Underlying Manifolds of High-Dimensional Morse–Smale Diffeomorphisms with Two Saddle Periodic Points

Abstract

The paper describes the topological structure of closed manifolds of dimension $\ge4$ that admit Morse–Smale diffeomorphisms whose nonwandering sets contain arbitrarily many sink periodic points, arbitrarily many source periodic points, and two saddle periodic points. The underlying manifolds of Morse–Smale diffeomorphisms with fewer saddle periodic points are also described.

Introduction

Morse–Smale systems are structurally stable dynamical systems with zero topological entropy. These systems are remarkable in that there is a deep relationship between their dynamical properties and the topological structure of the underlying manifolds (see the relatively recent survey [1]) and in that they exist on any (smooth) closed manifolds [2], [3]. In this article, we restrict ourselves to discrete-time Morse–Smale systems. A discrete-time dynamical system is generated by a diffeomorphism of the underlying manifold, and such a system is the set of iterations of the generating diffeomorphism. In the case of Morse–Smale systems, this generating diffeomorphism is called a Morse–Smale diffeomorphism. (See the next section for precise definitions.)

Clearly, the structure of the underlying manifold does not change if the original diffeomorphism is replaced by some of its iterations. Therefore, without loss of generality, we can (and will) consider Morse–Smale diffeomorphisms whose periodic points are fixed points. By $MS(M^n;a,b,c)$ we denote the set of Morse–Smale diffeomorphisms of a closed smooth $n$-dimensional manifold $M^n$ such that the nonwandering set $NW(f)$ of any diffeomorphism $f\in MS(M^n;a,b,c)$ consists of $a$ sink fixed points (sinks), $b$ source fixed points (sources), and $c$ saddle fixed points (saddles). Smale [2] proved that $a\ge1$ and $b\ge1$. In the dimension $n=1$, one always has $c=0$ and $a=b$, and the underlying manifold is the circle $M^1=S^1$. For the dimensions $n=2,3$, there are quite a few papers on the relationship between the topological structure of the underlying manifolds and the dynamics of Morse–Smale diffeomorphisms (see the surveys [1] and [4] and the book [5], which contains an extensive bibliography). Therefore, we consider underlying manifolds of dimension $n\ge4$ in what follows. (See the remark at the end of the article.)

It is well known that if $c=0$, then $a=b=1$ and the underlying manifold $M^n$ is the $n$-dimensional sphere $\mathbb{S}^n$ [6]. There exists a description of the topological structure of the underlying manifold $M^n$ and the triples $(a,b,c)$ for the case of $c=1$; see Proposition 4. The present research deals with the topological structure of closed underlying manifolds $M^n$, $n\ge4$, and triples $(a,b,c)$ in the case of $c=2$.

First, we consider the case in which both saddles of the diffeomorphism $f\in MS(M^n;a,b,2)$ are saddles of codimension 1. (That is, one of the invariant manifolds of the saddles is one-dimensional.) In the following theorem, $N\otimes S^1$ denotes the total space of a locally trivial bundle with fiber $N$ over $S^1$. The manifold $N\otimes S^1$ is obtained from $N\times[0;1]$ by identifying $N\times \{0\}$ with $N\times \{1\}$ via some diffeomorphism $\tau\colon N\to N$. Throughout the paper, $\mathbb{D}^k$ is the $k$-dimensional closed disk and $\mathbb{S}^k$ is the $k$-dimensional sphere.

Theorem 1.

Let both saddles of a diffeomorphism $f\in MS(M^n;a,b,2)$ , $n\ge4$ , be saddles of codimension 1. Then one of the following two conditions is satisfied:

• (1) $f\in MS(M^n;1,1,2)$ and $M^n$ is homeomorphic to the union of two copies of $\mathbb{D}^{n-1}\otimes S^1$ ;

• (2) $f\in MS(\mathbb{S}^n,2,2,2)\cup MS(\mathbb{S}^n,1,3,2) \cup MS(\mathbb{S}^n,3,1,2)$ .

The following result is proved for the case of only one saddle of codimension 1.

