Lowering topological entropy over subsets for amenable group actions

In this paper, we introduce the notions of lowerable, D-lowerable, P-lowerable, hereditarily lowerable, and hereditarily uniformly lowerable for countably infinite amenable group actions. We show that a system with finite entropy is lowerable, D-lowerable, and P-lowerable, and that asymptotic h-expansiveness is equivalent to hereditary uniform lowerability. Moreover, we prove a Bowen’s type theorem for amenable group actions.