| Russian Mathematical Surveys |
|
|
|
|
|
Morse-Smale diffeomorphisms with non-wandering points of pairwise different Morse indices on 3-manifolds O. V. Pochinkaa , E. A. Talanovab a HSE University b National Research Lobachevsky State University of Nizhny Novgorod
Russian version:
Uspekhi Matematicheskikh Nauk, 2024, Volume 79, Issue 1(475), DOI: https://doi.org/10.4213/rm10141 
Bibliographic databases:
  IntroductionOne important class of structurally stable dynemical systems consists of the Morse–Smale systems, a significant feature of which is a close link between the dynamical properties of the system and the topology of the supporting manifold. Morse–Smale systems were introduced by Smale in his pioneering paper [64]. Already in that paper he proved Morse’s inequalities, which connect the Betti numbers of the supporting manifold with the number of periodic orbits and their Morse indices. Remarkably, in such dynamical systems we can see various topological phenomena connected with the theory of knots and links. The investigation of dynamical systems of such types is quite interesting in the light of the fact that Morse–Smale systems are used to model many regular processes in natural sciences, as well as the cognitive and emotional functions of the brain [1]. Let us dwell in greater detail on the state of the art of studies in the field of Morse–Smale systems on orientable manifolds (also see [22] and [28]). In 1937 Andronov and Pontryagin [3] introduced the concept of a rough system in a bounded part of the plane and found a criterion for the roughness of such a system. Rough systems turn out to have a hyperbolic non-wandering set and no saddle connections (trajectories going from a saddle point to a saddle point). Moreover, they are dense in the space of all flows on the plane. This result was generalized to arbitrary closed surfaces by Peixoto [47], [48], who replaced the concept of roughness by structural stability. (Peixoto also showed there that these concepts are equivalent on the plane). In the early 1960s Smale [64], similarly to Andronov and Pontraygin, introduced the class of dynamical systems with finite non-wandering hyperbolic set whose invariant manifolds intersect transversally. He proved that the numbers of non-wandering orbits with different indices satisfy relation similar to Morse’s inequalities, after which these systems (as well as their discrete analogues) were called Morse–Smale systems. Subsequently, Smale and Palis [45], [46] proved that Morse–Smale dynamical systems (flows and cascades) are structurally stable. Although their non-wandering sets are trivially simple, the topological classification of such systems is still far from completion. Morse–Smale flows have exhaustively been classified, up to topological equivalence, on surfaces (by Leontovich and Meyer [36], [37], Peixoto [49], and Oshemkov and Sharko [44]). Let us list the known results related to the topologoical classification of various classes of Morse–Smale flows on manifolds of dimension 3 or higher. Fleitas [16] obtained a topological classification of polar flows (that is, Morse–Smale flows whose non-wandering set contains precisely two nodes and an arbitrary number of saddle periodic points) on 3-manifolds. Umanskii [65] obtained a topological classification of Morse–Smale flows on 3-manifolds that have a finite number of heteroclinic trajectories. Prishlyak [60] developed a full classification of three-dimensional gradient-like flows (Mores–Smale flows without periodic trajectories). Pilyugin [50] obtained a topological classification of Morse–Smale flows without heteroclinic intersections on a sphere of dimension at least 3. For some classes of multidimensional gradient-like flows classification results were obtained by Grines, Gurevich, Zhuzhoma, and Pochinka [20], [31], [22], [18], [34], [19]. Pochinka and Shubin classified non-singular three-dimensional flows with few periodic orbits [63], [53], [54]. By contrast with flows, Morse–Smale diffeomorphisms of surfaces admit saddle- to-saddle (heteroclinic) trajectories. Such doubly asymptotic motions complicate the dynamics strongly. A classification of Morse–Smale diffeomorphisms was obtained by Bonatti and Langevin [15] as part of their classification of structurally stable diffeomorphisms with zero-dimensional basis sets. With each Smale diffeomorphism they associated a finite combinatorial object, which is a set of the geometric types of Markov decompositions. However, they did not single out Morse–Smale diffeomorphisms for a separate discussion, so that for such diffeomorphims this classification turned out to be unduly laborious. When there are no heteroclinic points, a Morse–Smale diffeomorphism is gradient like, and various full topological invariants for such diffeomorphisms were found by Bezdenezhnykh, Grines, Kapkaeva, and Pochinka [4]–[6], [24]. Grines, Mitryakova, Morozov, and Pochinka [17], [41], [38] obtained a full topological classification of Morse–Smale surface diffeomorphisms with a finite number of heteroclinic orbits. Going over from two-dimensional to higher-dimensional manifolds is difficult not only because there can be heteroclinic orbits (discovered already by Poincaré), but also because separatrices of saddle periodic points can be wildly embedded (so that the closure of such a separatrix is not a submanifold of the ambient space). The existence of wild separatrices of three-dimensional Morse–Smale diffeomorphisms is largely due to the fact that, in general, these diffeomorphisms cannot even be included in a topological flow [21]. The first example of such a diffeomorphism of a 3-sphere was constructed by Pixton [51]. Grines and Bonatti showed that there exists an infinite countable set of pairwise non-conjugate Morse–Smale diffeomorphisms of the 3-sphere with four fixed points that have the same Peixoto graph. The paper [14] by Bonatti, Grines, and Pochinka, where they obtained a full topological classification of Morse–Smale diffeomorphisms of arbitrary closed 3-manifolds, was a completion of the long series of papers [23], [22], [7]–[14], [27], [52], [25], [30] due to Bonatti, Grines, Gurevich, Zhuzhoma, Laudenbach, V. Medvedev, Pécou, and Pochinka and approaching the solution of this problem. In the case when the dimension of the supporting manifold of a diffeomorphism is three, the heteroclinic set can be a non-empty disjoint union of curves. Non-compact hereoclinic curves play a special part in the study of deteriministic processes described by Morse–Smale systems; in the case of a flow they are trajectories, and in the case of a diffeomorphism they are curves invariant under some power of this diffeomorphism. In a series of papers by Priest and Forbes published since the end of the 20th century (see [58] and [59]), they devoted much attention to the description of the topology of the magnetic field in the Sun’s corona, for which so-called separators are important. It is heteroclinic trajectories and curves that provide a mathematical model for separators, and the question of their existence is one of the core problems in magneto-hydrodynamics. Bonatti, Grines, and V. Medvedev [10] found in 2002 an effective sufficient condition for the existence of heteroclinic trajectories and curves and, in conjunction with Zhuzhoma, T. Medvedev, and Pochinka [26], they managed to distinguish separators in the marnetic field of the Sun’s corona. In particular, it follows from the results in [9] that a 3-manifold admits a Morse–Smale diffeomorphism without heteroclinic curve if and only if it is homeomorphic to the connected sum of a finite number of copies of $\mathbb S^2\times\mathbb S^1$, which can explicitly be expressed in terms of the number of saddle and nodal periodic orbits of the diffeomorphism. Grines, Zhuzhoma, and V. Medvedev [29] showed that for a locally flat (tame) embedding of one-dimensional separatrices of saddle poibts the supporting manifold of a gradient-like 3-diffeomorphism has a Heegaard decomposition whose genus is well defined in terms of the numbers of saddle and nodal periodic orbits of the diffeomorphism. Whether or not a similar relation exists in the case when separatrices are wildly embedded, is so far an open question. In this paper we consider the class $G$ of gradient-like diffeomorphisms $f$ defined on oriented closed connected manifolds $M^3$ and whose non-wandering points have pairwise distinct indices. It follows from this definition that the non-wandering set of a diffeomorphism $f\in G$ consists of precisely four points $\omega_f$, $\sigma_f^{1}$, $\sigma_f^{2}$, and $\alpha_f$ with Morse indices $0,1,2$, and $3$, respectively. First examples of such diffeomorphisms were produced by Kruglov and Talanova [35]. Each diffeomorphism $f\in G$ has precisely two saddle points $\sigma_f^1$ and $\sigma_f^2$ with Morse indices 1 and 2, respectively, and the intersection of their two-dimensional invariant manifolds forms the heteroclinic set
$$
\begin{equation*}
H_f=W^{\rm s}_{\sigma_f^1}\cap W^{\rm u}_{\sigma_f^2}.
\end{equation*}
\notag
$$
It follows from the results in [29] that for a tame embedding of one-dimensional separatrices the supporting manifold of $f\in G$ is homeomorphic to a lens space $L_{p,q}$, and then the set $H_f$ contains at least $p$ non-compact heteroclinic curves. The opposite is also true: on each lens space there exists a diffeomorphism in $G$ with tamely embedded one-dimensional separatrices. The main aim of this paper is to show that the class under consideration contains diffeomorphisms with wild separactrices, to describe the topology of the supporting manifold of such a diffeomorphism, and to find topological invariants and construct quasi-energy functions for diffeomorphisms in certain subclasses of $G$. Given some $f\in G$, we define its heterclinic index $I_f$ as follows. If the set $H_f$ contains no non-compact curves, then we set $I_f=0$. Otherwise let $\widetilde H_f$ denote the subset consisting of non-compact curves. Since each curve $\gamma\subset \widetilde H_f$, together with a point $x\in\gamma$, contains also the point $f(x)$, we assume that $\gamma$ is oriented in the direction from $x$ to $f(x)$. We also fix orientations on the manifolds $W^{\rm s}_{\sigma_1}$ and $W^{\rm u}_{\sigma_2}$. Given a non-compact heteroclinic curve $\gamma$, let
$$
\begin{equation*}
v_\gamma=(\vec v^1_\gamma,\vec v^2_\gamma,\vec v^3_\gamma)
\end{equation*}
\notag
$$
denote the triple of vectors with initial point $x\in\gamma$ such that $\vec v^1_\gamma$ is the normal vector to $W^{\rm s}_{\sigma_1}$, $\vec v^2_\gamma$ is the normal vector to $W^{\rm u}_{\sigma_2}$, and $\vec v^3_\gamma$ is the tangent vector to the oriented curve $\gamma$. We call $v_\gamma$ a frame on the noncompact heteroclinic curve $\gamma$. It is obvious that the orientation of $v_\gamma$ (which can be positive or negative) is independent of the choice of the point $x$ on $\gamma$. Set $I_\gamma=+1$ ($I_\gamma=-1$) in the case when it is positive (negative. respectively). We call the quantity
$$
\begin{equation*}
I_{f}=\biggl|\sum_{\gamma\subset \widetilde H_f}I_\gamma\biggr|
\end{equation*}
\notag
$$