Theorem 2.

Assume that one saddle of a diffeomorphism $f\in MS(M^n;a,b,2)$ , $n\ge4$ , is a saddle of codimension 1 and the other saddle is not a saddle of codimension 1. If the closures of one-dimensional separatrices of the saddle of codimension 1 form a segment, then

• (1) the dimension $n$ of the underlying manifold can only take the values $n\in\{4,8,16\}$ ;

• (2) $f\in MS(M^n;1,2,2)\cup MS(M^n;2,1,2)$ ;

• (3) the manifold $M^n$ is the disjoint union of an open ball $\mathbb{B}^n$ and an $n/2$ -dimensional sphere $S^{n/2}$ topologically embedded in $M^n$ , and if $n\in\{8,16\}$ , then the embedding of $S^{n/2}$ in $M^n$ is locally flat.

We prove the following result for the case in which there are no saddles of codimension 1.

Theorem 3.

Let a diffeomorphism $f\in MS(M^n;a,b,2)$ , $n\ge4$ , have no saddles of codimension 1. Then $f\in MS(M^n;1,1,2)$ , and the manifold $M^n$ is simply connected.

Remark 1.

All sets of Morse–Smale diffeomorphisms in the above statements are nonempty:

$$\begin{gathered} \, MS(\mathbb{D}^{n-1}\otimes S^1\cup \mathbb{D}^{n-1}\otimes S^1;1,1,2)\ne\varnothing,\qquad MS(\mathbb{S}^n,2,2,2)\ne\varnothing, \\ MS(\mathbb{S}^n,1,3,2)\ne\varnothing,\qquad MS(\mathbb{S}^n,3,1,2)\ne\varnothing,\qquad MS(M^n;1,2,2)\ne\varnothing, \\ MS(M^n;2,1,2)\ne\varnothing,\qquad MS(\mathbb{S}^n,1,1,2)\ne\varnothing. \end{gathered}$$

The article is organized as follows. Section 1 contains the main definitions and auxiliary propositions needed to prove the main results. All the main results are proved in Sec. 2.

1. Auxiliary Statements

Let $f\colon M^n\to M^n$ be a diffeomorphism of a smooth closed $n$-dimensional manifold $M^n$ ($n\ge1$). Recall that a point $x\in M^n$ is said to be nonwandering if, for any neighborhood $U$ of $x$ and any positive integer $N_0$, there exists an $n_0\in\mathbb{Z}$ such that

$$|n_0|\ge N_0\qquad \text{and}\qquad f^{n_0}(U)\cap U \ne\varnothing.$$

The set of nonwandering points of $f$ is denoted by $NW(f)$. Obviously, any periodic point is nonwandering. A periodic point $x_0\in\operatorname{Per}(f)$, $f^q(x_0)=x_0$, is said to be hyperbolic if the derivative

$$Df^q(x_0)\colon T_{x_0}M^n\to T_{x_0}M^n,$$

viewed as a linear mapping of the tangent space to itself, has no eigenvalues with unit modulus. For a hyperbolic point $x_0$, there exists a stable manifold $W^s(x_0)$ and an unstable manifold $W^u(x_0)$, which are defined as the sets of points $y\in M^n$ such that

$$\varrho_M(f^{qk}x_0, f^{qk}y)\to 0\qquad \text{as}\quad k\to +\infty\quad \text{and}\quad k\to -\infty,\quad \text{respectively},$$

where $\varrho_M$ is a metric on $M^n$. Note that the unstable manifold $W^u(x_0)$ is a stable manifold with respect to $f^{-1}$. It is well known that $W^s(x_0)$ and $W^u(x_0)$ are homeomorphic (in the intrinsic topology) to the Euclidean spaces $\mathbb{R}^{\dim W^s(x_0)}$ and $\mathbb{R}^{\dim W^u(x_0)}$, respectively, and are injective immersions of the latter in $M^n$.

A diffeomorphism $f$ is called a Morse–Smale diffeomorphism if $NW(f)$ is hyperbolic and consists of finitely many periodic points and the invariant manifolds $W^s(x)$ and $W^u(y)$ intersect transversally (whenever the intersection is nonempty) for any points $x,y\in NW(f)$.

