On embedding Morse–Smale diffeomorphisms on the sphere in topological flows

One important indicator of the adequacy of a numerical solution of an autonomous system of differential equations is the topological conjugacy of the discrete model obtained to the time-one shift map of the original flow. The most significant results in this direction have been obtained for structurally stable flows. In particular, it was shown in [1] and [2] that the Runge–Kutta discretization of a Morse–Smale flow ($n\geqslant 2$) without periodic trajectories on the $n$-disk is topologically conjugate to the time-one shift (for a sufficiently small step size). In this connection the question, going back to Palis [3], of necessary and sufficient conditions for embedding a Morse–Smale diffeomorphism in a topological flow arises naturally.

Recall that a diffeomorphism $f$ on a closed manifold $M^n$ is called a Morse–Smale diffeomorphism if its non-wandering set $\Omega_f$ is finite and consists of hyperbolic periodic points, and for any two points $p,q\in \Omega_f$ the intersection of the stable manifold $W^{\mathrm{s}}_p$ of $p$ and the unstable manifold $W^{\mathrm{u}}_q$ of $q$ is transversal. In [3] the following necessary conditions for embedding a Morse–Smale diffeomorphism $f\colon M^n\to M^n$ in a topological flow were stated, and we call them the Palis conditions: 1) the non-wandering set $\Omega_f$ coincides with the set of fixed points; 2) the restriction of the diffeomorphism $f$ to each invariant manifold of each fixed point $p\in \Omega_f$ preserves its orientation; 3) if for any distinct saddle points $p,q\in \Omega_f$ the intersection $W^{\mathrm{s}}_p\cap W^{\mathrm{u}}_q$ is non-empty, then it contains no compact connected components.

According to [3], in the case when $n=2$ these conditions are not only necessary but also sufficient. In [4] examples of Morse–Smale diffeomorphisms on the three-dimensional sphere were constructed that satisfy the Palis conditions but do not embed in topological flows, and also necessary and sufficient conditions were obtained for embedding a three-dimensional Morse–Smale diffeomorphism in a topological flow. An additional obstruction to embedding such diffeomorphisms in topological flows is connected with the possibility of a non- trivial embedding of the separatrices of saddle points in the ambient manifold. In the present paper we show that for the class $G(S^n)$ of Morse–Smale diffeomorphisms without heteroclinic intersections defined on the sphere $S^n$ of dimension $n\geqslant 4$ and satisfying the Palis conditions no such obstruction exists and the following theorem holds.

Theorem 1.  Any diffeomorphism $f\in G(S^n)$, $n\geqslant 4$, is embedded in a topological flow.

The main tool of the proof is the scheme of a diffeomorphism, defined below. Let $f\in G(S^n)$. It follows from the connection between the dynamics of the diffeomorphism $f$ and the homologies of the sphere $S^n$ that for any saddle point of $f$ either the stable or the unstable manifold has dimension 1. Denote by $A_f$ and $R_f$ the unions of the closures of the unstable and the stable one-dimensional invariant manifolds of the saddle points, respectively; if $f$ has no saddle points with one-dimension unstable (stable) manifolds, then it has a unique sink (source) fixed point, which we also denote by $A_f$ ($R_f$). Let $V_f=M^n\setminus(A_f\cup R_f)$. According to [5], the sets $A_f$, $R_f$, and $V_f$ are connected, and $A_f$ is an attractor, $R_f$ is a repeller, and $V_f$ consists of the wandering points of $f$ going from $R_f$ to $A_f$ and contains all the saddle separatrices of codimension 1.

Denote by $\widehat V_f=V_f/f$ the orbit space of the $f$-action on $V_f$, by $p_f\colon {V_f\to \widehat V_f}$ the natural projection, and by $\eta_f\colon \pi_1(\widehat V_f)\to\mathbb Z$ the epimorphism induced by $p_f$. Let $\widehat{L}^{\mathrm{s}}_{f}$ and $\widehat{L}^{\mathrm{u}}_{f}$ denote the unions of the projections onto $\widehat V_f$ of all the stable and unstable separatrices of the saddle points, respectively. The set $S_{f}=(\widehat V_{f},\eta_f, \widehat{L}^{\mathrm{s}}_{f},\widehat{L}^{\mathrm{u}}_{f})$ is called the scheme of the diffeomorphism $f\in G(S^n)$. The schemes $S_f$ and $S_{f'}$ of diffeomorphisms $f,f'\in G(S^n)$ are said to be equivalent if there exists a homeomorphism $\widehat\varphi\colon\widehat V_f\to\widehat V_{f'}$ with the following properties: 1) $\eta_f=\eta_{f'}\widehat\varphi_*$; 2) $\widehat\varphi(\widehat{L}^ {\mathrm{s}}_{f})= \widehat{L}^{\mathrm{s}}_{f'}$ and $\widehat\varphi(\widehat{L}^{\mathrm{u}}_{f})=\widehat{L}^{\mathrm{u}}_{f'}$.

According to [6], the scheme is a complete topological invariant for diffeomorphisms in $G(S^n)$. The key and most non-trivial point for embedding a diffeomorphism $f\in G(S^n)$, $n\geqslant 4$, in a flow is the equivalence of the scheme $S_f$ to the following standard object. Let $a^t_0$ be a flow on $\mathbb{R}^n\setminus\{O\}$ defined by the formula $a^t_0(x_1,\dots,x_n)=(2^{-t}x_1,\dots,2^{-t}x_n)$, let $a_0$ be the time-one shift along trajectories of $a^t_0$, let $\widehat{V}_{a_0}$ be the orbit space of the $a_0$-action on $\mathbb{R}^n\setminus\{O\}$ (which is diffeomorphic to $\mathbb{S}^{n-1}\times \mathbb{S}^1$), and let $p_{\widehat{V}_{a_0}}\colon\mathbb{R}^n\setminus \{O\}\to \widehat{V}_{a_0}$ be the natural projection. On the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$ $k$ we choose smooth pairwise disjoint $(n-2)$-spheres $S^{n-2}_1,\dots,S^{n-2}_k$. Let $\widetilde{c}_i=\bigcup_{t\in \mathbb{R}} a^t_0(S^{n-2}_i)$ and ${c}_i=p_{\widehat{V}_{a_0}}(\widetilde{c}_i)$. We choose an integer $m\in[0,k]$ and let $\widehat{L}^{\mathrm{s}}_{a_0}=\bigcup_{i=1}^{m}{c}_i$ and $\widehat{L}^{\mathrm{u}}_{a_0}=\bigcup_{i=m+1}^{k}{c}_i$. The set $S_{a_0}=\{\widehat{V}_{a_0},\eta_{\widehat{V}_{a_0}}, \widehat{L}^{\mathrm{s}}_{a_0},\widehat{L}^{\mathrm{u}}_{a_0}\}$ is called the standard scheme.

Lemma.  The scheme $S_f$ of a diffeomorphism $f\in G(S^n)$, $n\geqslant 4$, is equivalent to the standard scheme for some $k$ and $m$.

This lemma allows one to use the method in [4] to construct a flow $X^t$ whose time-one shift has a scheme equivalent to $S_f$. Since the scheme is a complete invariant, there exists a homeomorphism $h\colon S^n\to S^n$ such that $f=hX^1h^{-1}$. Hence $f$ is embedded in the flow $Y^t=hX^th^{-1}$.