the heteroclinic index of the diffeomorphism $f$. For a non-negative integer $p$ let $G_p\subset G$ denote the subset consisting of $f\in G$ such that $I_{f}=p$. In a similar way we define a frame on a compact heteroclinic curve $\gamma$ bounding a disc $d_\gamma\subset W^{\rm s}_{\sigma_f^1}$ containing the saddle point $\sigma_f^1$. In this case $\gamma$ is oriented so that the disc $d_\gamma$ lies to the left of $\gamma$. For $f\in G_p$, $p>0$, we say that the set $H_f$ is orientable if it contains only non-compact surfaces and the frames on all curves in $H_f$ have the same orientation. For $f\in G_0$ we say that $H_f$ is orientable if it is either empty of contains only compact curves bounding discs in $W^{\rm s}_{\sigma_f^1}$ containing the saddle point $\sigma_f^1$, and the frames on all curves in $H_f$ have the same orientation. Let $G^+_p\subset G_p$, $p\geqslant 0$, denote the subset of diffeomorphisms $f\in G_p$ such that the set $H_f$ is orientable. The following result, proved in our paper, is the key to the description of the topology of manifolds admitting diffeomorphisms in the class $G$: The full topological invariant obtained in [14] for a Morse–Smale 3-diffeomorphism consists of a closed connected orientable simple 3-manifold and two laminations by tori and Klein bottles embedded in it and intersecting transversally. It [13] all admissible invariants were distinguished and a Morse–Smale diffeomorphism was realized for each of them. However, because there is no classification of simple 3-manifolds and laminations embedded in them, we cannot always realize a diffeomorphism with prescribed properties. In some special case invariants can be defined in a more natural way. For example, for diffeomorphisms with a unique saddle point (Pixton difeomorphisms) it was shown in [7] that their topological conjugacy is fully determined by the equivalence of the Hopf knots (knots $L\subset\mathbb S^2\times\mathbb S^1$ in the homotopy class of the standard knot $L_0=\{s\}\times\mathbb S^1$) obtained as the projections of one-dimensional unstable saddle separatrices onto the orbit space of the sink basin corresponding to the diffeomorphism in question. Hopf knots can be divided into trivial (equivalent to the standard knot) and non-trivial ones. It follows from the results of Akhmet’ev and Pochinka in [2] that there exist a countable number of pairwise inequivalent (not ambiently homeomorphic) non-trivial Hopf knots. By [51] and [7] each Hopf knot can be realized by means of a Pixton diffeomorphism of the 3-sphere. In this paper we obtain a similar result for diffeomorphisms in the class $G_1^+$: One of the central results in this paper is the following fact which we prove: We also present a constructive proof of the following result: Note that, previously, examples of diffeomorphisms in the above class with wildly embedded one-dimensional separatrices were only constructed on the 3-sphere. 1. Requisite information and facts1.1. Topology factsFor each subset $X$ of the topological space $Y$ let $i_X\colon X\,{\to}\,Y$ denote the inclusion map. For each continuous map $\phi\colon X\to Y$ from the topological space $X$ to the topological space $Y$ we let $\phi_*\colon \pi_1(X)\to\pi_1(Y)$ denote the induced homomorphism. By a $C^{r}$-embedding $(r\geqslant 0)$ of a manifold $X$ in a manifold $Y$ we mean a map $f\colon X \to Y$ such that $f\colon X\to f(X)$ is a $C^r$-diffeomorphism. A $C^0$-embedding is also called a topological embedding. A topological embedding $\lambda\colon X \to Y$ of an $m$-manifold $X$ in an $n$-manifold $Y$ ($m\leqslant n$) is said to be locally flat at a point $\lambda(x)$, $x \in X$, if the point $\lambda(x)$ lies in a chart $(U,\psi)$ on $Y$ such that either $\psi (U \cap \lambda(X))=\mathbb{R}^{m}$, where $\mathbb{R}^{m} \subset\mathbb{R}^{n}$ is the set of points with vanishing last $n-m$ coordinates, or $\psi(U \cap \lambda(X))=\mathbb{R}^{m}_{+}$, where $\mathbb{R}^{m}_{+} \subset\mathbb{R}^{m}$ is the set of points with non-negative last coordinate. The embedding $\lambda$ is tame, and $X$ is said to be tamely embedded, if $\lambda$ is locally flat at each point $\lambda(x)$, $x\in X$. Otherwise $\lambda$ is said to be wild and $X$ is wildly embedded. A point $\lambda(x)$ at which $\lambda$ is not locally flat is called a wild point. Let
$$
\begin{equation*}
\mathbb{D}^{n}=\biggl\{(x_{1},\dots, x_{n})\in \mathbb{R}^{n}\colon \sum_{i=1}^{n} x^{2}_{i} \leqslant 1\biggr\}
\end{equation*}
\notag
$$
be the standard $n$-disc (ball), where $\mathbb{D}^{0}=\{0\}$, and let
$$
\begin{equation*}
\mathbb{S}^{n-1}=\biggl\{(x_{1},\dots,x_{n})\in \mathbb{R}^{n}\colon \sum_{i=1}^{n} x^{2}_{i}=1\biggr\}
\end{equation*}
\notag
$$
be the standard $(n-1)$-sphere, where $\mathbb{S}^{-1}=\varnothing$. Proposition 1.1 ([10], Lemma 2.1). Let $\lambda\colon \mathbb{S}^2\to M^3$ be a topological embedding that is smooth everywhere away from a point $x_{0}=\lambda(s_{0}) $, $s_{0}\in \mathbb{S}^2$, let $\Sigma=\lambda(\mathbb{S}^2) $, let $y_0\in \Sigma\setminus\{x_0\}$ be a fixed point and $V$ be a fixed neighbourhood of the sphere $\Sigma$. Then there exists a smooth 3-ball $B$ in $V$ such that $x_0\in B$ and $\partial B$ intersects $\Sigma$ transversally along a single curve, which separates the points $x_0$ and $y_0$ in $\Sigma$. An $(n-1)$-sphere ${S}^{n-1}$ embedded topologically in an $n$-manifold $X$ is said to be cylindrical or embedded cylindrically if there exists a topological embedding $e\colon\mathbb{S}^{n-1}\times[-1,1]\to X$ such that $e(\mathbb{S}^{n-1}\times\{0\}) ={S}^{n-1}$. An $n$-manifold $X$ is irreducible if each $(n-1)$-sphere cylindrically embedded in $X$ bounds an $n$-ball there. A 3-manifold $X$ is said to be simple if it is either irreducible or homeomorphic to $\mathbb{S}^{2}\times{\mathbb{S}^{1}}$. A surface $F$ embedded topologically in a $3$-manifold $X$ is said to be propertly embedded if $\partial X \cap F=\partial F$. A surface $F$ properly embedded in $X$ is said to be compressible in $X$ in each of the following two cases: (1) there exist a non-contractible closed curve $c \subset \operatorname{int} F$ and an embedded 2-disc $D \subset \operatorname{int} X$ such that $D \cap F=\partial D=c$; (2) there exists a 3-ball $B \subset \operatorname{int} X$ such that $F=\partial B$. A surface $F$ is said to be incompressible in $X$ if it is not compressible there. Proposition 1.2 ([7], Theorem 4). Let $T$ be a 2-torus smoothly embedded in $\mathbb{S}^{2}\times \mathbb{S}^{1}$ so that $i_{T*}(\pi_1(T))\ne 0$. Then $T$ bounds a solid torus in $\mathbb{S}^{2}\times \mathbb{S}^{1}$. Proposition 1.3 ([7], Lemma 3.1). Let $S$ be a 2-sphere embedded cylindrically in $\mathbb{S}^{2}\times \mathbb{S}^{1}$. Then $S$ either bounds a 3-ball there or is ambiently isotopic to the sphere $\mathbb S^2\times\{s_0\}$, $s_0\in\mathbb S^1$. Proposition 1.4 ([43], Exercise 6). Each two-sided compressible torus $T$ in an irreducible 3-manifold $X$ either bounds a solid torus there or lies in a 3-ball. Proposition 1.5 ([32], Chap. 4, § 5, Corollary 1). No $n$-dimensional manifold can be disconnected by a subset of dimension $\leqslant n -2$. Recall that by a knot in a manifold $M^n$ we mean a smooth embedding $\gamma\colon\mathbb S^1\to M^n$ or its image $L=\gamma(\mathbb S^1)$. Two knots $L$ and $L'$ are said to be equivalent if there exists a homomorphism $h\colon M^n\to M^n$ such that $h(L)=L'$. We let $[L]$ denote the equivalence class of the knot $L$. Proposition 1.6 [42]. If two knots $L,L'\subset M^n$, $n\leqslant 3$, are equivalent, then there exists a diffeomorphism $h\colon M^n\to M^n$ such that $h(L)=L'$. Two knots $\gamma$ and $\gamma'$ are (smoothly) homotopic if there exists a (smooth) continuous map $\Gamma\colon\mathbb S^1\times[0,1]\to M^n$ such that
$$
\begin{equation*}
\Gamma(s,0)=\gamma(s)\quad\text{and}\quad \Gamma(s,1)=\gamma'(s) \quad\text{for each}\ \ s\in\mathbb S^1.
\end{equation*}
\notag
$$
If, in addition, $\Gamma|_{\mathbb S^1\times\{t\}}$ is an embedding for each $t\in[0,1]$, then the knots are said to be (smoothly) isotopic. A knot $L\subset\mathbb S^2\times\mathbb S^1$ is called 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 a group isomorphism $\pi_1(L)\cong\pi_1(\mathbb S^2\times\mathbb S^1)\cong\mathbb Z$. Each Hopf knot is smoothly homotopic to the standard Hopf knot $L_0=\{x\}\times\mathbb S^1$ (for instance, see [33]), but in general they are not isotopic or equivalent. Masur [39] constructed a Hopf knot $L_{\rm M}$ that is neither equivalent nor isotopic to $L_0$ (see Fig. 1). In [2] a countable family of Hopf knots $L_n$ was constructed (see Fig. 2) whose pairwise inequivalence was also proved there. Recall that a lens space is by defintiion obtained by gluing two solid tori by means of a homeomorphism between their boundries; it is denoted by $L_{p,q}$, $p,q\in\mathbb Z$, where ${\langle p,q\rangle}$ is the homotopy type of the image of the meridian under the gluing homeomorphism. Some well-known $3$-manifolds are in fact lens spaces, for example, the 3-sphere $\mathbb S^3=L_{1,0}$, the manifold $\mathbb S^2\times\mathbb S^1=L_{0,1}$, and the projective space $\mathbb RP^3=L_{1,2}$. 1.2. Morse–Smale diffeomorphismsLet $M^n$ be a smooth closed orientable $n$-manifold with metric $d$, and let $f\colon M^n\to M^n$ be an orientation-preserving diffeomorphism. To characterise the wandering property of trajectories of this diffeomorphism one usually uses the concept of chain recurrence. First we recall the following definition: an $\varepsilon$-chain of length $m\in\mathbb N$ of a diffeomorphism $f$ which connects a point $x$ with a point $y$ is a sequence of point $x=x_0,\dots,x_m=y$ such that $d(f(x_{i-1}),x_{i})<\varepsilon$ for $1\leqslant i\leqslant m$. A point $x\in M^n$ is a chain recurrent point of $f$ if for each $\varepsilon>0$ there exists $m$ depending on $\varepsilon>0$ and an $ \varepsilon$-chain of length $m$ that connects $x$ with itself. The set of chain recurrent points of $f$ is called the chain recurrent set of $f$; it is denoted by $\mathcal R_{f}$. We can introduce an equivalence relation on the chain recurrent set as follows: $x\sim y$ if for each $\varepsilon>0$ there exist $\varepsilon$-chains connecting $x$ with $y$ and $y$ with $x$. Then the chain recurrent set falls into equivalence classes, which are called chain components. If the chain recurrent set of a diffeomorphism $f$ is finite, then it consists of periodic points. A periodic point $p\in\mathcal R_f$ with period $m_p$ is called hyperbolic if all the eigenvalues of the Jacobian matrix $\biggl(\dfrac{\partial f^{m_p}}{\partial x}\biggr)\bigg|_{p}$ are distinct from 1 in modulus. If all eigenvalues are smaller (larger) than 1 in modulus, then $p$ is said to be a sink (respectively, a source) point. Both sink and source points are called nodal points. If a hyperbolic periodic point is not nodal, then it is called a saddle point. If a periodic point $p$ has the hyperbolic structure and the chain recurrent set is finite, then the stable manifold
$$
\begin{equation*}
W^{\rm s}_p=\Bigl\{x\in M^n\colon\lim_{k\to+\infty}d(f^{km_p}(x),p)=0\Bigr\}
\end{equation*}
\notag
$$
and the unstable manifold
$$
\begin{equation*}
W^{\rm u}_p=\Bigl\{x\in M^n\colon \lim_{k\to+\infty}d(f^{-km_p}(x),p)=0\Bigr\}
\end{equation*}
\notag
$$
of $p$ are smooth submanifolds diffeomorphic to $\mathbb R^{n-q_p}$ and $\mathbb R^{q_p}$, respectively, where $q_p$ is the number of eigenvalues of the Jacobian matrix with modulus greater than 1 (the Morse index of $p$). Stable and unstable manifolds are called invariant manifolds. A connected component of the set $W^{\rm u}_p\setminus p$ (or $W^{\rm s}_p\setminus p$) is called an unstable (respectively, stable) separatrix of $p$. A diffeomorphism $f\colon M^n\to M^n$ is called a Morse–Smale diffeomorphism if the following conditions are satisfied: (1) the chain recurrent set $\mathcal R_f$ consists of a finite number of hyperbolic points; (2) for any points $p,q\in\mathcal R_f$ the manifolds $W^{\rm s}_p$ and $W^{\rm u}_q$ intersect transversally. Let $\operatorname{MS}(M^{n})$ denote the set of orientation-preserving Morse–Smale diffeomorphisms defined on an orientable $n$-manifold $M^n$. Note that by hyperbolicity the chain recurrent set of each Morse–Smale diffeomorphism coincides with the non- wandering set $\Omega_f$. Recall that a point $x\in M^n$ is called a wandering point of a diffeomorphism $f\colon M^n\to M^n$ if it has a neighbourhood $U_x\subset M^n$ such that
$$
\begin{equation*}
f^k(U_x)\cap U_x=\varnothing\quad\text{for each } k\ne 0.
\end{equation*}
\notag
$$
The complement to the set of wandering points is called the non-wandering set of $f$. Given a set $P\subset\Omega_f$, we use the notation
$$
\begin{equation*}
W^{\rm u}_P=\bigcup_{p\in P} W^{\rm u}_p\quad\text{and}\quad W^{\rm s}_P=\bigcup_{p\in P} W^{\rm s}_p.
\end{equation*}
\notag
$$
A compact $f$-invariant set $A\subset M^n$ is called an attractor of the diffeomorphism $f$ if it has a compact neighbourhood $U_{A}$ such that
$$
\begin{equation*}
f(U_{A})\subset \operatorname{int} U_{A}\quad\text{and}\quad A=\bigcap_{k\geqslant 0} f^{k}(U_{A}).
