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, 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, 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, F_n(g), has the modified uniform h-shadowing property. We also show that several uniform dynamical properties are preserved under semi-conjugacies.