On aperiodicity and hypercyclic weighted translation operators

We find several equivalent characterizations of aperiodicity of an element on a locally compact group G, and give an intuition for “how strong the aperiodicity of an element affects the existence of hypercyclic weighted translation operators.” In fact, if a is an aperiodic element in G, then there exists a mixing, chaotic and frequently hypercyclic weighted translation Ta,w on Lp(G).