\end{equation*}
\notag
$$
In this case $U_{A}$ is called a trapping (or isolating) neighbourhood. A repeller is defined as an attractor for the diffeomorphism $f^{-1}$. In the case when $p$ is a hyperbolic periodic point of $f\in \operatorname{MS}(M^n)$ we use the following notation:
Proposition 1.7 ([25], Theorem 2.1.1). Let $f\in\operatorname{MS}(M^{n})$. Then (1) $M^{n}=\bigcup_{p\in{\mathcal R}_{f}}W^{\rm u}_{p}$; (2) $W^{\rm u}_{p}$ is a smooth submanifold of $M^{n}$, which is diffeomorphic to $\mathbb{R}^{q_{p}}$ for each periodic point $p\in{\mathcal R}_{f}$; (3) $\operatorname{cl}(\ell^{\rm u}_{p})\setminus(\ell^{\rm u}_{p}\cup p)= \bigcup_{r\in{\mathcal R}_{f}: \ell^{\rm u}_{p}\cap W^{\rm s}_{r}\ne\varnothing}W^{\rm u}_{r}$ for each unstable separatrix $\ell^{\rm u}_{p}$ of the periodic point $p\in{\mathcal R}_{f}$. A similar result for stable manifold is proved by going over to the diffeomorphism $f^{-1}$. If $\sigma_{1}$ and $\sigma_{2}$ are distinct saddle periodic points of a diffeomorphism $f\in \operatorname{MS}(M^{n})$ such that $W^{\rm s}_{\sigma_{1}}\cap W^{\rm u}_{\sigma_{2}}\ne\varnothing$, then the intersection $W^{\rm s}_{\sigma_{1}}\cap W^{\rm u}_{\sigma_{2}}$ is called a heteroclinic intersection. In this case zero-dimensional path-connected components of the heteroclinic intersection are called heteroclinic points, one-dimensional components are called heteroclinic curves, and components of greater dimension are called heteroclinic manifolds. A diffeomorphism $f \in \operatorname{MS}(M^n)$ is called gradient like if the condition $W^{\rm s}_{\sigma_1} \cap W^{\rm u}_{\sigma_2} \ne \varnothing$ for distinct points $\sigma_1, \sigma_2 \in {\mathcal R}_f$ implies that $\dim W^{\rm u}_{\sigma_1}< \dim W^{\rm u}_{\sigma_2}$ (which is equivalent to the absence of heteroclinic points of $f$). Proposition 1.8 ([25], Proposition 2.1.3). If a separatrix $\ell^{\rm u}_p$ of a saddle point $p$ of a diffeomorphism $f\in \operatorname{MS}(M^{n})$ is not involved in a heteroclinic intersection, then there exists a unique sink point $\omega$ such that In this case $\operatorname{cl}(\ell^{\rm u}_p)$ is homeomorphic to a line segment for $q_p=1$, or to the sphere $\mathbb S^{q_p}$ for $q_p>1$. Set $\widehat{W}^{\rm u}_{p}=(W^{\rm u}_{p}\setminus {p})/f^{m_p}$, and let $p_{\widehat{{W}}^{\rm u}_{p}}\colon W^{\rm u}_{p} \setminus {p} \to \widehat{{W}}^{\rm u}_{p} $ denote the natural projection. Proposition 1.9 ([25], Theorem 2.1.3). The projection $p_{\widehat{W}^{\rm u}_{p}}$ is a covering, which induces the structure of a smooth $q_{p}$-manifold on the orbit space $\widehat{W}^{\rm u}_{p}$. Moreover, Let $f\in \operatorname{MS}(M^n)$, and let $\Omega_f^i$ denote the set of periodic points of $f$ with Morse index $i$. We partition $\Omega^1_f\cup\Omega^2_f\cup\cdots\cup\Omega^{n-1}_f$ into two disjoint subsets $\Sigma_{A}$ and $\Sigma_{R}$ such that the sets
$$
\begin{equation*}
A=\Omega^0_f\cup W^{\rm u}_{\Sigma_{A}}\quad\text{and}\quad R=\Omega^n_f\cup W^{\rm s}_{\Sigma_{R}}
\end{equation*}
\notag
$$
are closed and invariant. By construction $A$ and $R$ contain all periodic points of $f$. We call the greatest dimension of the unstable (stable) manifolds of periodic points in $A$ (in $R$, respectively) the dimenson of $A$ (of $R$, respectively). Proposition 1.10 ([30], Theorem 1). Let $f\in \operatorname{MS}(M^n)$. Then $A$ is an attractor and $R$ is a repeller of the diffeomorphism $f$. Moreover, if the dimension of $A$ (of $R$) is at most $n-2$, then the repeller $R$ (respectively, the attractor $A$) is connected. Following [30], we call $A$ and $R$ the dual attractor-repeller pair of the Morse–Smale diffeomorphism $f\in \operatorname{MS}(M^n)$, and we call $V=M^n\setminus(A\cup R)$ the characteristic set. Let denote the set of orbits of the action of the group $F=\{f^k,k\in\mathbb{Z}\}$ on $V$, the characteristic space, which coincides with the set of orbits of $f$ on $V$. Let be the natural projection, which assigns to a point $x\in V$ its $f$-orbit and endows the set $\widehat V$ with the quotient topology.Proposition 1.11 ([30], Theorem 2). For each dual attractor-repeller pair $(A,R)$ of an orientation-preserving Mortse–Smale diffeomorphism $f\in \operatorname{MS}(M^n)$ the following results hold: (1) the characteristic set $\widehat V$ is a closed smooth orientable $n$-manifold each of whose connected components is either irreducible or homeomorphic to $\mathbb{S}^{n-1}\times\mathbb{S}^1$; (2) the projection $p_{\widehat V}\colon V\to\widehat V$ is a covering map; (3) the map $\eta_{\widehat V}$ assigning to each homotopy class $[\widehat c]$ of loops $\widehat c\subset\widehat V$ through a point $\widehat x$ the integer $n$ such that the lift of a loop $\widehat c$ to $V$ connects a point $x\in p^{-1}_{\widehat V}(\widehat x)$ with the point $f^n(x)$, induces a homomorphism of the fundamental group of each connected component of the space $\widehat V$; (4) if the attractor $A$ and the repeller $R$ have dimension at most $n-2$, then the sets $V$ and $\widehat V$ are connected, and the map $\eta_{\widehat V}\colon \pi_1(\widehat V)\to\mathbb Z$ is an epimorphism. We say that a submanifold $\widehat X\subset\widehat V$ is $\eta_{\widehat V}$-essential if
$$
\begin{equation*}
\eta_{\widehat V}(i_{\widehat X*}(\pi_1(\widehat X)))\ne \{0\}.
\end{equation*}
\notag
$$
Let $ U_{A}$ be a trapping neighbourhood of the attractor $A$ of a Morse–Smale diffeomorphism $f\colon M^n\to M^n$, and let $R$ be the repeller dual to $A$. Set then $\operatorname{cl}(F_{A})$ is a fundamental domain of the restriction of $f$ to $V$. Set $\widehat V_{A}=\operatorname{cl}(F_{A})/f$; then $\widehat V_{A}$ is a smooth closed $n$-manifold obtained from $\operatorname{cl}(F_{A})$ by identifying the boundaries by means of $f$. Let $p_{A}\colon \operatorname{cl}(F_{A})\to \widehat V_{A}$ denote the natural projection.Consider the family $E_f\in \operatorname{Diff}(M^n)$ of diffeomorphisms such that ${\mathcal R}_{f'}= {\mathcal R}_{f}$ for each diffeomorphism $f'\in E_f$ and $f'$ coincides with $f$ in $U_{A}$ and in a neighbourhood of the repeller $R$. For each diffeomorphism $f'\in E_f$ set
$$
\begin{equation*}
\widehat l^{\rm s}_{f'}=p_{A}(W^{\rm s}_{\Sigma_{A}}\cap F_{A})\quad\text{and}\quad \widehat l^{\rm u}_{f'}=p_{A}(W^{\rm u}_{\Sigma_{R}}\cap F_{A}).
\end{equation*}
\notag
$$
Proposition 1.12 ([62], Lemma 1). Let $\widehat h\colon \widehat V_{A} \to \widehat V_{A}$ be a diffeomorphism isotopic to the identity map. Then there exists a smooth arc ${\varphi}_{t}\subset E_f$ such that 1.3. Classification of Morse–Smale 3-diffeomorphismsLet $f\in \operatorname{MS}(M^3)$. Set
$$
\begin{equation*}
A_f=W^{\rm u}_{\Omega_f^0\cup\Omega_f^1},\qquad R_f=W^{\rm s}_{\Omega_f^2\cup\Omega_f^3},\quad\text{and}\quad V_f=M^3\setminus(A_f\cup R_f).
\end{equation*}
\notag
$$
By Proposition 1.11 the set $A_f$ (or $R_f$) is a connected attractor (repeller) of topological dimension at most 1, $V_f$ is a connected 3-manifold, and
$$
\begin{equation*}
V_f=W^{\rm s}_{A_f\cap \Omega_f}\setminus A_f= W^{\rm u}_{R_f\cap \Omega_f}\setminus R_f.
\end{equation*}
\notag
$$
Moreover, the characteristic space $\widehat V_f=V_f/f$ is a connected closed orientable 3-manifold, and the natural projection $p_{f}\colon V_f\to\widehat V_f$ induces an epimorphism $\eta_{f}\colon \pi_1(\widehat V_f)\to\mathbb Z$ that assigns to each homotopy class $[c]\in\pi_1(\widehat V_f)$ of a closed curve $c\subset\widehat V_f$ the integer $n$ such that the lift of $c$ to $V_f$ connects some point $x\in V_f$ with $f^n(x)$. Set
$$
\begin{equation*}
\widehat{L}^{\rm s}_f=p_{f}(W^{\rm s}_{\Omega_f^1}\setminus A_f)\quad\text{and}\quad \widehat{L}^{\rm u}_f=p_{f}(W^{\rm u}_{\Omega_f^2}\setminus R_f).
\end{equation*}
\notag
$$
The set $S_{f}=(\widehat V_{f},\eta_{_{f}},\widehat{L}^{\rm s}_{f},\widehat{L}^{\rm u}_{f})$ is called the scheme of the diffeomorphism $f\in \operatorname{MS}(M^3)$. Proposition 1.13 ([14], Theorem 1). Two diffeomorphisms $f,f'\in \operatorname{MS}(M^3)$ are topologically conjugate if and only if their schemes are equivalent, that is, there exists a heomeomorphism $\widehat\varphi\colon\widehat V_{f}\to\widehat V_{f'}$ such that (a) $\eta_{f}=\eta_{_{f'}}\widehat\varphi_*$; (b) $\widehat\varphi(\widehat L^{\rm s}_f)=\widehat L^{\rm s}_{f'}$, $\widehat\varphi(\widehat L^{\rm u}_f)=\widehat L^{\rm u}_{f'}$. To solve the realization problem we must distinguish the set of abstract schemes admitting realization (that is, the construction of a Morse–Smale diffeomorphism whose scheme is equivalent to the abstract scheme in question). Let $\widehat V$ be a simple smooth 3-manifold with fundamental group admitting an epimorphism $\eta\colon\pi_1(\widehat V)\to \mathbb Z$, and let $\widehat{\ell}\subset\widehat V$ be an $\eta$-essential smooth torus and $N_{\widehat{\ell}}\subset\widehat V$ be a tubular neighbourhood of this torus. Let $\widehat Y=\mathbb D^2\times\mathbb S^1$, and let $\widehat\mu$ be a meridian of the solid torus $\widehat Y$ (a closed curve contractible in $\widehat Y$ and not contractible on $\partial\widehat Y$), and $\zeta_\ell\colon\partial\widehat Y\times\mathbb S^0\to \partial N_{\widehat{\ell}}$ be a diffeomorphism such that $\eta(\zeta_\ell(\widehat{\mu}\times\mathbb S^0))=0$. We say that the space
$$
\begin{equation*}
\widehat V_{\widehat{\ell}}=(\widehat V\setminus \operatorname{int}N_{\widehat{\ell}}) \,\cup_{\zeta_\ell}(\widehat{Y}\times\mathbb S^0)
\end{equation*}
\notag
$$
is obtained from the manifold $\widehat V$ by means of a surgery along the torus $\widehat{\ell}$. The structure of a closed 3-manifold is induced on the set $\widehat V_{\widehat{\ell}}$ by the natural projeciton $p_{\widehat{\ell}}\colon(\widehat V\setminus \operatorname{int} N_{\widehat{\ell}}) \sqcup(\widehat{Y}\times\mathbb S^0)\to\widehat V_{\widehat{\ell}}$. Since each homeomorphism of the boundary of the solid torus taking a meridian to a meridian extends to the solid torus [61], this surgery is well defined, that is, it is independent (up to a homomorphism) on the choice of the tubular neighbourhood and the homeomorphism $\zeta_\ell$. In a similar was we can define a surgery of $\widehat V$ along an $\eta$-essential smooth Klein bottle $\widehat{\ell}\subset\widehat V$, which consists in gluing a solid torus $\widehat Y$ to the boundary of the manifold $\widehat V\setminus \operatorname{int} N_{\widehat{\ell}}$. This surgery can also be generalized to the set $\widehat L\subset\widehat V$, which is a disjoint union of smooth $\eta$-essential tori and Klein bottles. As a result, we obtain a manifold denoted by $V_{\widehat L}$. By Propositions 1.9 and 1.11, for a gradient-like diffeomorphism $f\in \operatorname{MS}(M^3)$ each connected component $\widehat\ell^{\rm s}$ (or $\widehat\ell^{\rm u}$) of the set $\widehat L^{\rm s}_{f}$ (of $\widehat L^{\rm u}_f$, respectively) is a torus or a Klein bottle which is $\eta_{f}$-essentially embedded in the manifold $\widehat V_f$. The scheme of each gradient-like diffeomorphism $f\in \operatorname{MS}(M^3)$ is an abstract scheme in the sense of the following definition. A set ${\mathcal S}=(\widehat V,\eta,\widehat{L}^{\rm s},\widehat{L}^{\rm u})$ is called an abstract scheme whenever (a) $\widehat V$ is a simple manifold whose fundamental group admits an epimorphism $\eta\colon\pi_1(\widehat V)\to \mathbb Z$; (b) the sets $\widehat{L}^{\rm s},\widehat{L}^{\rm u}\subset \widehat V$ are transversally intersecting disjoint unions of smooth $\eta$-essential tori and Klein bottles; (c) each connected component of the manifolds $\widehat V_{\widehat{L}^{\rm s}}$ and $\widehat V_{\widehat{L}^{\rm u}}$ is homeomorphic to $\mathbb S^2\times\mathbb S^1$. Proposition 1.14 ([13], Theorem 1). For each abstract scheme ${\mathcal S}=(\widehat V,\eta,\widehat{L}^{\rm s},\widehat{L}^{\rm u})$ there exists a gradient-like diffeomorphism $f\in \operatorname{MS}(M^3)$ such that its scheme $S_f$ is equivalent to ${\mathcal S}$. 1.4. The topology of 3-manifolds admitting Morse–Smale diffeomorphisms with prescribed structure of the non-wandering setLet $f\in \operatorname{MS}(M^3)$. Set where $r_{f}$ is the number of saddle periodic points of $f$ and $l_{f}$ is the number of nodal ones. By [25], $g_{f}$ is a non-negative integer for each diffeomorphism $f\in \operatorname{MS}(M^3)$.Let $f\in \operatorname{MS}(M^3)$ be a gradient-like diffeomorphism. Then by Proposition 1.8 the closure $\operatorname{cl}(\ell^{\rm u}_\sigma)$ of each one-dimensional unstable separatrix $\ell^{\rm u}_\sigma$ of a saddle point $\sigma$ of $f$ is homeomorphic to a line segment and consists of this separatrix and two points, $\sigma$ and the sink $\omega$. Let ${L}_\omega$ be the union of the unstable one-dimensional separatrices of saddle points whose closures contain $\omega$. By Proposition 1.9, $W^{\rm s}_\omega$ is homeomorphic to $\mathbb R^3$, and the set $L_\omega\cup\omega$ is a union of simple arcs with the only common point $\omega$, and so (by analogy with a pencil of arcs in $\mathbb R^3$) the set $L_\omega\cup\omega$ is called a pencil of one-dimensional unstable separatrices. By [29] a pencil of separatrices $L_\omega\cup\omega$ is said to be tame if there exists a homeomorphism $\psi_\omega\colon W^{\rm s}_\omega \to \mathbb R^3$ such that $\psi_\omega(L_\omega\cup\omega)$ is a pencil of rays in $\mathbb R^3$ with initial point $O(0,0,0)$. Otherwise such a pencil is said to be wild (see Fig. 3). If $\alpha$ is a source of the diffeomorphism $f$, then we can similarly define a tame (a wild) pencil $L_\alpha$ of one-dimensional stable separatrices. Proposition 1.15 ([10], Theorem 1). Let $f\in \operatorname{MS}(M^3)$ be a Morse–Smale diffeomorphism without heteroclinic curves. Then the following results hold: (1) if $g_{f}=0$, then $M^3$ is a 3-sphere; (2) if $g_{f}>0$, then $M^3$ is a connected sum of $g_{f}$ copies of $\mathbb S^2\times\mathbb S^1$. Conversely, for any non-negative integers $r$ and $l$ such that $g=(r-l+2)/2$ is a non-negative integer, there exists a diffeomorphism $f\in \operatorname{MS}(M^3)$ without heteroclinic curves with the following properties: (a) $M^3$ is a 3-sphere for $g=0$, or $M^3$ is a connected sum of $g$ copies of $\mathbb S^2\times\mathbb S^1$ for $g>0$; (b) the non-wandering set of $f$ consists of $r$ saddle and $l$ nodal points. 2. Dynamics of a diffeomorphism in the class $G$In this section we establish several dynamical properties of diffeomorphisms $f\colon M^3\to M^3$ in the class $G$. Recall that $G$ consists of the diffeomorphisms $f\in \operatorname{MS}(M^3)$ with precisely four non-wandering points $\omega_f$, $\sigma_f^{1}$, $\sigma_f^{2}$, and $\alpha_f$ whose Morse indices are $0,1,2$, and $3$, respectively. Since $f$ has no heteroclinoc points, the closure of each one-dimensional saddle manifold contains a unique nodal point (see Proposition 1.8). Namely,
$$
\begin{equation*}
\operatorname{cl}(W^{\rm u}_{\sigma_f^1})= W^{\rm u}_{\sigma_f^1} \cup \omega_f\quad\text{and}\quad \operatorname{cl}(W^{\rm s}_{\sigma_f^2})= W^{\rm s}_{\sigma_f^2}\cup \alpha_f.
