In my previous research, during my PhD and several years as a postdoc, I studied endomorphisms of the projective line. More specifically, I focused on rational functions that can be represented as the quotient of two single-variable polynomials, working primarily over number fields or function fields (often defined over finite fields). The main objects of interest were the periodic and preperiodic points of these rational functions. Periodic points are those whose forward orbits form cycles, while preperiodic points have finite forward orbits but are not necessarily part of a cycle. As we will see, many concepts arising in this context can be naturally interpreted within the theory of elliptic curves. In addition, I investigated particular families of rational functions characterized by a natural notion of good reduction, which enables us to reduce the problem to a setting over finite fields. Throughout this work, moduli spaces play a central role in understanding the structure and classification of such dynamical systems.