In linear dynamics, bounded linear operators over infinite-dimensional Banach spaces have been shown to be able to exhibit interesting characteristics including topological transitivity, topological mixing, and even chaos in the sense of Devaney. This talk will examine weighted $\ell^p$ sequence spaces together with the shift action as the operator. In the case the shift action is over the semi-group $\mathbb{N}$, the above topological properties have been characterized by conditions on the weight sequence associated with a given $\ell^p$ space. In this talk I will present recent results for new characterizations of these properties when we instead consider the group action over a countable group. I will also highlight other open questions.

This is a joint work with Kevin McGoff and William Brian.