\end{equation*}
\notag
$$
Moreover, by Proposition 1.9 the sets $A_f=\operatorname{cl}(W^{\rm u}_{\sigma_f^1})$ and $R_f=\operatorname{cl}(W^{\rm s}_{\sigma_f^2})$ are pairwise disjoint topologically embedded circles (see Fig. 4), which can be wild at nodal points (see Fig. 5). Set
$$
\begin{equation*}
H_f=W^{\rm s}_{\sigma_f^1}\cap W^{\rm u}_{\sigma_f^2}.
\end{equation*}
\notag
$$
If the set $H_f$ is non-empty, then by Proposition 1.7
$$
\begin{equation*}
\operatorname{cl}(W^{\rm s}_{\sigma_f^1})= W^{\rm s}_{\sigma_f^1} \cup R_f\quad\text{and}\quad \operatorname{cl}(W^{\rm u}_{\sigma_f^2})= W^{\rm u}_{\sigma_f^2} \cup A_f.
\end{equation*}
\notag
$$
Otherwise, by Proposition 1.8 the sets
$$
\begin{equation*}
\operatorname{cl}(W^{\rm s}_{\sigma_f^1})= W^{\rm s}_{\sigma_f^1} \cup \alpha_f\quad\text{and}\quad \operatorname{cl}(W^{\rm u}_{\sigma_f^2})= W^{\rm u}_{\sigma_f^2} \cup \omega_f
\end{equation*}
\notag
$$
are top[ologically embedded disjoint 2-spheres (see Fig. 6). 2.1. A consistent system of neighbourhoodsFor $t\in(0,1]$ set
$$
\begin{equation*}
\begin{aligned} \, \mathcal N_{1}^t&=\{(x_1,x_2,x_3)\in\mathbb{R}^3\colon x_1^2(x_2^2+x_3^2)\leqslant t\}, \\ \mathcal N_{2}^t&=\{(x_1,x_2,x_3)\in\mathbb{R}^3\colon (x_1^2+x_2^2)x_3^2\leqslant t\}, \end{aligned}
\end{equation*}
\notag
$$
and for $i\in\{1,2\}$ set $\mathcal N_{i}=\mathcal N^1_{i}$. In a neighbourhood of $\mathcal N_{1}$ we define a pair of transversal foliations $\mathcal{F}^{\rm u}_1$ and $\mathcal{F}^{\rm s}_{1}$ by
$$
\begin{equation*}
\mathcal{F}^{\rm u}_1=\bigcup_{(c_{2},c_3)\in Ox_2x_3}\{(x_1,x_2,x_3)\in \mathcal N_{1}\colon (x_{2},x_3)=(c_{2},c_3)\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathcal{F}^{\rm s}_{1} =\bigcup_{c_1\in Ox_1}\{(x_1,x_2,x_3)\in \mathcal N_{1} \colon x_1=c_1\}.
\end{equation*}
\notag
$$
In a neighbourhood of $\mathcal N_{2}$ we define a pair of transversal foliations $\mathcal{F}^{\rm u}_2$ and $\mathcal{F}^{\rm s}_{2}$ by
$$
\begin{equation*}
\mathcal{F}^{\rm u}_2=\bigcup_{c_3\in Ox_3}\{(x_1,x_2,x_3)\in \mathcal N_{2} \colon x_3=c_3\},
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathcal{F}^{\rm s}_{2}=\bigcup_{(c_{1},c_2)\in Ox_1x_2}\{(x_1,x_2,x_3)\in \mathcal N_{2} \colon (x_{1},x_2)=(c_{1},c_2)\}.
\end{equation*}
\notag
$$
Let $\nu_i\colon\mathbb R^3\to\mathbb R^3$, $i \in \{1,2\}$, be the diffeomorphisms defined by
$$
\begin{equation*}
\nu_1(x_1,x_2,x_3)=\biggl(2x_1,\frac{x_2}{2}\,,\frac{x_3}{2}\biggr)\quad\text{and}\quad \nu_2=\nu_1^{-1}.
\end{equation*}
\notag
$$
Note that for $i\in\{1,2\}$ the set $\mathcal N_{i}^t$ is invariant under the diffeomorphism $\nu_i$, which takes leaves of $\mathcal{F}^{\rm u}_i$ (of $\mathcal{F}^{\rm s}_{i}$) to leaves of the same foliation. By [14] each saddle point $\sigma_f^i$ of a diffeomorhism $f\in G$ has a linearlizing neighbourhood $N_f^{i}$ endowed with a homeomorphism ${\mu}_{i}\colon N_f^{i}\to {\mathcal N}_{i}$ that conjugates $f\big|_{{N}_f^{i}}$ to the diffeomorphism $\nu_i\big|_{{\mathcal N}_{i}}$ and is a diffeomorphism on $N_f^i\setminus(W^{\rm s}_{\sigma_f^i}\cup W^{\rm u}_{\sigma_f^i})$. The foliations $\mathcal{F}^{\rm u}_{i}$ and $\mathcal{F}^{\rm s}_{i}$ induce (by means of ${\mu}_{i}^{-1}$) $f$-invariant foliaitons ${F}^{\rm u}_{i}$ and ${F}^{\rm s}_{i}$ on the linearizing neighbourhood $N_f^{i}$. For each point $x\in N_f^i$ we let ${F}^{\rm u}_{i,x}$ (or ${F}^{\rm s}_{i,x}$) denote the unique leaf of ${F}^{\rm u}_{i}$ (of ${F}^{\rm s}_{i}$, respectively) that passes through $x$. If the set $H_f$ is empty, then we call a pair $N_f$ of disjoint linearizing neigbourhoods $N_f^1$, $N_f^2$ of the saddle points of $f$ a compatible system of neighbourhoods, and we call $F^{\rm s}_i$ and $F^{\rm u}_i$ ($i=1,2$) compatible foliations. If $H_f\ne\varnothing$, then we select an $f$-invariant tubular neighbourhood $N_{H_f}\subset M^3$ of the curves in $H_f$ which carries an $f$-invariant $C^{1,1}$-foliation $F$ by two-dimensional discs transversal to $H_f$. For each point $x\in N_{H_f}$ we let ${F}_{x}$ denote the unique leaf of ${F}$ that passes through this point. We call the union $N_f$ of linearizing neighbourhoods $N_f^1$ and $N_f^2$ of saddle points of $f$ a compatible system of neighbourhoods, and we call $F^{\rm s}_i$ and $F^{\rm u}_i$ ($i=1,2$) compatible foliations if the following conditions hold for each point $x\in(N_f^1\cap N_f^2\cap N_{H_f})$ and the leaf $F_x$ of $F$ that passes through $x$ (see Fig. 7):
$$
\begin{equation*}
{F}^{\rm s}_{1,x}\cap F_x={F}^{\rm s}_{2,x}\cap(N_f^{1}\cap N_{H_f})\quad\text{and}\quad {F}^{\rm u}_{2,x}\cap F_x={F}^{\rm u}_{1,x}\cap (N_f^{2}\cap N_{H_f}).
\end{equation*}
\notag
$$
Proposition 2.1 ([14], Theorem 1). Each diffeomorphism $f\in G$ has a compatible system of neighbourhoods. 2.2. Spaces of wandering orbitsConsider the spaces
$$
\begin{equation*}
V_{\omega_f}=W^{\rm s}_{\omega_f}\setminus\omega_f\quad\text{and}\quad \widehat V_{\omega_f}=V_{\omega_f}/f.
\end{equation*}
\notag
$$
Let $p_{\omega_f}\colon V_{\omega_f}\to\widehat V_{\omega_f}$ be the natural projection. Set (see Fig. 8) Proposition 2.2 ([25], Theorem 2.1.3). For each diffeomorphism $f\in G$ the following results hold: (1) the space $\widehat V_{\omega_f}$ is diffeomorphic to $\mathbb{S}^2\times \mathbb{S}^1$; (2) the projection $p_{\omega_f}$ is a covering map; (3) the set $\widehat A_f$ consists of a pair of disjoint knots $L^1_f\sqcup L^2_f$ in $\widehat V_{\omega_f}$ such that the map is an isomorphism.Proposition 2.3 ([25], Lemma 4.4). The set $p_{\omega_f}(N^1_f)$ is a disjoint union of two solid tori $N_{L^1_f}$ and $N_{L^2_f}$ that are tubular neighbourhoods of the knots $L^1_f$ and $L^2_f$, respectively. If at least one of the sets $\widehat V_{\omega_f}\setminus \operatorname{int}N_{L^1_f}$ and $\widehat V_{\omega_f}\setminus \operatorname{int}N_{L^2_f}$ is not homeomorphic to a solid torus, then the manifold $W^{\rm u}_{\sigma^1_f}$ is wildly embedded in the ambient manifold $M^3$. By Proposition 1.10 the sets $A_{f}$ and $R_{f}$ are the dual attractor and repeller for the diffeomorphism $f$, respectively. Set By Proposition 1.11 the orbit space $\widehat{V}_f=V_f/f$ is a smooth closed orientable 3-manifold, and the natural projection $p_{f}\colon V_f \to \widehat{V}_f$ is a covering map and induces an epimorphism which assigns to an element $[\widehat{c}]\in\pi_1( \widehat{V}_{f})$ an integer $n\in\mathbb{Z}$ such that each lift of $\widehat{c}$ connects a point $x\in V_{f}$ with $f^{n}(x)$. Set (see Fig. 9)
$$
\begin{equation*}
T^{\rm s}_f=p_{f}(W^{\rm s}_{\sigma_f^1}\setminus\sigma_f^1),\qquad T^{\rm u}_f=p_{f}(W^{\rm u}_{\sigma_f^{2}}\setminus\sigma_f^2),\quad\text{and}\quad C_f=p_{f}(H_f).
\end{equation*}
\notag
$$
By Propositions 1.9 and 1.11 the sets $T^{\rm s}_f$ and $T^{\rm u}_f$ are 2-tori smoothly embedded in $\widehat V_f$ and such that
$$
\begin{equation*}
\eta_{f}(i_{T^{\rm s}_f*}(\pi_1(T^{\rm s}_f)))= \eta_{f}(i_{T^{\rm u}_f*}(\pi_1(T^{\rm u}_f)))\cong\mathbb Z.
\end{equation*}
\notag
$$
The central result in this section is the following lemma. Lemma 2.1 ([62], Lemma 2). For each diffeomorphism $f\in G$ the following holds: (1) the set $\widehat V_f$ is a connected irreducible closed 3-manifold, and the tori $T^{\rm s}_f$ and $T^{\rm u}_f$ are incompressible in it; (2) the set $C_f$ consists of a finite number of knots $c$, and $\eta_{f}([c])=0$ if and only if the knot $c\subset C_f$ is the projection of a compact heteroclinic curve; (3) each knot $c\subset C_f$ such that $\eta_{f}([c])=0$ is contractible (or not) on both $T^{\rm s}_f$ and $T^{\rm u}_f$ simultaneously (see Fig. 10). Proof. We prove all assertions of the lemma in succession.