Let $f\colon M^n\to M^n$ be a Morse–Smale diffeomorphism. A hyperbolic fixed point $p\in NW(f)$ is called a node if either $\dim W^s(p)=n$ (in this case, $p$ is a sink) or $\dim W^u(p)=n$ (in this case, $p$ is a source). A hyperbolic fixed point $\sigma\in NW(f)$ is called a saddle if its stable and unstable manifolds have nonzero topological dimension. If $\dim W^u(\sigma)=i$, then we will refer to each component of the set $W^u(\sigma)\setminus\{\sigma\}$ as an $i$-dimensional unstable separatrix and to each component of the set $W^s(\sigma)\setminus\{\sigma\}$ as an $(n-i)$-dimensional stable separatrix. A saddle $\sigma\in NW(f)$ is called a saddle of codimension $1$ if one of its separatrices is one-dimensional. Since the removal of a point disconnects one-dimensional Euclidean space but does not disconnect Euclidean spaces of higher dimension, it follows that a one-dimensional stable or unstable manifold of a saddle periodic point consists of the saddle point itself and two one-dimensional separatrices, while an $i$-dimensional manifold for $i\ge2$ consists of the saddle point and one $i$-dimensional separatrix.

Let $W^{\tau}(\sigma)$, where $\tau=u$ or $s$, be an $i$-dimensional invariant manifold, $i\ge1$, of a saddle $\sigma$. If $i\ge2$, then we denote the separatrix of $\sigma$ lying in $W^{\tau}(\sigma)$ by $W^{\tau}_{\rm sep}(\sigma)$. If $i=1$ (that is, the invariant manifold $W^{\tau}(\sigma)$ is one-dimensional), then $W^{\tau}_{\rm sep}(\sigma)$ denotes one of the two separatrices; where necessary, we will denote them by $W^{\tau}_{\rm sep,1}(\sigma)$ and $W^{\tau}_{\rm sep,2}(\sigma)$. We say that a separatrix $W^{\tau}_{\rm sep}(\sigma)$ has no heteroclinic intersections if it does not meet other separatrices. We need the following well-known statements (see [1]).

Proposition 1.

Let $W^{u(s)}_{\rm sep}(\sigma)$ be a $d$ -dimensional separatrix of a saddle $\sigma$ of a Morse–Smale diffeomorphism, and assume that $W^{u(s)}_{\rm sep}(\sigma)$ has no heteroclinic intersections. Then $W^{u(s)}_{\rm sep}(\sigma)$ lies in the attraction domain $W^s(p)$ (respectively, the repulsion domain $W^u(p)$ ) of exactly one sink (respectively, source) periodic point $p$ .

Moreover, if $d\ge2$ , then the topological closure $\operatorname{clos}W^{u(s)}_{\rm sep}(\sigma)$ of $W^{u(s)}_{\rm sep}(\sigma)$ coincides with

$$\operatorname{clos}W^{u(s)}_{\rm sep}(\sigma)= W^{u(s)}_{\rm sep}(\sigma)\cup\{p\}$$

and is a topologically embedded $d$ -sphere.

If $d=1$ and each one-dimensional separatrix $W^{u(s)}_{\rm sep,j}(\sigma)$ , $j=1,2$ , lies in the attraction domain $W^s(p_j)$ (respectively, the repulsion domain $W^u(p_j)$ ) of exactly one sink (respectively, source) periodic point $p_j$ , then the topological closure $\operatorname{clos}(W^{u(s)}_{\rm sep,1}(\sigma)\cup W^{u(s)}_{\rm sep,2}(\sigma))$ is either a topologically embedded closed segment (for $p_1\ne p_2$ ) or a topologically embedded circle (for $p_1=p_2$ ).

Proposition 2.

