On various definitions of shadowing with average error in tracing

When computing a trajectory of a dynamical system, influence of noise can lead to large perturbations which can appear, however, with small probability. Then when calculating approximate trajectories, it makes sense to consider errors small on average, since controlling them in each iteration may be impossible. Demand to relate approximate trajectories with genuine orbits leads to various notions of shadowing (on average) which we consider in the paper. As the main tools in our studies we provide a few equivalent characterizations of the average shadowing property, which also partly apply to other notions of shadowing. We prove that almost specification on the whole space induces this property on the measure center which in turn implies the average shadowing property. Finally, we study connections among sensitivity, transitivity, equicontinuity and (average) shadowing.