(1) By Propositions 1.10 and 1.11 the manifold $\widehat V_f$ is a connected closed simple 3-manifold. Since $T^{\rm s}_f$ is an $\eta_{f}$-essential torus in $\widehat V_f$, it does not lie in a 3-ball. We show by contradiction that $T^{\rm s}_f$ does not bound a solid torus in $\widehat V_f$. By Proposition 1.2 this will mean that $\widehat V_f$ is not homeomorphic to $\mathbb{S}^2\times\mathbb{S}^1$, so that it is irreducible, and then by Proposition 1.4 the torus $T^{\rm s}_f$ is incompressible in $\widehat V_f$. Suppose that $T^{\rm s}_f$ bounds a solid torus in $\widehat V_f$. Then $\widehat V_f\setminus T^{\rm s}_f$ consists of two connected components. On the other hand, by Proposition 1.7
$$
\begin{equation*}
M^3=W^{\rm s}_{\omega_f}\cup W^{\rm s}_{\sigma_f^{1}}\cup W^{\rm s}_{\sigma_f^{2}}\cup W^{\rm s}_{\alpha_f}.
\end{equation*}
\notag
$$
Then $V_f\setminus W^{\rm s}_{\sigma_f^1}=W^{\rm s}_{\omega_f}\setminus A_f$, and therefore the manifolds $\widehat V_f\setminus T^{\rm s}_f$ and $\widehat V_{\omega_f}\setminus\widehat A_f$ are homeomorphic. Since a one-dimensional submanifold cannot disconnect a manifold of dimension 3 (see Proposition 1.5), the set $\widehat V_{\omega_f}\setminus\widehat A_{f}$ is connected (see Fig. 8). This is a contradiction: a connected manifold is homeomorphic to a disconnected one.
(2) It follows immediately from the definition of the diffeomorphism $\eta_{f}$ that $\eta_{f}([c])=0$ if and only if $c \subset C_{f}$ is the projection of a compact heteroclinic curve. (3) Suppose that a knot $c\subset C_f$ is contractible on $T^{\rm u}_f$ and essential on $T^{\rm s}_f$. Then by definition the torus $T^{\rm s}_f$ is compressible in $\widehat V_f$, which contradicts the result established in (1). The proof is complete. Let $ C^0_f$ denote the subset of $C_f$ that consists of the curves contractible on $T^{\rm u}_f$ and $T^{\rm s}_f$. We call curves in the set $H^0_f=p_f^{-1}( C^0_f)$ inessential; other heteroclinic curves are called essential. 3. Trivializing the dynamics of diffeomorphisms in the class $G$Recall that for each diffeomorphism $f\in G$ we define its heteroclinic index $I_f$ as follows. If the set $H_f$ contains no non-compact curves, then we set $I_f=0$. Otherwise we let $\widetilde H_f$ denote the subset consisting of the non-compact curves. Since each curve $\gamma\subset \widetilde H_f$, together with any point $x\in\gamma$, contains the point $f(x)$, we assume that $\gamma$ is oriented from $x$ to $f(x)$. We also fix orientations on the manifolds $W^{\rm s}_{\sigma_1}$ and $W^{\rm u}_{\sigma_2}$. For a non-compact heteroclinic curve $\gamma$ let
$$
\begin{equation*}
v_\gamma=(\vec v^1_\gamma,\vec v^2_\gamma,\vec v^3_\gamma)
\end{equation*}
\notag
$$
denote a triple of vectors with initial point $x\in\gamma$ such that
It is obvious that the (left or right) orientation of $v_\gamma$ is independent of the choice of the point $x$ on $\gamma$. Set $I_\gamma=+1$ ($I_\gamma=-1$) if this orientation is positive (respectively, negative). We call the integer $I_{f}=\bigl|\sum_{\gamma\subset \widetilde H_f}I_\gamma\bigr|$ the heteroclinic index of the diffeomorphism $f$. Given a non-negative integer $p\geqslant 0$, let $G_p\subset G$ denote the subset of diffeomorphisms $f\in G$ such that $I_{f}=p$. For example, in Fig. 6 we show the phase portrait of a diffeomorphism in $G_0$, and in Fig. 12 we show the phase portrait of a diffeomorphism in $G_1$. Note that each essential compact heteroclinic curve $\gamma\subset H_f$ bounds a disc $d_\gamma\subset W^{\rm s}_{\sigma_f^1}$ which contains the saddle point $\sigma_f^1$. We assume that such a curve is oriented so that the disc $d_\gamma$ lies to the left of this curve. Then, similarly to the case of a non-compact curve, we can define a frame $v_\gamma$ on $\gamma$. We say that a set $H_f$ is orientable if it consists only of essential heteroclinic curves and, the frames on all of these have the same orientation. Otherwise we say that $H_f$ is non-orientable (see Fig. 13). Let $G^+_p\subset G_p$, $p\geqslant 0$, denote the subset of diffeomorphisms $f\in G_p$ with orientable set $H_f$ (see Fig. 12). Thus, for a diffeomorphism $f\in G^+_p$ the set $H_f$ is either empty, or it contains only non-compact or only compact heteroclinic curves (see Fig. 14). The main result of this section is the following theorem. Theorem 1 ([55], Theorem 1). For each diffeomorphism $f\colon M^3\to M^3$ in the class $G_p$, $p\geqslant 0$, there exists a diffeomorphism $f_+\in G_p^+$ isotopic to it. This result follows directly from Lemmas 3.1 and 3.2 established below. 3.1. A scenario of the elimination of non-essential heteroclinic curvesLet $\widetilde G_p\subset G_p$ denote the subclass of diffeomorphisms $f$ whose sets $H^0_f$ are empty. The central result in this subsection is as follows. Lemma 3.1 ([62], Theorem 1). For each diffeomorphism $f\colon M^3\to M^3$ in the class ${G}_p$ there exists an arc in the set $\operatorname{Diff}(M^3)$ that connects $f$ with a diffeomorphism $\widetilde f\in\widetilde G_p$. Proof. Let $f\in G_p$. If $H^0_f=\varnothing$, then the proof is complete. Otherwise, by Lemma 2.1, for each knot $c\subset C^0_f$ there exists a unique disc $d^{\rm s}_{c}$ such that $d^{\rm s}_{c} \subset T^{\rm s}_{f}$, $c=\partial d^{\rm s}_{c}$; a similar disc $d^{\rm u}_{c} \subset T^{\rm u}_{f}$ is bounded by the knot $c=\partial d^{\rm u}_{c}$ on the torus $T^{\rm u}_f$ (see Fig. 15, (a)).
Among the curves in $C^0_f$ we select an extreme curve $c$, that is, a curve such that Since $d^{\rm s}_c\cap d^{\rm u}_c=c$, the set $d_c^{\rm s} \cup d_c^{\rm u}$ is a 2-sphere embedded cylindrically in $\widehat V_f$. By Lemma 2.1 the manifold $\widehat V_f$ is irreducible, and therefore this sphere bounds a single 3-ball $b_c$ in it. Let $T^{\rm u}_{f,c}$ be a 2-torus obtained by smoothing the torus $(T^{\rm u}_f\setminus d^{\rm u}_c)\cup d^{\rm s}_c$ (see Fig. 15, (b)). Then there exists a diffeomorphism $\widehat h\colon \widehat V_f\to\widehat V_f$ isotopic to the identity and such that $\widehat h(T^{\rm u}_f)=T^{\rm u}_{f,c}$. By Proposition 1.12 there exists an arc ${\zeta}_{t}\subset E_f$ such that
$$
\begin{equation*}
{\zeta}_{0}=f\quad\text{and}\quad T^{\rm u}_{\zeta_1}=T^{\rm u}_{f, c},\quad T^{\rm s}_{\zeta_1}=T^{\rm s}_f.
\end{equation*}
\notag
$$
Repeating this process for each extreme curve we obtain the required diffeomorphism $\widetilde f\in\widetilde G_p$. The proof is complete. 3.2. A scenario of the elimination of non-orientable heteroclinic curvesThe central result in this subsection is as follows. Lemma 3.2 ([55], Theorem 1). For each diffeomorphism ${f}\colon M^3\to M^3$ in the class $\widetilde{G}_p$ there exists an arc in $\operatorname{Diff}(M^3)$ that connects $f$ with a diffeomorphism $f_+\in G^+_p$. Proof. Let ${f}\in\widetilde{G}_p$.
If the set $H_f$ is empty or orientable, then the proof is complete. Otherwise $C_f$ consists of non-contractible knots which are pairwise homotopic on each of the tori $T^{\rm s}_f$ and $T^{\rm u}_f$ (see, for example, [61]), including knots $c_+$ and $c_-$ with positive and negative oriented frames, respectively (see Figs. 16 and 17). We show that we can reduce the number of curves in $H_f$ by two at least. To do this we set
$$
\begin{equation*}
Y_{f}=p_{\omega_f}(W^{\rm u}_{\sigma^2_f}\cap V_{\omega_f})\quad\text{and}\quad \widehat N^1_f=p_{\omega_f}(N^1_f\cap V_{\omega_f}).
\end{equation*}
\notag
$$
Then $\widehat N^1_f$ is a disjoint union of two solid tori $N_{L^1_f}$ and $N_{L^2_f}$ that are tubular neighbourhoods of Hopf knots $L^1_f$ and $L^2_f$, respectively. In this case $Y_f\setminus\operatorname{int}\widehat N^1_{f}$ consists of a finite number of annuli with boundaries on the tori $T^1_f=\partial N_{L^1_f}$ and $T^2_f=\partial N_{L^2_f}$. As $H_f$ is non-orientable, the set $Y_f\setminus \operatorname{int}\widehat N^1_{f}$ has a connected component $K^{\rm u}$ whose boundary circles lie on the same connected component of $\partial\widehat N^1_f$ (for definiteness, let it be $T^1_f$). Then the circles $\partial K^{\rm u}$ partition $T^1_f$ into two annuli, and the union of each of these annuli (denoted by $K^{\rm s}$) with $K^{\rm u}$ is a 2-torus $T_{K^{\rm u}}$. We claim that we can select $K^{\rm s}$ so that this torus $T_{K^{\rm u}}$ bounds a solid torus $Q_{K^{\rm u}}$ whose interior is disjoint from $\widehat N^1_f$ in $\widehat V_{\omega_f}$ (see Figs. 18 and 19).  Figure 18.The projections of invariant saddle manifolds onto the space $\widehat V_{\omega_f}$ in accordance with Fig. 16.  Figure 19.The projections of invariant saddle manifolds onto the space $\widehat V_{\omega_f}$ in accordance with Fig. 17. Since the torus $T^1_f$ is homotopically non-trivial in $\widehat V_{\omega_f}$, we can select $K^{\rm u}$ so that the torus $T_{K^{\rm u}}$ is also homotopically non-trivial. By Proposition 1.2 the torus $T_{K^{\rm u}}$ bounds a solid torus $Q_{K^{\rm u}}$ in $\widehat V_{\omega_f}$. If $N_{L^1_f}\subset Q_{K^{\rm u}}$, then by construction $\operatorname{cl}(Q_{K^{\rm u}})\setminus N_{L^1_f}$ is also a solid torus, which is bounded by the 2-torus constructed using the second annulus $K^{\rm s}$. For this reason, in what follows we assume that $Q_{K^{\rm u}}\cap \widehat N_{L^1_f}=K^{\rm s}$. Thus, with each annulus $K^{\rm u}$ we associate the 2-torus $T_{K^{\rm u}}$ bounding the solid torus $Q_{K^{\rm u}}$ in $\widehat V_{\omega_f}$. By Proposition 1.4 each 2-torus that is homotopically non- trivially embedded in a solid torus bounds in it a unique solid torus, and therefore among all such solid tori $Q_{K^{\rm u}}$ there exists a torus $Q_{K^{\rm u}_0}$ whose interior is disjoint from the annuli $K^{\rm u}$. Then the interior of $Q_{K^{\rm u}_0}$ is disjoint from the tori $\widehat N^1_f$ and $Q_{K^{\rm u}_0}\cap Y_f=K^{\rm u}_0$. Let $K^{\rm s}_0$ denote the second half of the torus $T_{K^{\rm u}_0}$. Set
$$
\begin{equation*}
N_{T^{\rm s}_f}=p_{f}(N^1_f\cap V_f) \quad\text{and}\quad N_{T^{\rm u}_f}=p_{f}(N^2_f\cap V_f).
\end{equation*}
\notag
$$
Let $\mathcal K^{\rm u}_0$ denote the connected component of the set $T^{\rm u}_f\setminus C_f$ such that $p_{f}(p_{\omega_f}^{-1}(K^{\rm u}_0))\subset \mathcal K^{\rm u}_0$, and let $\mathcal K^{\rm s}_0$ denote the connected component of $T^{\rm s}_f\setminus C_f$ such that the annuli $p_{f}(p_{\omega_f}^{-1}(K^{\rm s}_0)$) and $\mathcal K^{\rm s}_0$ lie in the same connected component $N^{\rm s}_0$ of $N_{T^{\rm s}_f}\setminus T^{\rm u}_f$. Let $\mathcal K'^{\rm s}_0$ denote the connected component of the set $\partial N_{T^{\rm s}_f}\cap N^{\rm s}_0$ distinct from $p_{f}(p_{\omega_f}^{-1}(K^{\rm s}_0))$. Set
$$
\begin{equation*}
\mathcal K'^{\rm u}_0=\mathcal K^{\rm u}_0\cup (\operatorname{cl}(N^{\rm s}_0)\cap T^{\rm u}_f).
