  We prove that the two-sided limit shadowing property is among the strongest
known notions of pseudo-orbit tracing. It implies shadowing, average shadowing,
asymptotic average shadowing and specification properties. We also introduce a
weaker notion that is called two-sided limit shadowing with a gap and prove
that it implies shadowing and transitivity. We show that those two properties
allow to characterize topological transitivity and mixing in a class of
expansive homeomorphisms and hence they characterize transitive (mixing) shifts
of finite type.
