Abstract: 
The weak shadowing property is really weaker than the
shadowing property. It is proved that every element of the $C^1$
interior of the set of all diffeomorphisms on a $C^\infty $
closed surface having the weak shadowing property satisfies
Axiom A and the no-cycle condition (this result does not
generalize to higher dimensions), and that the non-wandering set
of a diffeomorphism $f$ belonging to the $C^1$ interior is
finite if and only if $f$ is Morse--Smale.

1. Department of Mathematics
   Kanagawa University
   Yokohama 221-8686, Japan