Average shadowing and gluing property

The purpose of this work is threefold: (i) extend shadowing theory for discontinuous and non-invertible systems, (ii) consider more general classes of perturbations (for example, small only on average), (iii) establish a general theory based on the property that the shadowing holds for the case of a single perturbation. The "gluing" construction used in the analysis of the last property turns out to be the key point of this theory.