The construction of a homeomorphism $\widehat h_1\colon\mathbb S^2\times\mathbb S^1\to\mathbb S^2\times\mathbb S^1$ such that $\widehat h_1(U_f)=U_{f'}$. By virtue of Lemma 1.1, $U_f$ is a tubular neighbourhood of the knot $L^1_f$. Let $\widetilde U_f\supset U_f$ be another tubular neighbourhood of $L^1_f$. Let $U=\widehat h_0^{-1}(U_{f'})$ and $\widetilde U=\widehat h_0^{-1}(\widetilde U_{f'})$, and choose a tubular neighbourhood $V$ of $L^1_f$ such that $V\subset \operatorname{int}(U_f\cap U)$ (see Figure 8). Since the sets $\widetilde U_f\setminus \operatorname{int}U_f$ and $U_f\setminus \operatorname{int}V$ are homeomorphic to $\mathbb T^2\times[0,1]$, there exists a homeomorphism $\widehat\psi_f\colon\mathbb S^2\times\mathbb S^1\to\mathbb S^2\times\mathbb S^1$ that coincides with the identity outside $\widetilde U_f$ and such that $\widehat\psi_f(U_f)=V$. In a similar way we construct a homeomorphism $\widehat\psi\colon \mathbb S^2\times\mathbb S^1\to\mathbb S^2\times\mathbb S^1$ that coincides with the identity outside $\widetilde U$ and such that $\widehat\psi(U)=V$ (see Figure 8). Then the required homeomorphism $\widehat h_1$ is
$$
\begin{equation*}
\widehat h_1=\widehat h_0\widehat\psi^{-1}\widehat\psi_f.
\end{equation*}
\notag
$$
Step 2. The construction of a homeomorphism $\widehat h_2\colon\mathbb S^2\times\mathbb S^1\to\mathbb S^2\times\mathbb S^1$ coinciding with $\widehat h_1$ outside $\widehat U_f$ and such that $\widehat h_2(C_f)=C_{f'}$. Set $H=\mu_1(H_f)$ and $H'=\mu'_1(H_{f'})$. Then $H$ and $H'$ are $a_1$-invariant curves on the plane $Ox_2x_3$. Thus, there exists a homeomorphism $\xi_s\colon Ox_2x_3\to Ox_2x_3$ commuting with $a_1|_{Ox_2x_3}$ and such that $\xi_s(H)=H'$ (see, for example, [9]). Define a homeomorphism $\xi\colon\mathbb R^3\to\mathbb R^3$ by (see Figure 9).We choose $\tau\in(0,1)$ so as to have $\xi(\mathcal N^\tau_1)\subset \operatorname{int}\mathcal N_1$. Let $N^\tau_1=\mu_1(\mathcal N^\tau_1)$ and $\xi_1 = \mu'^{-1}_1 \xi \mu_1|_{N^\tau_1}$. The definition of a compatible system of neighbourhoods implies that $\xi_1(N^\tau_1\cap W^{\mathrm{u}}_{\sigma^2_f})\subset W^{\mathrm{u}}_{\sigma^2_{f'}}$. Let $N^\tau_2=\mu_2(\mathcal N^\tau_2)$, $U^{\tau}_1=p_{\omega_f}(N^\tau_1)$, $U^{\tau}_2=p_{\omega_f}(N^\tau_2)$, $U^\tau=U^\tau_1\cup U^\tau_2$, $U'^{\tau}_2=p_{\omega_{f'}}(N^\tau_2)$ and $\widehat\xi_1=p_{\omega_{f'}}\xi_1p_{\omega_f}^{-1}|_{U^\tau_1}$. Also let $U'^\tau_1=\widehat\xi_1(U^\tau_1)$, $U'^\tau=U'^\tau_1\cup U'^{\tau}_2$, $K=C_f\setminus \operatorname{int}U^\tau$, $K'=C_{f'}\setminus \operatorname{int}U^\tau_1$, $U_K=U^\tau\setminus U^\tau_1$ and $U_{K'}=U'^\tau\setminus U'^\tau_1$ (see Figure 10). Since the sets $K$ and $K'$ are homeomorphic to two-dimensional annuli and the sets $U_K$ and $U_{K'}$ are tubular neighbourhoods of these sets, the homeomorphism $\widehat\xi_1$ can be extended to a homeomorphism $\widehat\xi\colon U^\tau\to U'^\tau$ such that $\widehat\xi(C_f)=C_{f'}$. Thus, the homeomorphism $\widehat\psi_1=\widehat h_1^{-1}\widehat\xi$ possesses the property $\widehat\psi_1(U^\tau)\subset \operatorname{int} U_f$, and the homeomorphism $\widehat\xi_1|_{\partial U^\tau}$ is homotopic to the identity. Since the sets $U_f\setminus \operatorname{int}U^\tau$ and $U_f\setminus \operatorname{int}\widehat\psi_1(U^\tau)$ are homeomorphic to $\mathbb T^2\times[0,1]$, $\widehat\psi_1$ extends to a homeomorphism $\widehat\psi_1\colon\mathbb S^2\times\mathbb S^1\to\mathbb S^2\times\mathbb S^1$, which is the identity outside $U_f$. Then the required homeomorphism $\widehat h_2$ is Step 3. The construction of the required homeomorphism $h$. By the construction of $\widehat h_2$ there exists a lift $h_2\colon V_{\omega_f}\to V_{\omega_{f'}}$ of it conjugating $f|_{V_{\omega_f}}$ with $f'|_{V_{\omega_{f'}}}$ and extending to $W^{\mathrm{s}}_{\sigma_1}$ as the homeomorphism $\xi_1$. So the conjugating homeomorphism is defined everywhere except the closures of the one-dimensional manifolds of saddle points. By Theorem 1 in [1] such a homeomorphism can be extended to a required homeomorphism $h$. Theorem 1.1 is proved. § 5. Realization of diffeomorphisms in the class $G$The Fox-Artin arc was first encountered in dynamics in [6] by Pixton. In that paper a Morse-Smale diffeomorphism on a 3-sphere with a unique saddle point whose invariant manifolds form the Fox-Artin arc was constructed. In [5] an arbitrary Hopf knot in $\mathbb S^{2}\times\mathbb S^1$ was realized by means of a Morse-Smale diffeomorphism with unique saddle point on the 3-sphere (see, for example, [9] and [11]). In this section we present a similar realization for diffeomorphisms in the class $G$. Let $L\subset \mathbb S^2\times\mathbb S^1$ be a Hopf knot and $U_L$ be a tubular neighbourhood of it. Then the set $\overline L=p^{-1}(L)$ is an $a$-invariant curve in $\mathbb R^3$, and $U_{\overline L}=p^{-1}(U_L)$ is an $a$-invariant neighbourhood of it diffeomorphic to $\mathbb{D}^{2}\times\mathbb R^1$ (see Figure 11). We define a flow $g^t\colon C\to C$ on the cylinder $C=\{\mathbf{x}\in\mathbb R^3\colon x_2^2+x_{3}^2\leqslant 4\}$ by Then there exists a diffeomorphism $\zeta\colon U_L\to C$ that conjugates $a|_{U_L}$ and $g=g^1|_C$. We define a flow $\phi^t$ on $C$ by
$$
\begin{equation*}
\begin{aligned} \, \dot{x}_1 &= \begin{cases} 1-\dfrac{1}{9}(\|\mathbf x\|-4)^2, &\|\mathbf x\| \leqslant 4, \\ 1, &\|\mathbf x\| > 4, \end{cases} \\ \dot{x}_2 &=\begin{cases} \dfrac{x_2}{2}\biggl(\sin\biggl(\dfrac{\pi}{2}(\|\mathbf x\|-3) \biggr)-1\biggr), &2<\|\mathbf x\|\leqslant 4, \\ -x_2,&\|\mathbf x\|\leqslant 2, \\ 0, &\|\mathbf x\| > 4, \end{cases} \\ \dot{x}_3&= \begin{cases} -\dfrac{x_3}{2}\biggl(\sin\biggl(\dfrac{\pi}{2}(\|\mathbf x\|-3)\biggr)-1\biggr), &2<\|\mathbf x\|\leqslant 4, \\ x_3,&\|\mathbf x\|\leqslant 2, \\ 0, &\|\mathbf x\| > 4. \end{cases} \end{aligned}
\end{equation*}
\notag
$$
The construction of the diffeomorphism $\phi=\phi^1$ implies that it has two hyperbolic fixed saddles, the point $P_1(-1,0,0)$ with Morse index $1$ and the point $P_2(1,0,0)$ with Morse index $2$ (see Figure 12). The noncompact heteroclinic curve $W^{\mathrm{s}}_{P_1}\cap W^{\mathrm{u}}_{P_2}$ coincides with the open interval $\bigl\{\mathbf x\in\mathbb R^3\colon |x_1|<1,\,x_2=x_3=0\bigr\}$. Note that $\phi$ coincides with the diffeomorphism $g=g^1$ outside the ball $\{\mathbf{x}\in C\colon ||\mathbf{x}||\leqslant 4\}$. We define a diffeomorphism $\overline f_L\colon\mathbb R^3\to\mathbb R^3$ so that $\overline{f}_{L}$ coincides with $a$ outside $U_L$ and with $\zeta^{-1}\phi\zeta$ on $U_L$. Then $\overline f_L$ has two hyperbolic fixed points in $U_L$: the saddle $\zeta^{-1}(P_1)$ and the saddle $\zeta^{-1}(P_2)$ (see Figure 13). We denote the North pole of the sphere $\mathbb S^3$ by $N(0,0,0,1)$ and the standard stereographic projection by $\vartheta\colon\mathbb R^3\to(\mathbb{S}^3\setminus\{N\})$. The construction of $\overline{f}_L$ implies that it coincides with $a$ in a neighbourhood of the point $O$ and outside another neighbourhood of this point. Thus, it induces a Morse-Smale diffeomorphism $f_L$ on $\mathbb{S}^3$:
$$
\begin{equation*}
f_L(s)= \begin{cases} \vartheta(\overline f_{L}(\vartheta^{-1}(s))), &s\neq N; \\ N, &s=N. \end{cases}
\end{equation*}
\notag
$$
It follows directly from the construction that the nonwandering set of the diffeomorphism $f_L$ consists of four hyperbolic fixed points: a sink $\omega$, the two saddles $\sigma^1=\vartheta({\zeta}^{-1}(P_1))$, $\sigma^2=\vartheta({\zeta}^{-1}(P_2))$ and a source $\alpha$. This diffeomorphism belongs to the class $G$ and $\mathcal L_{f_L}=[L]$.
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