(untitled) (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 1993)
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 Ω of such a system are obtained. Namely, the following inequalities are proved. a) cat(Ω,M) ≥ 1s cat M b)cat Ω ≥ cat M where s is the number of the basic sets Ω_i. Some applications of this result are obtained.