The asymptotic average shadowing property and transitivity

We introduce the notion of the asymptotic average shadowing property (Abbrev. AASP) and investigate the relation between the AASP and transitivity. It is shown that a continuous map on XX with the AASP is chain transitive and a LL-hyperbolic homeomorphism on XX with the AASP is topologically transitive, where XX is a compact metric space.