\end{equation*}
\notag
$$
By construction $\partial \mathcal K^{\rm s}_0= \partial \mathcal K^{\rm u}_0=c_+\sqcup c_-$, where $c_+\subset C_f$ and $c_-\subset C_f$ are non-contractible curves with positive and negative oriented frames, respectively. In addition, the 2-torus $\mathcal T_0=\mathcal K^{\rm u}_0\cup\mathcal K^{\rm s}_0$ bounds a solid torus $\mathcal Q_0$ in $\widehat V_f$, whose interior is disjoint from the set $T^{\rm u}_f\cup T^{\rm s}_f$. Let $T'^{\rm u}_{f}$ denote a 2-torus obtained by smoothing $(T^{\rm u}_f\setminus \mathcal K'^{\rm u}_0)\cup\mathcal K'^{\rm s}_0$. Since $\mathcal T'_0=\mathcal K'^{\rm u}_0\cup\mathcal K'^{\rm s}_0$ bounds a solid torus $\mathcal Q'_0$ in $\widehat V_f$ whose interior is disjoint from $T^{\rm u}_f$, there exists a diffeomorphism $\widehat h\colon\widehat V_f\to\widehat V_f$ isotopic to the identity and such that $\widehat h(T^{\rm u}_f)=T'^{\rm u}_{f}$. Then by Proposition 1.12 there exists an arc ${\zeta}_{t}\subset E_f$ such that
$$
\begin{equation*}
{\zeta}_{0}=f,\quad {\zeta}_{1}=f'\quad\text{and}\quad T^{\rm u}_{f'}=T'^{\rm u}_{f},\quad T^{\rm s}_{f'}=T^{\rm s}_f.
\end{equation*}
\notag
$$
Thus, the diffeomorphism $f'\in G$ is defined on the same manifold $M^3$ as $f$, but it has two heteroclinic curves fewer. Continuing in this way we construct an arc connecting $f$ with a diffeomorphism $f_+\in G^+_p$, which completes the proof. 4. Hopf knot as a full invariant of a diffeomorphism in the class $G^+_1$. Topological classification of diffeomorphisms in $G^+_1$In this section we present a full topological classification, including a realization, of diffeomorphisms in the class $G^+_1$. Let $f\in G^+_1$. We denote the unstable separatrices of the point $\sigma_f^{1}$ by $\ell_f^1$ and $\ell_f^2$. By Proposition 1.8 the closure $\operatorname{cl}(\ell_f^i)$ ($i=1,2$) of a one-dimensional unstable separatrix of $\sigma_f^{1}$ is a closed simple arc and consists of this separatrix and two points, $\sigma_f^{1}$ and the sink $\omega_f$ (see Fig. 12). By Proposition 1.7 the orbit space of the sink basin $\widehat V_{\omega_f}$ is diffeomorphic to the manifold $\mathbb S^{2}\times\mathbb S^1$ (throughout what follows we identify these manifolds), and the subset $L_f^{i}=p_{\omega_f}(\ell_f^i)$, $i=1,2$, is a Hopf knot in this manifold. 4.1. The equivalence of the knots $L^1_f$ and $L^2_f$Lemma 4.1 ([57], Lemma 1.1). For each diffeomorphism $f\in G^+_1$ the knots $L_f^1$ and $L_f^2$ are isotopic. By Proposition 1.9 the orbit space $(W^{\rm u}_{\sigma_f^{2}}\setminus\sigma_f^{2})/f$ is homeomorphic to a 2-torus, and the orbit space $H_f/f$ is homeomorphic to a circle, which forms a non-trivial knot on the torus $(W^{\rm u}_{\sigma_f^{2}}\setminus\sigma_f^{2})/f$. Since
$$
\begin{equation*}
W^{\rm u}_{\sigma_f^{2}}\cap V_{\omega_f}= W^{\rm u}_{\sigma_f^{2}}\setminus (H_f\cup \sigma_f^{2}),
\end{equation*}
\notag
$$
the set $Y_f$ is homeomorphic to an annulus; furthermore, the homeomorphism $i_{Y_f*}$ induced by the inclusion $i_{Y_f}\colon Y_f\to\mathbb S^2\times\mathbb S^1$ is a group isomorphism
$$
\begin{equation*}
\pi_1(Y_f)\cong\pi_1(\mathbb S^2\times\mathbb S^1)\cong\mathbb Z.
\end{equation*}
\notag
$$
We have $N^1_f\cap V_{\omega_f}=N^1_f\setminus W^{\rm s}_{\sigma_f^{1}}$, so that the set is a disjoint union of two solid tori that are tubular neighbourhoods of the knots $L^1_f$ and $L^2_f$: $\widehat N^1_f=N_{L^1_f}\sqcup N_{L^2_f}$. Set (see Fig. 21)
$$
\begin{equation*}
T^i_f=\partial N_{L^i_f}\quad\text{and}\quad S^i_f=T^i_f\cap Y_f.
\end{equation*}
\notag
$$
Since $H_f=W^{\rm s}_{\sigma_f^{1}}\cap{W^{\rm u}_{\sigma_f^{2}}}$ consists of a unique non-compact $f$-invariant curve, the set $\partial N^{1}_{f}\cap{W^{\rm u}_{\sigma_f^{2}}}$ consists of two non-compact $f$-invariant curves such that their projections onto $\widehat V_{\omega_f}\cong\mathbb S^2\times\mathbb S^1$ are the knots $S^1_f\sqcup S^2_f$. Hence $S^1_f$ and $S^2_f$ are isotopic Hopf knots. Thus, the knots $S^i_f$ and $L^i_f$ are generators on the solid torus $N_{L^i_f}$. Hence they bound a 2-dimensional annulus in it and therefore are isotopic. The proof is complete. 4.2. The equivalence class a of a Hopf knot as a full invariant of topological conjugacy in the class $G$It follows from Lemma 4.1 and Thom’s isotopy lemma (for instance, see [40]) that the Hopf knots $L_f^1$ and $L_f^2$ are equivalent. Let $\mathcal L_f=[L_f^1]=[L_f^2]$ denote their class of equivalence. Theorem 2 ([57], Theorem 1.1). Two diffeomorphisms $f,f'\in G^+_1$ are topologically conjugate if and only if $\mathcal L_f=\mathcal L_{f'}$. Proof. ($\Rightarrow$) Assume that two diffeomorphisms
$$
\begin{equation*}
f\colon M^3\to M^3\quad\text{and}\quad f'\colon M'^3\to M'^3,\quad f,f'\in G^+_1,
\end{equation*}
\notag
$$
are topologically conjugate by means of a homeomorphism $h\colon M^3\to M'^3$. Since $h$ takes the invariant manifolds of fixed points of $f$ to the invariant manifolds of fixed points of $f'$ preserving the stability properties, it follows that Since $hf=f'h$, the homeomorphism $h$ defines a homeomorphism $\widehat h\colon\widehat V_{\omega_f}\to\widehat V_{\omega_{f'}}$ by Hence $\widehat h(L^i_f)=L^i_{f'}$, and therefore the Hopf knots $L^i_f$ and $L^i_{f'}$ are equivalent.
($\Leftarrow$) Let $\mathcal L_f=\mathcal L_{f'}$. We construct a homeomorphism $h\colon M^3\to M'^3$ conjugating $f$ to $f'$ in several steps. In the construction we use the notation from Lemma 4.1 and § 2.1, putting primes in the case of the diffeomorphism $f'$. Set (see Fig. 22) Step 1: we construct a homeomorphism $\widehat h_1\colon\widehat V_{\omega_f}\to\widehat V_{\omega_{f'}}$ such that $\widehat h_1(\widehat N_f)=\widehat N_{f'}$. We choose a non-contractible knot $L_f$ in the annulus $Y_f\setminus\widehat N_f^1$ and a non-contractible knot $L_{f'}$ in $Y_{f'}\setminus\widehat N_{f'}^1$. It follows from Lemma 4.1 that $[L_f]=\mathcal L_f$ and $[L_{f'}]= \mathcal L_{f'}$. Since $\mathcal L_f=\mathcal L_{f'}$, by Thom’s lemma there exists a homeomorphism $\widehat h_0\colon\widehat V_{\omega_f}\to\widehat V_{\omega_{f'}}$ such that $\widehat h_0(L_f)=L_{f'}$. In view of Proposition 1.6 we can assume that it is in fact a diffeomorphism. Since $Y_f$ and $Y_{f'}$ are smooth annuli, we can also assume without loss of generality that $\widehat h_0(Y_f)\subset Y_{f'}$ in a neighbourhood of $L_f$. Let
$$
\begin{equation*}
\widetilde N_f=\widehat h_0^{-1}(\widehat N_{f'}),\quad \widetilde N^1_f=\widehat h_0^{-1}(\widehat N^1_{f'}),\quad\text{and}\quad \widetilde Y_f=\widehat h_0^{-1}(Y_{f'}),
\end{equation*}
\notag
$$
and let $J_{f}\subset \widehat V_f$ be a smooth two-dimensional annulus that is transversal to $\widehat N_f$ and $\widetilde N_f$ and such that $\widehat J_f=J_f\cap\widehat N_f$ and $\widetilde J_f=J_f\cap \widetilde N_f$ are annuli in $\widehat N_f\setminus\widehat N^1_f$ and $\widetilde N_f\setminus \widetilde N^1_f$ that contain the knot $L_f$ in their interiors (see Fig. 23). By Lemma 4.1, $\widehat N_f$ and $\widetilde N_f$ are tubular neighbourhoods of the knot $L_f$. We take tubular neighbourhoods $W$, $U$, and $\widetilde U$ of $L_f$ such that
$$
\begin{equation*}
W\subset \operatorname{int}(\widehat N_f\cap\widetilde N_f),\qquad \widehat N_f\subset \operatorname{int}U,\quad\text{and}\quad \widetilde N_f\subset \operatorname{int} \widetilde U
\end{equation*}
\notag
$$
and such that each of them intersects $J_f$ in an annulus. Since the sets
$$
\begin{equation*}
\widehat N_f\setminus \operatorname{int} W,\qquad U\setminus \operatorname{int} \widehat N_f,\qquad \widetilde N_f\setminus \operatorname{int} W,\quad\text{and}\quad \widetilde U\setminus \operatorname{int}\widetilde N_f
\end{equation*}
\notag
$$
are homeomorphic to $\mathbb T^2\times[0,1]$, there exists a homeomorphism
$$
\begin{equation*}
\widehat\psi\colon\widehat V_{\omega_f}\to\widehat V_{\omega_f},
\end{equation*}
\notag
$$
equal to the identity on $W$ and outside $U$ and such that
$$
\begin{equation*}
\widehat\psi(\widehat N_f)=W\quad\text{and}\quad \widehat\psi(\widehat J_f)=\widehat J_f,
\end{equation*}
\notag
$$
and there exists a homeomorphism
$$
\begin{equation*}
\widetilde\psi\colon\widehat V_{\omega_f}\to\widehat V_{\omega_f},
\end{equation*}
\notag
$$
equal to the identity on $W$ and outside $\widetilde U$ and such that
$$
\begin{equation*}
\widetilde\psi(\widetilde N_f)=W\quad\text{and}\quad \widetilde\psi(\widetilde J_f)=\widetilde J_f.
\end{equation*}
\notag
$$
Then the required homeomorphism $\widehat h_1$ has the following form:
$$
\begin{equation*}
\widehat h_1=\widehat h_0\widetilde\psi^{-1}\widehat\psi.
\end{equation*}
\notag
$$
Step 2: we construct a homeomorphism $\widehat h_2\colon\widehat V_{\omega_f}\to\widehat V_{\omega_{f'}}$ that coincides with $\widehat h_1$ outside $\widehat N_f$ and satisfies $\widehat h(Y_f)=Y_{f'}$. To do this we recall that the linearizing neighbourhood $N^1_f$ is endowed with the homeomorphism where
$$
\begin{equation*}
\mathcal N_{1}^t=\{(x_1,x_2,x_3)\in\mathbb{R}^3\colon x_1^2(x_2^2+x_3^2)< t\},\qquad t\in(0,1],
\end{equation*}
\notag
$$
and such that $\mu_1$ conjugates $f\big|_{N^1_f}$ to the diffeomorphism $\nu_1\big|_{\mathcal N_1}$ defined by $\nu_1(x_1,x_2,x_3)=(2x_1,x_2/2,x_3/2)$. We introduce the analogous notation (with primes) for the diffeomorphism $f'$. Set $H=\mu_1(H_f)$ and $H'=\mu'_1(H_{f'})$. Then $H$ and $H'$ are $\nu_1$-invariant curves on the $Ox_2x_3$-plane. Hence there exists a homeomorphism $\xi_{\rm s}\colon Ox_2x_3\to Ox_2x_3$ that commutes with $\nu_1\big|_{Ox_2x_3}$ and satisfies $\xi_{\rm s}(H)=H'$ (for instance, see [25]). Let $\xi\colon\mathbb R^3\to\mathbb R^3$ be the homeomorphism defined by (see Fig. 24). We choose $\tau\in(0,1)$ so that $\xi(\mathcal N^\tau_1)\subset \operatorname{int}\mathcal N_1$ and set
$$
\begin{equation*}
N^\tau_1=\mu_1(\mathcal N^\tau_1)\quad\text{and}\quad \xi_1=(\mu'_1)^{-1}\xi\mu_1\big|_{N^\tau_1}.
\end{equation*}
\notag
$$
It follows from the definition of a consistent system of neighbourhoods that
$$
\begin{equation*}
\xi_1(N^\tau_1\cap W^{\rm u}_{\sigma^2_f})\subset W^{\rm u}_{\sigma^2_{f'}}.
\end{equation*}
\notag
$$
Set
$$
\begin{equation*}
\widehat N^{\tau}_1=p_{\omega_f}(N^\tau_1),\quad \widehat\xi_1=p_{\omega_{f'}}\xi_1 p_{\omega_f}^{-1}\big|_{\widehat N^\tau_1},\quad\text{and}\quad \widehat N'^\tau_1=\widehat\xi_1(\widehat N^\tau_1).