Suppose that $\sigma$ is a codimension-1 saddle of a Morse–Smale diffeomorphism $f\colon M^n\to M^n$ , $n\ge4$ , and assume that neither of the one-dimensional separatrices $\operatorname{Sep}_1(\sigma)$ and $\operatorname{Sep}_2(\sigma)$ of $\sigma$ has heteroclinic intersections.

If $\operatorname{clos}(\operatorname{Sep}_1(\sigma) \cup\operatorname{Sep}_2(\sigma))$ is a topologically embedded circle $S_0$ , then $S_0$ has a closed neighborhood $T$ homeomorphic to $\mathbb{D}^{n-1}\otimes S^1$ and containing only two fixed points, the saddle $\sigma$ and some node $n_0$ .

Moreover, if $n_0$ is a sink, then $T$ is forward invariant, and if $n_0$ is a source, then $T$ is backward invariant.

Proposition 3.

Suppose that $\sigma$ is a codimension-1 saddle of a Morse–Smale diffeomorphism $f\colon M^n\to M^n$ , $n\ge4$ , and assume that neither of the one-dimensional separatrices $\operatorname{Sep}_1(\sigma)$ and $\operatorname{Sep}_2(\sigma)$ $\sigma$ has heteroclinic intersections.

If $\operatorname{clos}(\operatorname{Sep}_1(\sigma) \cup\operatorname{Sep}_2(\sigma))$ is a topologically embedded segment $I$ , then $I$ has a closed neighborhood $B$ homeomorphic to an $n$ -dimensional disk and containing only three fixed points, the saddle $\sigma$ and two nodes $n_1$ and $n_2$ .

Moreover, if $n_1$ and $n_2$ are sinks, then $B$ is forward invariant, and if $n_1$ and $n_2$ are sources, then $B$ is backward invariant.

The following proposition, which we give here for reference and which can be extracted from [7] and [8], holds for a Morse–Smale diffeomorphism $f\colon M^n\to M^n$, $n\ge4$, with exactly one saddle (i.e., for $f\in MS(M^n;a,b,1)$).

Proposition 4.

Let $f\colon M^n\to M^n$ be a Morse–Smale diffeomorphism of a closed $n$ -dimensional manifold $M^n$ , $n\ge4$ , and assume that the nonwandering set $NW(f)$ consists of $a$ sinks $\omega_1,\dots,\omega_{a}$ , $b$ sources $\alpha_1,\dots,\alpha_{b}$ , and one saddle $\sigma$ . Then one of the following cases takes place:

• (1) $a+b=3$ (i.e., either $a=1$ and $b=2$ or $a=2$ and $b=1$ ) and the manifold $M^n$ is the sphere $\mathbb{S}^n$ ; moreover, the unstable Morse index of $\sigma$ is $n-1$ for $a=1$ and $b=2$ and 1 for $a=2$ and $b=1$ ;

• (2) $a=b=1$ , and the dimension $n$ of the manifold takes one of the values $n\in\{4,8,16\}$ ; moreover, $M^n$ is the disjoint union of an open ball $\mathbb{B}^n$ and an $n/2$ -dimensional sphere $S^{n/2}$ topologically embedded in $M^n$ , the embedding of $S^{n/2}$ in $M^n$ is locally flat if $n\in\{8,16\}$ , and the saddle $\sigma$ has $n/2$ -dimensional separatrices.

2. Proofs of the Main Results

In this section, the main results of the article are proved. We use the following operation of cutting a manifold $M^n$ along a submanifold of codimension 1. Let $N^{n-1}\subset M^n$ be an $(n-1)$-dimensional submanifold. By cutting $M^n$ along $N^{n-1}$ we mean removing a sufficiently small neighborhood $U$ of $N^{n-1}$ homeomorphic to $N^{n-1}\times(0;1)$ from $M^n$. The resulting (possibly disconnected) manifold $\operatorname{clos}(M^n\setminus U)$ has two additional boundary components, each of which is homeomorphic to $N^{n-1}$; see [9] for a rigorous justification of the possibility of this operation.

Proof of Theorem 1.

