Uniform dynamics - Good, Chris; Macías, Sergio (Topology Proceedings, 2022)
We study uniform dynamics properties for compact Hausdorff spaces. Mainly, the uniform shadowing and the uniform $h$-shadowing properties of induced maps between hyperspaces. It is known that if $f$ is a map between compact metric spaces with the shadowing property, then it is not necessarily true that, for $n ≥ 3$, the induced map of $f, \mathcal{F}_n(f)$, between the $n$-fold symmetric products, has the shadowing property. We modify the definition of the uniform shadowing property of an induced map between symmetric products to obtain the following: a map $g$ between compact Hausdorff spaces has the uniform shadowing property if and only if given a positive integer $n$, the induced map of $g, \mathcal{F}_n(g)$, has the modified uniform shadowing property. We do a similar modification of the uniform $h$-shadowing property for an induced map between symmetric products to have that: a map $g$ between compact Hausdorff spaces has the uniform $h$-shadowing property if and only if given a positive integer $n$, the induced map of $g, \mathcal{F}_n(g)$, has the modified uniform $h$-shadowing property. We also show that several uniform dynamical properties are preserved under semi-conjugacies.