This article is about the shadowing property of homeomorphisms on compact metric
spaces with the focus on the map associating a point of the space to each
pseudoorbit — this is what we call a “shadowing map”. Based on some particular
dynamical properties, such as expansivity, we develop a brief theory and a hierarchy
of such maps. We consider examples as odometers, shifts on infinite spaces,
topologically hyperbolic homeomorphisms and north–south dynamics. We revisit a
well-known technique for proving shadowing of expansive homeomorphisms with
canonical coordinates due to R. Bowen to obtain a shadowing map with
the property we call “self-tuning” from a hyperbolic bracket. This notion
is introduced as part of the hierarchy of shadowing maps studied herein.