First, consider the case in which the codimension-$1$ separatrices of both saddles do not meet the one-dimensional separatrices. Let $\operatorname{Sep}_1(\sigma_i)$ and $\operatorname{Sep}_2(\sigma_i)$ be the one-dimensional separatrices of the saddle $\sigma_i$, $i=1,2$. By assumption,

$$\begin{aligned} \, (\operatorname{Sep}_1(\sigma_1)\cup \operatorname{Sep}_2(\sigma_1))\cap (W^s(\sigma_2)\cup W^u(\sigma_2))&=\varnothing, \\ (\operatorname{Sep}_1(\sigma_2)\cup \operatorname{Sep}_2(\sigma_2))\cap (W^s(\sigma_1)\cup W^u(\sigma_1))&=\varnothing. \end{aligned}$$

By Proposition 1, the following three cases are possible:

• (a) $\operatorname{clos}(\operatorname{Sep}_1(\sigma_1)\cup \operatorname{Sep}_2(\sigma_1))=S_1$ and $\operatorname{clos}(\operatorname{Sep}_1(\sigma_2)\cup \operatorname{Sep}_2(\sigma_2))=S_2$ are circles;

• (b) $\operatorname{clos}(\operatorname{Sep}_1(\sigma_1)\cup \operatorname{Sep}_2(\sigma_1))$ and $\operatorname{clos}(\operatorname{Sep}_1(\sigma_2)\cup \operatorname{Sep}_2(\sigma_2))$ are segments;

• (c) $\operatorname{clos}(\operatorname{Sep}_1(\sigma_1)\cup \operatorname{Sep}_2(\sigma_1))$ is a circle and $\operatorname{clos}(\operatorname{Sep}_1(\sigma_2)\cup \operatorname{Sep}_2(\sigma_2))$ is a segment.

In case (a), the circle $S_i$, $i=1,2$, has a neighborhood $T_i$ homeomorphic to $\mathbb{D}^{n-1}\otimes S^1$ by Proposition 2. Let us show that one of the $T_i$ is forward invariant and the other is backward invariant.

Assume the contrary. To be definite, we assume that both $T_1$ and $T_2$ are backward invariant; i.e., $f^{-1}(T_i)\subset T_i$, $i=1,2$. Since the Morse–Smale diffeomorphism $f$ has only two stable separatrices $\operatorname{Sep}_1(\sigma_i)$ and $\operatorname{Sep}_2(\sigma_i)$, $i=1,2$, lying in $T_1\cup T_2$, it follows that there exist no sources in $M^n\setminus(T_1\cup T_2)$. The set $M^n\setminus(T_1\cup T_2)$ is connected and hence contains exactly one sink. Therefore, $f$ has three nodes. Since the unstable manifolds $W^u(\sigma_1)$ and $W^u(\sigma_2)$ do not meet, it follows by [10] that the number of nodes of $f$ must be even. The resulting contradiction proves that one of the $T_i$ is forward invariant and the other is backward invariant.

To be definite, assume that $T_1$ is backward invariant and $T_2$ is forward invariant. Then $T_1$ contains a source $\alpha\in S_1\subset T_1$ and $T_2$ contains a sink $\omega\in S_2\subset T_2$. Let us show that the set $M^n\setminus(T_1\cup T_2)$ contains no fixed points of $f$.

Assume the contrary. Without loss of generality, we can assume that there exists a sink $\omega_0\in M^n\setminus(T_1\cup T_2)$. Note that since the invariant manifold $W^u(\sigma_1)$ is simply connected, it follows that its limit set $\operatorname{Lim}(W^u(\sigma_1))$ is connected. By assumption, $f$ has only two unstable manifolds. This, together with the inclusions $W^u(\sigma_2)\subset T_2$, $\omega_0\in M^n\setminus(T_1\cup T_2)$, implies the inclusion $\omega_0\in\operatorname{Lim}(W^u(\sigma_1))$. Since $\operatorname{Lim}(W^u(\sigma_1))$ is connected, it follows that $\operatorname{Lim}(W^u(\sigma_1))$ cannot contain $W^u(\sigma_2)\cup\{\omega\}\subset T_2$. Consequently,

