Dynamics of regular topological flows

O. V. Pochinka ; S. Kh. Zinina

Taurida Journal of Computer Science Theory and Mathematics, no. 3 (2020), pp. 77-91

Résumé

It is well known that for dimensions 4 and greater there are topological manifolds admitting no smooth structure. Therefore, dynamical systems as well as functions on such manifolds may only be considered as topological and continuous, respectively. Nevertheless, these systems and functions have the same properties as the smooth ones and they are closely related to the topology of the ambient manifold. In this paper, we introduce a class $G$ of continuous flows $f^t$ on closed topological n-manifold $M$ that generalize the concept of Morse-Smale flows. Such flows have a hyperbolic (in the topological sense) chain recurrent set $R_{f^t}$ consisting of a finite number of orbits (chain components). Each non-wandering orbit is either a fixed point or a periodic orbit $\mathcal O$ for which the concept of stable $W^s_{\mathcal O}$ and unstable $W^u_{\mathcal O}$ manifolds is correctly defined. It is shown that the chain components of the considered flows do not form cycles and, therefore, can be completely ordered $$O_1 \prec \dots \prec \mathcal O_k$$ with the Smale relation preserved: $$W^s_{\mathcal O_i}\cap W^u_{\mathcal O_j}\neq\emptyset\Rightarrow i$$ We establish the following main dynamic properties of flows from the class $G$. Let $f^t\in G$. Then $M=\bigcup\limits_{i=1}^kW^u_{ \mathcal O_i}=\bigcup\limits_{i=1}^kW^s_ {\mathcal O_i}$; for any fixed point $\mathcal O_i$ there is a number $\lambda_i\in\{0,\dots, n\}$ $($Morse index of the point $\mathcal O_i)$ such that its unstable manifold $W^u_{\mathcal O_i}$ is a topological submanifold of $M$, homeomorphic to $\mathbb R^{\lambda_{i}}$, and the stable manifold $W^s_{\mathcal O_i}$ is a topological submanifold of $M$, homeomorphic to $\mathbb R^{n-\lambda_{i}}$; for a periodic orbit $\mathcal O_i$ there is a number $\lambda_i\in\{0,\dots, n-1\}$ $($Morse index of the orbit $\mathcal O_i)$ and a pair of numbers $\mu_i,\nu_i\in\{-1,+1\}$ $($orbit type $\mathcal O_i)$ such that its unstable manifold $W^u_{\mathcal O_i}$ is a topological submanifold of the manifold $M$, homeomorphic to $\mathbb R^{\lambda_{i}}\times\mathbb S^1$ for $\mu_i=+1$ and $\mathbb R^{\lambda_{i}}\widetilde{\times}\mathbb S^1$ for $\mu_i=-1$; the stable manifold $W^s_{\mathcal O_i}$ is a topological submanifold of the manifold $M$, homeomorphic to $\mathbb R^{n-\lambda_{i}}\times\mathbb S^1$ for $\nu_i=+1$ and $\mathbb R^{n-\lambda_{i}}\widetilde{\times}\mathbb S^1$ for $\nu_i=-1$; $(cl(W^u_{\mathcal O_i})\setminus W^u_{\mathcal O_i})\subset\bigcup\limits_{j=1}^{i-1}W^u_{ \mathcal O_j}\,\,\,\,\,\, ((cl(W^s_{ \mathcal O_i})\setminus W^s_{\mathcal O_i})\subset\bigcup\limits_{j=i+1}^{k}W^s_{ \mathcal O_j})$.

Détail

Keywords: manifold, topological flow, regular dynamics, hyperbolic set, chain recurrent set.