Characterizations of topological dynamical IP* sets

Abstract

ABSTRACT 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 .