$$W^u(\sigma_1)\cap W^s(\sigma_2)=\varnothing.$$

By Proposition 1, the union $W^u(\sigma_1)\cup\{\omega_0\} =S^{n-1}_0$ is a topologically embedded $(n-1)$-dimensional sphere, which we denote by $S^{n-1}_0$. Since its codimension is strictly greater than 2, it follows that this embedding is locally flat [11], [12]. Removing it does not disconnect the manifold $M^n$, because the circle $\operatorname{clos}(\operatorname{Sep}_1(\sigma_1) \cup\operatorname{Sep}_2(\sigma_1))$ meets $S^{n-1}_0$ at exactly one point and the intersection is transversal. Therefore, cutting $M^n$ along $S^{n-1}_0$ gives a connected manifold $\widehat{M}^n$ with two boundary components $M_1$ and $M_2$, each of which is homeomorphic to $S^{n-1}_0$. We glue $M_1$ and $M_2$ with $n$-dimensional balls $B^n_1$ and $B^n_2$, respectively.

Then we obtain a closed manifold $\widetilde{M}^n$. Since the sphere $S^{n-1}_0$ had a backward invariant neighborhood in the original manifold $M^n$, it follows that $f$ extends to $\widetilde{M}^n$ as a Morse–Smale diffeomorphism $\widetilde{f}\colon\widetilde{M}^n\to\widetilde{M}^n$ with two sinks $\omega_i\in B^n_i$, $i=1,2$. Then the nonwandering set of $\widetilde{f}$ consists of the saddle $\sigma_2$, the source $\alpha$, and at least three sinks $\omega$ and $\omega_i$, $i=1,2$, which contradicts Proposition 4.

Thus,

$$NW(f)=\{\alpha\}\cup\{\omega\}\cup\{\sigma_1\} \cup\{\sigma_2\}\in T_1\cup T_2.$$

Moreover, the set $S_1\subset T_1$ is repulsive, and the set $S_2\subset T_2$ is attracting. Since $\partial T_1$ is compact, it follows that there exists a $k\in\mathbb{N}$ such that $f^k(\partial T_1)\subset T_2$. Consequently, $f\in MS(M^n;1,1,2)$, and the manifold $M^n$ is homeomorphic to the union of two copies of $\mathbb{D}^{n-1}\otimes S^1$.

In case (b), we denote the segments $\operatorname{clos}(\operatorname{Sep}_1(\sigma_1) \cup\operatorname{Sep}_2(\sigma_1))$ and $\operatorname{clos}(\operatorname{Sep}_1(\sigma_2) \cup\operatorname{Sep}_2(\sigma_2))$ by $I_1$ and $I_2$, respectively. First, consider the case in which $I_1$ is a repelling set and $I_2$ is an attracting set. Then there exist neighborhoods $U_i\supset I_i$, $i=1,2$, such that $U_1\subset f(U_1)$ and $f(U_2)\subset U_2$. Therefore, one can modify $f$ in $U_1\cup U_2$ so as to obtain a Morse–Smale diffeomorphism $\widetilde{f}\colon M^n\to M^n$ that has one source $\alpha_0\in U_1$ and one sink $\omega_0\in U_2$ and coincides with $f$ outside $U_1\cup U_2$. Thus, $\widetilde{f}$ is a Morse–Smale diffeomorphism without saddle periodic points. Consequently, $\widetilde{f}\in MS(\mathbb{S}^n,1,1,0)$ and $f\in MS(\mathbb{S}^n,2,2,2)$.

Now assume that $I_1$ and $I_2$ are repelling sets. Using the above saddle removal method and Proposition 4, one can show that $\widetilde{f}\in MS(\mathbb{S}^n,1,2,1)\cup MS(\mathbb{S}^n,2,1,1)$ and hence

$$f\in MS(\mathbb{S}^n,1,3,2)\cup MS(\mathbb{S}^n,3,1,2).$$

Let us prove that case (c) cannot be realized. Assume the contrary. Removing the saddle $\sigma_2$ lying in the segment $\operatorname{clos}(\operatorname{Sep}_1(\sigma_2) \cup\operatorname{Sep}_2(\sigma_2))$, we obtain a Morse–Smale diffeomorphism $\widetilde{f}\in MS(M^n;1,1,1)$. By Proposition 4, both separatrices of the remaining saddle $\sigma_1$ must be $n/2$-dimensional. This contradicts the assumption that $\sigma_1$ is a saddle of codimension $1$.

