Various shadowing properties for positively expansive maps

In this paper, various shadowing properties are considered for a positively expansive map on a compact metrizable space. We show that the Lipschitz shadowing property, the s-limit shadowing property and the strong shadowing property are all equivalent to the (usual) shadowing property for a positively expansive map. Furthermore, for a positively expansive open map, the average shadowing property is shown.