Rate of approach to minima and sinks—the Morse-Smale case

Abstract:

The dynamical systems herein are Morse-Smale diffeomorphisms and flows on ${C^\infty }$ compact manifolds. We show the asymptotic rate of approach of orbits to the sinks of the systems to be bounded by an expression of the form $K\;\exp ( - DN)$, where $D$ may be any number smaller than $C = {\min _p}\{ 1/m\;\log \;\operatorname {Jac}\;{D_P} {f^m}|{W^u}(P)\}$. Here the minimum is taken over all nonsink $P$ in the nonwandering set of $f$, and $m$ is the period of $P$. We extend our theorems to the entire manifold, so that there is no restriction on the location of the initial points of trajectories.