\end{equation*}
\notag
$$
Then the sets $K=Y_f\setminus \operatorname{int}\widehat N^\tau$ and $K'=Y_{f'}\setminus \operatorname{int}\widehat N'^\tau_1$ are homeomorphic to (two-dimensional) annuli. We choose tubular neighbourhoods
$$
\begin{equation*}
N_K\subset (\operatorname{int}\widehat N^2_f\setminus \operatorname{int} \widehat N^\tau_1)\quad\text{and}\quad N_{K'}\subset(\operatorname{int}\widehat N^2_{f'}\setminus \operatorname{int}\widehat N'^\tau_1)
\end{equation*}
\notag
$$
of $K$ and $K'$ so that the intersection of $N_K$ with each connected component of $\partial \widehat N^\tau_1$ is an annulus and
$$
\begin{equation*}
\widehat\xi_1(N_K\cap \partial{\widehat N}^\tau_1)= N_{K'}\cap \partial{\widehat N}'^\tau_1.
\end{equation*}
\notag
$$
Set $\widehat N^\tau=\widehat N^\tau_1\cup N_K$ and $\widehat N'^\tau=\widehat N'^\tau_1\cup N_{K'}$ (see Fig. 25). Then $\widehat\xi_1$ extends to a homeomorphism $\widehat\xi\colon\widehat N^\tau\to \widehat N'^\tau$ such that $\widehat\xi(Y_f)=Y_{f'}$. By construction the homeomorphism $\widehat\psi_1=\widehat h_1^{-1}\widehat\xi$ has the property $\widehat\psi_1(\widehat N^\tau)\subset \operatorname{int} \widehat N_f$, and the homeomorphism $\widehat\psi_1\big|_{\partial{\widehat N}^\tau}$ is homotopic to the identity map. Since the sets $\widehat N_f\setminus \operatorname{int}\widehat N^\tau$ and $\widehat N_f\setminus \operatorname{int}\widehat\psi_1(\widehat N^\tau)$ are homeomorphic to $\mathbb T^2\times[0,1]$, $\widehat\psi_1$ extends to a homeomorphism $\widehat\psi_1\colon\widehat V_{\omega_f}\to\widehat V_{\omega_f}$ equal to the identity outside $\widehat N_f$. Then the required homeomorphism $\widehat h_2$ has the form Step 3: we construct the required homeomorphism $h$. It follows from the construction of $\widehat h_2$ that it can be lifted to a map $h_2\colon V_{\omega_f}\to V_{\omega_{f'}}$ which conjugates the diffeomorphisms $f\big|_{V_{\omega_f}}$ and $f'\big|_{V_{\omega_{f'}}}$ and extends to a homeomorphism $\xi_1$ of $W^{\rm s}_{\sigma_f^1}$. Thus we have constructed the conjugation homeomorphism everywhere outside the closures of the one-dimensional invariant manifolds of saddle points. By Proposition 1.13 such a homeomorphism extends to the required homeomorphism $h$. Theorem 2 is proved. Thus, the equivalence class $\mathcal L_f$ of a Hopf knot is a full topological invariant of the diffeomorphism $f\in G^+_1$. Moreover, the following realization theorem holds. 4.3. Realizing diffeomorphisms in the class $G^+_1$ by Hopf knotsTheorem 3 ([57], Theorem 1.2). For each equivalence class $\mathcal L$ of Hopf knots in $\mathbb S^{2}\times \mathbb S^1$ there exists a diffeomorphism $f\in G^+_1$ such that $\mathcal L_{f}=\mathcal L$. Proof. Recall that and ${\nu}\colon\mathbb R^3\to\mathbb R^3$ is the diffeomorphism defined by ${\nu}({\mathbf x})={\mathbf x}/2$. We have defined the projection $p\colon\mathbb R^3\setminus O\to\mathbb S^{2}\times\mathbb S^1$ by
Now we realize an arbitrary Hopf knot by a diffeomorphism $f\in G^+_1$. 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 a ${{\nu}}$-invariant arc in $\mathbb R^3\setminus O$, and $U(\overline L)=p^{-1}(U(L))$ is a ${{\nu}}$-invariant neighbourhoods of it which is diffeomorphic to $\mathbb{D}^{2}\times\mathbb R^1$ (see Fig. 26). Set
$$
\begin{equation*}
C=\{(x_1,x_2,x_3)\in\mathbb R^3\colon x_2^2+x_{3}^2\leqslant 4\}
\end{equation*}
\notag
$$
and let $g^t\colon C\to C$ be the flow defined by Then there exists a diffeomorphism $\zeta\colon U(L)\to C$ that conjugates the diffeomorphisms $\nu\big|_{U(L)}$ and $g=g^1\big|_C$. Let $\phi^t$ be the flow on $C$ defined by the formulae
$$
\begin{equation*}
\begin{aligned} \, \dot{x}_1&=\begin{cases} 1-\dfrac{1}{9}(x_1^2+x_2^2+x_3^2-4)^2,& x_1^2+x_2^2+x_3^2 \leqslant 4, \\ 1, & x_1^2+x_2^2+x_3^2 > 4; \end{cases} \\ \dot{x}_2&=\begin{cases} \dfrac{x_2}{2}\biggl(\sin\biggl(\dfrac{\pi}{2} (x_1^2+x_2^2+x_3^2-3)\biggr)-1\biggr), & 2<x_1^2+x_2^2+x_3^2\leqslant 4, \\ -x_2,& x_1^2+x_2^2+x_3^2\leqslant 2, \\ 0, & x_1^2+x_2^2+x_3^2 > 4; \end{cases} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \dot{x}_3&=\begin{cases} -\dfrac{x_3}{2}\biggl(\sin\biggl(\dfrac{\pi}{2} (x_1^2+x_2^2+x_3^2-3)\biggr)-1\biggr), & 2<x_1^2+x_2^2+x_3^2\leqslant 4, \\ x_3,& x_1^2+x_2^2+x_3^2\leqslant 2, \\ 0, & x_1^2+x_2^2+x_3^2 > 4. \end{cases} \end{aligned}
\end{equation*}
\notag
$$
By construction $\phi=\phi^1$ has two fixed hyperbolic saddle points, the saddle $P_1(-1,0,0)$ with Morse index 1 and the saddle $P_2(1,0,0)$ with Morse index 2 (see Fig. 27). The non-compact heteroclinic curve $W^{\rm s}_{P_1}\cap W^{\rm u}_{P_2}$ coincides with the open interval $\{{\mathbf x}\in\mathbb R^3\colon |x_1|<1,\ x_2=x_3=0\}$. Note that $\phi$ coincides with $g=g^1$ outside the ball $\{{\mathbf x}\in C\colon\|{\mathbf x}\|\leqslant 4\}$. Let $\overline f_L\colon\mathbb R^3\to\mathbb R^3$ be a diffeomorphism that coincides with $\nu$ outside $U(L)$ and with ${\zeta}^{-1}\phi{\zeta}$ on ${U({L})}$. Then $\overline f_{L}$ has two fixed points on $U(L)$, namely, the saddle points ${\zeta}^{-1}(P_1)$ and ${\zeta}^{-1}(P_2)$. Let $N(0,0,0,1)$ be the North pole of the sphere $\mathbb S^3=\{{\mathbf x}=(x_1,x_2,x_3,x_{4})\colon\!\! \|{\mathbf x}\|=1\}$ and $\vartheta\colon\mathbb R^3\to(\mathbb{S}^3\setminus\{N\})$ be the standard stereographic projection. By construction $\overline{f}_{L}$ coincides with $\nu$ in a neighbourhood of the point $O$ and in a neighbourhood of the point at infinity, so that it induces on $\mathbb{S}^3$ the Morse–Smale diffeomorphism
$$
\begin{equation*}
f_{L}({\mathbf s})=\begin{cases} \vartheta(\overline f_{L}(\vartheta({\mathbf s}))),&{\mathbf s}\ne N; \\ N,& {\mathbf s}=N. \end{cases}
\end{equation*}
\notag
$$
We see directly from the construction that the non-wandering set of $f_{L}$ consists of four fixed hyperbolic points: the sink $\omega=S$, the two saddles $\sigma^1=\vartheta({\zeta}^{-1}(P_1))$ and $\sigma^2=\vartheta({\zeta}^{-1}(P_2))$, and the source $\alpha=N$. This diffeomorphism belongs to the class $G^+_1$; we say that such a diffeomorphism is model (see Fig. 28). The proof is complete. It is immediate from Theorems 2 and 3 that for diffeomorphisms in $G^+_1$ the supporting manifold is the 3-sphere $\mathbb S^3$. 5. Topology of manifolds admitting diffeomorphisms in the class $G$5.1. A lens space as the supporting space of diffeomorphisms in $G$In this section we prove the following result. Theorem 4 (see [56], Theorem 1). For each diffeomorphism $f\in G_p$, $p\geqslant 0$, its supporting space is homeomorphic to a lens space $L_{p,q}$. Proof. By Lemma 3.2 we can assume without loss of generality that $f\in G^+_p$, that is, the set $H_f$ of heteroclinic curves of $f$ is orientable and the heteroclinic index is $p\geqslant 0$. We consider two cases separately: (1) $p=0$; (2) $p>0$.
(1) If $p=0$, then $H_f$ is either empty or contains only compact curves bounding discs on $W^{\rm s}_{\sigma_f^1}$, which contain the saddle points $\sigma_f^1$, and on all curves in $H_f$ frames have the same orientation (see Fig. 14). If the set $H_f$ is empty, then by Proposition 1.15 the supporting manifold is homeomorphic to $\mathbb S^2\times\mathbb S^1=L_{0,1}$ (see Fig. 6). Throughout what follows we use the notation introduced in the proof of Lemma 3.1. If $H_f$ is not empty, then each connected component $K^{\rm u}$ of the set $Y_f\setminus \operatorname{int}\widehat N^1_f$ is a smooth annulus one of whose boundary components lies on the torus $T^1_f$ and the other lies on $T^2_f$, and these circles are meridians of the solid tori $N_{L^1_f}$ and $N_{L^2_f}$, respectively. Let $d_1\subset N_{L^1_f}$ and $d_2\subset N_{L^2_f}$ be 2-discs bounded by these meridians and intersecting the knots $L^1_f$ and $L^2_f$, respectively, in one point (see Fig. 29). Then the set $S=K^{\rm u}\cup d_1\cup d_2$ is a 2-sphere embedded cylindrically in the manifold $\widehat V_{\omega_f}$. Since the sphere $S$ intersects both $L^1_f$ and $L^2_f$ in one point, by Proposition 1.3 it is ambiently isotopic to the sphere $\mathbb S^2\times\{s_0\}$, $s_0\in\mathbb S^1$. Let $\widetilde S$ be a sphere close to $S$ and ambiently istopic to $\mathbb S^2\times\{s_0\}$, $s_0\in\mathbb S^1$, such that the intersection $\widetilde S\cap N_{L^i_f}$, $i=1,2$, is a 2-disc $\widetilde d_i$ with a unique point of intersection with the knot $L^i_f$ and $(\widetilde S\cap Y_f)\subset \operatorname{int}(\widetilde d_1\sqcup\widetilde d_2)$. Then the sphere $\overline S$ that is a connected component of the set $p_{\omega_f}^{-1}(\widetilde S)$ bounds a 3-ball $B\subset W^{\rm s}_{\omega_f}$ of which $\omega_f$ is an interior point. Moreover, the intersection $\overline S\cap W^{\rm u}_{\sigma^2_f}$ lie in the disjoint union of the two discs $\Delta_1\subset p_{\omega_f}^{-1}(\widetilde d_1)$ and $\Delta_2\subset p_{\omega_f}^{-1}(\widetilde d_2)$ (see Fig. 30). Set $I=W^{\rm u}_{\sigma_f^1}\setminus \operatorname{int}B$. Since heteroclinic curves are orientable, it follows from the properties of compatible systems of neighbourhoods that there exists a tubular neighbourhood $N_I$ of the arc $I$ such that the intersection $\partial N_I\cap W^{\rm u}_{\sigma_f^2}$ consists of a single closed curve $\mu_2$. Then the set $Q_1=B\cup N_I$ is homeomorphic to a solid torus and $\mu_2$ is a meridian of this torus. Since $\mu_2$ is homotopic to the heteroclinic curves of $f$ on $W^{\rm u}_{\sigma_f^2}\setminus \sigma_f^2$, it bounds a disc $\delta_2$ containing the saddle point $\sigma_f^2$. Consider a tubular neighbourhood $N_{\delta_2}\subset M^3\setminus \operatorname{int}Q_1$ of $\delta_2$ such that $N_{\delta_2}\cap W^{\rm u}_{\sigma_f^2}=\delta_2$ and $N_{\delta_2}\cap\partial Q_1$ is an annulus on $\partial Q_1$ which is a tubular neighbourhood of $\mu_2$. Since $\mu_2$ is an essential curve on the torus $\partial Q_1$, the set $S_\alpha=\partial (Q_1\cup N_{\delta_2})$ is homeomorphic to a 2-sphere. By construction the sphere $S_\alpha$ is disjoint from the unstable manifolds of saddle points, so that by Proposition 1.7 it lies in $W^{\rm u}_\alpha$, where it bounds a 3-ball $B_\alpha$. Thus the set $Q_2=M^3\setminus \operatorname{int}Q_1$ has the following feature: cutting it along the disc $\delta_2$ we obtain a 3-ball. Hence $Q_2$ is a solid torus, $\mu_2$ is a meridian on it, and $M^3=Q_1\cup Q_2$ is the lens space $L_{0,1}\cong\mathbb S^2\times\mathbb S^1$ (see Fig. 14). 2) If $p>0$, then since $H_f$ is orientable, each connected component $K^{\rm u}$ of the set $Y_f\setminus \operatorname{int}\widehat N^1_f$ is an annulus with boundary circles on distinct tori in the set $\partial{\widehat N}_f^1$. Furthermore, these circles are Hopf knots. Then each boundary component of the set $\widehat N_f$ is a 2-torus which is homotopically non-trivially embedded in $\mathbb S^2\times\mathbb S^1$ (see Fig. 31). By Proposition 1.2 such a torus must bound a solid torus in $\widehat V_{\omega_f}$, and therefore $\widehat N_f$ lies in the interior of a solid torus $\widehat J\subset\widehat V_{\omega_f}$. Set $J=p_{\omega^{-1}_f}(\widehat J)$. Since $J$ is an $f$-invariant solid torus with boundary disjoint from the invariant manifolds of saddle points, from Proposition 1.7 we obtain $\partial J\subset W^{\rm u}_{\alpha_f}$. We select a 2-disc $d\subset W^{\rm u}_{\alpha_f}\setminus \alpha_f$ so that $\partial d\subset\partial J$ and $\partial d$ divides $\partial J$ into two connected components. Fix $y_0\in \operatorname{int} d$, and let $S_{\omega_f}$ denote the closure of the connected component of the set $\partial J\setminus\partial d$ that contains $\omega_f$. By construction $S_{\omega_f}$ is a 2-sphere, which is smooth everywhere with the possible exception of the points $\omega_f$. By Proposition 1.1 there exists a smooth 3-ball $B\subset M^3$ such that $\omega_f\in \operatorname{int} B$ and $\partial B$ intersects $S_{\omega_f}$ transversally in a single curve, which separates the points ${\omega_f}$ and $y_0$ in $S_{\omega_f}$. We assume without loss of generality that $\partial B$ intersects the cylinder $J$ in a disc $\Delta$ which intersects $N_f^1$ transversally in two discs and intersects $N_f^2\setminus \operatorname{int} N_f^1$ in $p$ discs (see Fig. 32). We set $I=W^{\rm u}_{\sigma_f^1}\setminus \operatorname{int}B$. By the properties of compatible systems of neighbourhoods and since heteroclinic curves are orientable, there exists a tubular neighbourhood $N_I$ of the arc $I$ such that $N_I\cap\Delta\subset N_f^1\cap\Delta$, $W^{\rm s}_{\sigma_f^1}$ intersects $Q_1=B\cup N_I$ in one 2-disc whose boundary $\mu_1$ intersects $W^{\rm u}_{\sigma_f^2}$ in precisely $p$ points, and the intersection $\partial N_I\cap W^{\rm u}_{\sigma_f^2}$ consists of precisely $p$ curves. By construction the set $Q_1$ is homeomorphic to a solid torus, and
$$
\begin{equation*}
W^{\rm u}_{\sigma_f^2}\cap \partial Q_1= W^{\rm u}_{\sigma_f^2}\cap (\Delta\cup\partial N_I).
