Continuous Morse-Smale flows with three equilibrium positions (Q2821893)

Let \(f^t\) be a continuous flow on a closed \(n\)-dimensional topological manifold \(M^n\), \(n\geq2\). If \(N_1\) and \(N_2\) are invariant sets of the flows \(f_1^t\) and \(f_2^t\), respectively, then we say that these flows are locally topologically equivalent on \(N_1\), \(N_2\), if there exist neighborhoods \(U(N_1)\), \(U(N_2)\) and a homeomorphism \(\phi:U(N_1) \rightarrow U(N_2)\) such that \(\phi(N_1)=N_2\), and \(\phi\) maps every trajectory in \(U(N_1)\) to a trajectory in \(U(N_2)\). Furthermore, the endpoints of arcs of trajectories in \(U(N_1)\) must go to the endpoints of arcs of trajectories in \(U(N_2)\).NEWLINENEWLINEThe authors define \(f^t\) as a continuous Morse-Smale flow if the following conditions hold: NEWLINE{\parindent=7mmNEWLINE\begin{itemize}\item[(i)] The nonwandering set of \(f^t\) consists of a finite set of equilibrium points and periodic trajectories, and the \(\alpha\)- and \(\omega\)-limit sets of any trajectory are contained in the nonwandering set of \(f^t\). NEWLINE\item[(ii)] In a neighborhood of every trajectory in the nonwandering set of \(f^t\), the flow is locally equivalent to a flow which has either a hyperbolic periodic trajectory or a hyperbolic equilibrium position. NEWLINE\item[(iii)] The stable and unstable manifolds of trajectories in the nonwandering set are topologically transversal.NEWLINENEWLINE\end{itemize}}NEWLINENEWLINEA manifold \(M^n\) is said to be projective-like if the following holds: NEWLINE{\parindent=7mm\begin{itemize}\item[(i)] \(n \in \{2,4,8,16\}\); \item[(ii)] \(M^n\) is a disjoint union of an \(n/2\)-dimensional sphere \(S^{n/2}\) that is locally flatly embedded in \(M^n\) and an open \(n\)-dimensional ball \(B^n\), \(M^n =S^{n/2}\cup B^n\), \(S^{n/2}\cap B^n = \emptyset\).NEWLINENEWLINE\end{itemize}}NEWLINENEWLINESuppose that \(f_i^t\), \(i=1,2\), are continuous Morse-Smale flows on a closed \(n\)-dimensional topological manifold \(M_i^n\), and that the nonwandering set of each flow consists of three equilibrium positions. The authors prove that: (1) if \(n\geq2\) then \(M^n\) is a projective-like manifold. Furthermore \(M^2\) is a projective plane \(M^2 = \mathbb{P}^2\) (a nonorientable surface of genus 1 with fundamental group \(\pi_1 (M^2) = \mathbb{Z}_2\)), and for \(n\geq 4\) \(\pi_1(M^n) = \hdots = \pi_{n/{2-1}}(M^n)=0\), and consequently, \(M^n\) is orientable; (2) if \(n=8,16\) then the flows \(f_i^t, \, i=1,2,\) are topologically equivalent if and only if one of the following conditions holds: NEWLINE{\parindent=7mm\begin{itemize}\item[(i)] the stable manifolds of the saddles of the flows \(f_i^t, i=1,2,\) are locally equivalently embedded; \item[(ii)] the unstable manifolds of the saddles of the flows \(f_i^t, \,i=1,2,\) are locally equivalently embedded;NEWLINENEWLINE\end{itemize}} (3) if \(n=2, 4\), \(f_1^t\) and \(f_2^t\) are topologically equivalent. In particular, the manifolds \(M_1^4\) and \(M_2^4\) are homeomophic, and for dimension \(n=2\), the manifolds \(M^2_1\) and \(M^2_2\) are homeomorphic to the two-dimensional projective plane \(\mathbb{P}^2\).