Characterizations of topological dynamical IP* sets

We introduce the concept of topological dynamical IP∗ set in a semigroup. We prove that every dynamical IP∗ set in (N,+) is a topological dynamical IP∗ set, and construct a topological dynamical IP∗ set in (N,+) which is not a dynamical IP∗ set. In addition, we generalize the combined zigzag structure of dynamical IP∗ sets to topological dynamical IP∗ sets in weak rings. Furthermore, we prove that topological dynamical IP∗ sets in (N,+) are invariant under the nonhomogeneous spectra maps, that is, if a subset A of N is a topological dynamical IP∗ set, then {⌊nα+γ⌋:n∈A} is also a topological dynamical IP∗ set for any α>0 and 0<γ<1.