\end{equation*}
\notag
$$
Because
$$
\begin{equation*}
\operatorname{cl}(W^{\rm u}_{\sigma_f^2})\setminus W^{\rm u}_{\sigma_f^2}= \operatorname{cl}(W^{\rm u}_{\sigma_f^1})\subset \operatorname{int}Q_1,
\end{equation*}
\notag
$$
$W^{\rm u}_{\sigma_f^2}\cap\partial Q_1$ consists of closed curves. Since the intersection of the disc $W^{\rm u}_{\sigma_f^2}$ with $\partial Q_1$ is orientable, it consists of a single curve $\mu_2$ (see Fig. 33). The curve $\mu_2$ intersects all heteroclinic curves of $f$ in an orientable way, and thus it bounds a disc $\delta_2$ containing the saddle point $\sigma_f^2$. Arguing like for $p=0$ we see that $M^3=Q_1\cup Q_2$ is the lens space $L_{p,q}$, where $\langle p,q\rangle$ is the homotopy type of the curve $\mu_2$ on the torus $\partial Q_1$. The proof is complete. 5.2. Constructing diffeomorphisms with wildly embedded separatrices on each lens spaceIn this subsection we establish constructively the following result. Theorem 5 ([56], Theorem 2). On each lens space $L_{p,q}$ there exists a diffeomorphism $f\in G$ with wildly embedded one-dimensional saddle separatrices. 5.2.1. The construction on $L_{0,1}\cong\mathbb S^2\times\mathbb S^1$Let $L^1,L^2\subset \mathbb S^2\times\mathbb S^1$ be two disjoint Hopf knots, a standard knot and a Masur knot, respectively. Let $N_{L^1}$ and $N_{L^2}$ be disjoint tubular neighbourhoods of these knots. On each torus $T_i=\partial N_{L^i}$, $i=1,2$, we choose generators $\lambda_i$ and $\mu_i$ so that the latitude cycle $\lambda_i$ is a Hopf knot and $\mu_i$ is a meridian of the solid torus $N_{L^i}$. Let $\widetilde N_{L^1}\supset N_{L^1}$ be another tubular neighbourhood of $L^1$ disjoint from $N_{L^2}$, and let $\widetilde T=\partial\widetilde N_{L^1}$ (see Fig. 34). Let $\widehat V$ denote the manifold obtained from $\mathbb S^2\times\mathbb S^1\setminus \operatorname{int}(N_{L^1} \cup N_{L^2})$ by identifying the boundary tori via a diffeomorphism taking the meridian $\mu_1$ to $\mu_2$. Let
$$
\begin{equation*}
q\colon\mathbb S^2\times\mathbb S^1\setminus \operatorname{int}(N_{L^1} \cup N_{L^2})\to\widehat V
\end{equation*}
\notag
$$
be the natural projection. We write $\widehat L^{\rm s}=q(\widetilde T)$ and $\widehat L^{\rm u}=q(T_2)$. Note that the fundamental group $\pi_1(\widehat V)$ admits an epimorphism $\eta\colon\pi_1(\widehat V)\to \mathbb Z$ which assigns to the homotopy class of a closed curve in $\widehat V$ the number of circuits made by this curve around the generator $q(\lambda_1)$. Furthermore, the tori $\widehat L^{\rm s}$ and $\widehat L^{\rm u}$ are $\eta$-essential. We write
$$
\begin{equation*}
\mathcal S=(\widehat V,\eta,\widehat L^{\rm s},\widehat L^{\rm u}).
\end{equation*}
\notag
$$
By construction $\widehat V_{\widehat L^{\rm s}}$ is homeomorphic to the original manifold $\mathbb S^2\times\mathbb S^1$. Since $\widetilde T$ bounds two solid tori in $\mathbb S^2\times\mathbb S^1$, the manifold $\widehat V_{\widehat L^{\rm u}}$ is also homeomorphic to $\mathbb S^2\times\mathbb S^1$. Thus, the scheme $\mathcal S$ is an abstract scheme. By Proposition 1.14, $\mathcal S$ can be realized by a gradient-like diffeomorphism $f\in \operatorname{MS}(M^3)$ such that the schemes $S_f$ and $\mathcal S$ are equivalent. As the sets $\widehat L^{\rm s}$, $\widehat L^{\rm u}$, $\widehat V_{\widehat L^{\rm s}}$, and $\widehat V_{\widehat L^{\rm u}}$ are connected, $f$ has precisely four non-wandering points with pairwise distinct Morse indices, that is, $f\in G$. Since the tori $\widehat L^{\rm s}$ and $\widehat L^{\rm u}$ are disjoint, the set $H_f$ is empty. By Theorem 4 the supporting manifold of $f$ is homeomorphic to the lens space $L_{0,1}\cong\mathbb S^2\times\mathbb S^1$. By Proposition 2.3 the manifold $W^{\rm u}_{\sigma^1_f}$ is wildly embedded in this supporting manifold. 5.2.2. The construction on $L_{p,q}$, $p\ne 0$Let $p\ne 0$, and let $q\ne 0$ be coprime with $p$. On the three-dimensional torus
$$
\begin{equation*}
\mathbb T^3=\mathbb S^1\times\mathbb S^1\times\mathbb S^1= \{(e^{i2\pi x},e^{i2\pi y},e^{i2\pi z})\colon x,y,z\in\mathbb R\}
\end{equation*}
\notag
$$
we consider the generators
$$
\begin{equation*}
\begin{gathered} \, a=\mathbb S^1\times\{e^{i2\pi 0}\}\times\{e^{i2\pi 0}\},\qquad b=\{e^{i2\pi 0}\}\times\mathbb S^1\times\{e^{i2\pi 0}\}, \\ \text{and}\quad c=\{e^{i2\pi 0}\}\times\{e^{i2\pi 0}\}\times\mathbb S^1. \end{gathered}
\end{equation*}
\notag
$$
Let $\widetilde T^{\rm s},\widetilde T^{\rm u}\subset\mathbb T^3$ be the 2-tori defined by
$$
\begin{equation*}
\widetilde T^{\rm s}=\{(e^{i2\pi x},e^{i2\pi y},e^{i2\pi z}) \colon z=0\}\quad\text{and}\quad \widetilde T^{\rm u}=\biggl\{(e^{i2\pi x},e^{i2\pi y},e^{i2\pi z})\colon z=\frac {p}{q}\,y\biggr\}.
\end{equation*}
\notag
$$
Consider tubular neighbourhoods $N_{\widetilde T^{\rm s}}$ and $N_{\widetilde T^{\rm u}}$ of these tori. By construction the closure of each connected component of $\mathbb T^3\setminus (N_{\widetilde T^{\rm s}}\cup N_{\widetilde T^{\rm u}})$ is a solid torus with generator homotopic to the knot $a$. Let $W$ be one such component, and let $\mu_{W}$ denote the meridian of the solid torus $W$ (see Fig. 35). Let $L\subset\mathbb S^2\times\mathbb S^1$ be a Masur knot and $N_L$ be a tubular neighbourhood of it with meridian $\mu_{N_L}$, and let $\zeta\colon \partial N_L\to\partial W$ be a diffeomorphism taking $\mu_{N_L}$ to the meridian $\mu_{W}$. Set
$$
\begin{equation*}
\widehat V=(\mathbb T^3\setminus \operatorname{int}W)\cup_{\zeta} (\mathbb S^2\times\mathbb S^1\setminus \operatorname{int}N_L),
\end{equation*}
\notag
$$
and let
$$
\begin{equation*}
q\colon (\mathbb T^3\setminus \operatorname{int}W)\sqcup (\mathbb S^2\times\mathbb S^1\setminus \operatorname{int}N_L)\to\widehat V
\end{equation*}
\notag
$$
denote the natural projection. Let $\widehat L^{\rm s}=q(\widetilde T^{\rm s})$ and $\widehat L^{\rm u}=q(\widetilde T^{\rm u})$. Note that the fundamental group $\pi_1(\widehat V)$ admits an epimorphism $\eta\colon\pi_1(\widehat V)\to \mathbb Z$ that assigns to the homotopy class of a closed curve in $\widehat V$ the number of circuits it makes around the generator $q(a)$. Furthermore, the tori $\widehat L^{\rm s}$ and $\widehat L^{\rm u}$ are $\eta$-essential. Set
$$
\begin{equation*}
\mathcal S=(\widehat V,\eta,\widehat L^{\rm s},\widehat L^{\rm u}).
\end{equation*}
\notag
$$
Now we verify that the abstract scheme is admissible by showing that the manifold $\widehat V_{\widehat L^{\rm s}}$ is homeomorphic to $\mathbb S^2\times\mathbb S^1$ (the proof for $\widehat V_{\widehat L^{\rm u}}$ is similar). By construction the manifold $\mathbb T^3\setminus \operatorname{int}N_{\widetilde T^{\rm s}}$ is homeomorphic to $\mathbb T^2\times[0,1]$. Gluing a solid torus to each connected component of this manifold so that the meridian of the solid torus is glued to a curve homotopic to the generator $b$, we obtain $\mathbb S^2\times\mathbb S^1$. In addition, the resulting manifold is glued of the solid tori $W$ and $\mathbb S^2\times\mathbb S^1\setminus \operatorname{int}W$ along their boundaries. Hence the manifold $\widehat V_{\widehat L^{\rm s}}$ is obtained by gluing $\mathbb S^2\times\mathbb S^1\setminus \operatorname{int}W$ to $\mathbb S^2\times\mathbb S^1\setminus \operatorname{int}N_L$ along the boundary via a diffeomorphism taking $\mu_{N_L}$ to $\mu_{W}$. Since $\mathbb S^2\times\mathbb S^1\setminus \operatorname{int}W$ is a solid torus, $\widehat V_{\widehat L^{\rm s}}$ is homeomorphic to $\mathbb S^2\times\mathbb S^1$. Thus, $\mathcal S$ is an abstract scheme. By Proposition 1.14, $\mathcal S$ can be realized by a gradient-like diffeomorphism $f\in \operatorname{MS}(M^3)$ such that the schemes $S_f$ and $\mathcal S$ are equivalent. Since the sets $\widehat L^{\rm s}$, $\widehat L^{\rm u}$, $\widehat V_{\widehat L^{\rm s}}$, and $\widehat V_{\widehat L^{\rm u}}$ are connected, $f$ has precisely four non- wandering points with pairwise distinct Morse indices, that is, $f\in G$. Since the tori $\widehat L^{\rm s}$ and $\widehat L^{\rm u}$ intersect along $p$ $\eta$-essential curves in an orientable way, the set $H_f$ is orientable and consists of $p$ non-compact heteroclinic curves. By Theorem 4 the supporting manifold of $f$ is homeomorphic to a lens space $L_{p,q}$. By Proposition 2.3, $W^{\rm u}_{\sigma^1_f}$ is wildly embedded in the supporting manifold.
Citation in format AMSBIB
\Bibitem{PocTal24} |
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|