This project aims to foster interactions among the different parts of the broad French holomorphic dynamics community, focusing on the study of moduli spaces associated to algebraic dynamical systems. By their nature, these objects can be approached from multiple perspectives (algebraic, analytic, or arithmetic), and sophisticated techniques in dynamics, whether real, complex, one-dimensional, or multivariate, can be required. Given the rapid development of this topic in recent years, both in France and internationally, our goal is to be a driving force playing a prominent role in this surge. This ambition is grounded in the diverse expertise of our members. The range of skills indeed spans from non-Archimedean geometry to pluripotential theory, through height theory, as well as a deep understanding of bifurcation phenomena such as parabolic implosion or blenders. This project seeks to significantly enhance the interactions among these experts through numerous meetings and three international conferences. It will also promote extended visits by researchers, which are crucial for the development of our international collaborations, and fund young researchers, doctoral students or postdoctoral fellows.