Shadowing of the induced map for contracting homeomorphisms

Abstract

Let $f:X\to X$ be a contracting homeomorphism of a metric space with positive diameter. We prove that the induced map $f_{⁎}$ in the space of probability measures equipped with the Prokhorov metric does not have the shadowing property. However, if X is Polish, then the restriction of $f_{⁎}$ to the Wasserstein space has the generalized shadowing property as per Boyarsky and Gora [4], concerning the Kantorovich-Rubinstein and Prokhorov metrics.