Topological Stability in Paired Dynamical Systems

We study the classical topological dynamical notions of shadowing and topological stability from a viewpoint of paired dynamical system $(f,g)$, where $f$ and $g$ are uniform equivalences on a metric space $X$. We observe that if $g$ is equicontinuous and commutes with $f$, then the study of shadowing for $(f,g)$ is reduced to the study of the classical shadowing for $g^{-1}f$. The fact that these assumptions are sufficient is justified through examples. Finally, we prove that if $f$ is an expansive homeomorphism on a relatively compact metric space, then any pair $(f,g)$ with shadowing is topologically stable.