Now consider the case in which the codimension-$1$ separatrix of one of the saddles, say $\sigma_1$, meets the one-dimensional separatrix of the other saddle $\sigma_2$. Let us show that then $f\in MS(\mathbb{S}^n,2,2,2)$. To be definite, assume that the one-dimensional separatrices of the saddles $\sigma_i$, $i=1,2$, are stable separatrices. Since periodic points do not form cycles in the Smale graph, it follows that

$$W^u(\sigma_2)\cap W^s(\sigma_1)=\varnothing.$$

This, together with Proposition 1, implies that $\operatorname{clos}(W^u(\sigma_2))=S^{n-1}$ is a topologically embedded $(n-1)$-dimensional sphere containing the sink $\omega$. Since $n\ge4$, it follows that the embedding of $S^{n-1}$ is locally flat [11], [12]. Cutting $M^n$ along $S^{n-1}$, we obtain a manifold $\widehat{M}^n$ with two boundary components $M_1$ and $M_2$, each of which is homeomorphic to $\mathbb{S}^{n-1}$. By gluing $n$-dimensional balls $B^n_1$ and $B^n_2$ to $M_1$ and $M_2$, respectively, we obtain a closed manifold $\widetilde{M}^n$. Since $S^{n-1}$ is an attracting set, it follows that $f$ extends to $\widetilde{M}^n$ as a Morse–Smale diffeomorphism $\widetilde{f}\colon\widetilde{M}^n\to\widetilde{M}^n$ with sinks $\omega_i\in B^n_i$, $i=1,2$. Note that $\widetilde{f}$ has only one saddle $\sigma_1$ and does not have the fixed points $\sigma_2$ and $\omega$.

If $\widetilde{M}^n$ is connected, then the nonwandering set $NW(\widetilde{f}\,)$ contains only one saddle $\sigma_1$ with unstable Morse index $(n-1)$ and at least two sinks $\omega_i$, $i=1,2$. This contradicts Proposition 4. If the manifold $\widetilde{M}^n$ is disconnected, then one of its components $\widetilde{M}^n_2$ does not contain saddles. Therefore, $\widetilde{M}^n_2=\mathbb{S}^n$, and the nonwandering set $NW(\widetilde{f}\,)\cap\widetilde{M}^n_2$ in $\widetilde{M}^n_2$ consists of a sink and a source. It follows that $M^n=\widetilde{M}^n_1\mathbin{\sharp}\mathbb{S}^n$ is homeomorphic to $\widetilde{M}^n_1$ and $NW(\widetilde{f}\,)$ contains only one source. By Proposition 4, $\widetilde{f}\in MS(\mathbb{S}^n;1,2,1)$. Therefore, $f\in MS(\mathbb{S}^n,2,2,2)$. □

Proof of Theorem 2.

To be definite, we assume that $f$ has a saddle $\sigma$ of codimension $1$ with stable one-dimensional separatrices and a saddle $\sigma_0$ that is not a saddle of codimension $1$. Since the invariant manifolds of saddle periodic points of a Morse–Smale diffeomorphism must intersect transversally, it follows that the one-dimensional separatrices of $\sigma$ do not have heteroclinic intersections. By Proposition 3, the topological closure $\operatorname{clos}W^s(\sigma)$ is a segment $I$ with sources $\alpha_1$ and $\alpha_2$ at the endpoints.

