Ljusternik-Schnirelman category of the non-wandering set

Abstract

The paper deals with dynamical systems in some manifold $M$ satisfying some condition, which is more general tahn axiom A + no-cycle condition and consequently is fulfilled for Morse-Smale systems. Some low estimates for the Ljusternik-Schnirelman category of the non-wandering set $\mathrm{\Omega }$ of such a system are obtained. Namely, the following inequalities are proved.

a) cat$(\mathrm{\Omega },M)\ge \frac{1}{s}$ cat $M$

b)cat $\mathrm{\Omega }\ge$ cat $M$

where $s$ is the number of the basic sets ${\mathrm{\Omega }}_i$. Some applications of this result are obtained.