There has been some substantial work on studying the structure of a group by analyzing the behavior of primitive elements, sometimes under strong assumptions by work of Platonov, Potapchik, Shpilrain and many others. We formulate and study a conjecture of Platonov and Potapchik for general group actions via analyzing the dynamics of primitive elements for a given action. Such studies led us to further afield to produce results that combines computational, dynamical, geometric, and purely algebraic viewpoints.