Further, $I$ has a closed backward invariant neighborhood $B$ homeomorphic to an $n$-dimensional disk; i.e., $\operatorname{int}f(B)\subset B$. Therefore, one can modify $f$ in $B$ so as to obtain a Morse–Smale diffeomorphism $\widetilde{f}\colon M^n\to M^n$ that has one source $\alpha_0$ in $B$ and coincides with $f$ outside $B$. In other words, $\widetilde{f}$ has one source less and one saddle less than $f$. Since $\widetilde{f}$ has exactly one saddle of codimension greater than $1$, it follows by Proposition 4 that the dimension of $M^n$ can only take one of the values $n\in\{4,8,16\}$, that $M^n$ is the disjoint union of an open ball $\mathbb{B}^n$ and an $n/2$-dimensional sphere $S^{n/2}$, and that if $n\in\{8,16\}$, then the embedding of $S^{n/2}$ in $M^n$ is locally flat; in addition, the saddle $\sigma_0$ has $n/2$-dimensional separatrices. Moreover, since $\widetilde{f}\in MS(M^n;1,1,1)$, it follows that $f\in MS(M^n;1,2,2)$.

Clearly, if we assume that the saddle $\sigma$ of codimension $1$ has unstable one-dimensional separatrices, then we obtain $f\in MS(M^n;2,1,2)$.

Proof of Theorem 3.

It is well known that if a Morse–Smale diffeomorphism does not contain saddles of codimension $1$, then it is a polar diffeomorphism; i.e., $a=b=1$ [1], [6]. It remains to prove that the manifold $M^n$ is simply connected. Consider a mapping $\gamma\colon S^1\to M^n$ representing an element of the fundamental group $\pi_1(M^n)$. Without loss of generality, we can assume $\gamma$ to be a smooth embedding [13]. Further, by deforming $\gamma$, one can ensure that the image $\gamma(S^1)$ contains no fixed points of $f$.

Since the invariant manifolds of the saddles are the images of smooth immersions of Euclidean spaces and there are finitely many of them, we can ensure by successive deformations of $\gamma$ that $\gamma(S^1)$ intersects all the invariant manifolds of the saddles transversally. Indeed, first, we can ensure the transversality of the intersection with separatrices that do not have heteroclinic intersections, because, outside some neighborhoods of fixed points, these separatrices are embeddings of compact domains of Euclidean spaces [13]. Outside some neighborhoods of these separatrices, the remaining separatrices (with heteroclinic intersections) are embeddings of compact domains of Euclidean spaces as well. Thus, we can assume that $\gamma(S^1)$ intersects all the invariant manifolds of the saddles transversally.

By assumption, the codimension of all invariant manifolds of the saddles is greater than $1$. Since $\gamma(S^1)$ is a circle, we see that the transversality of the intersection of $\gamma(S^1)$ with the invariant manifolds of the saddles means that $\gamma(S^1)$ belongs to the attraction or repulsion domain of a sink or source, respectively. Since such a domain is homeomorphic to an open ball, it follows that the curve $\gamma(S^1)$ is contractible to a point. Consequently, the manifold $M^n$ is simply connected.

Remark 2.

An analysis of the proof of Theorem 3 shows that its statement remains true for an arbitrary number of saddles. We thank the referee for bringing this to our attention.

Remark 3.

For the reader’s convenience, we present a list of closed two- and three-dimensional manifolds admitting Morse–Smale diffeomorphisms with one and two saddles. The only two-dimensional manifolds that admit Morse–Smale diffeomorphisms with one saddle are the sphere and the projective plane. The only 3-manifold admitting Morse–Smale diffeomorphisms with one saddle is the sphere. The only two-dimensional manifolds that admit Morse–Smale diffeomorphisms with two saddles are the sphere, the torus, the Klein bottle, and the projective plane. The 3-manifolds admitting Morse–Smale diffeomorphisms with two saddles are lenses, the sphere, and the direct and skew products of the two-dimensional sphere and the circle.

Funding

This work, except for the proof of Theorem 2, was supported by the Laboratory of Dynamical Systems and Applications at National Research University Higher School of Economics (grant of the Ministry of Science and Higher Education of the Russian Federation, agreement no. 075-15-2019-1931). The proof of Theorem 2 was supported by the Russian Science Foundation under grant 17-11-01041.