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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.
Let X be a compact Hausdorff space, with uniformity U , and let f : X → X be a continuous function. For D ∈ U , a D-pseudo-orbit is a sequence (x i ) for which (f (x i ), x i+1 ) ∈ D for all indices i. In this paper we show that pseudo-orbits trap ω-limit sets in a neighbourhood of prescribed accuracy after a uniform time period. A consequence of this is a generalisation of a result of Pilyugin et al : every system has the second weak shadowing property. By way of further applications we give a characterisation of minimal systems in terms of pseudo-orbits and show that every minimal system exhibits the strong orbital shadowing property.
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