  We look at the preservation of various notions of shadowing in discrete
dynamical systems under inverse limits, products, factor maps and the induced
maps for symmetric products and hyperspaces. The shadowing properties we
consider are the following: shadowing, h-shadowing, eventual shadowing, orbital
shadowing, strong orbital shadowing, the first and second weak shadowing
properties, limit shadowing, s-limit shadowing, orbital limit shadowing and
inverse shadowing.
