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In this paper, we develop the basic theory of the shadowing property forlocal homeomorphisms of metric locally compact spaces, with a focus onapplications to edge shift spaces connected with C*-algebra theory. For thelocal homeomorphism (the Deaconu-Renault system) associated with a directgraph, we completely characterize the shadowing property in terms of conditionson sets of paths. Using these results, we single out classes of graphs forwhich the associated system presents the shadowing property, fully characterizethe shadowing property for systems associated with certain graphs, and showthat the system associated with the rose of infinite petals presents theshadowing property and that the Renewal shift system does not present theshadowing property.

Shadowing for local homeomorphisms, with applications to edge shift spaces of infinite graphs