Explore the third lecture in a series on dynamics, focusing on methods for studying abelian actions and centralizers. Delve into two key approaches: linearization and non-linearization techniques, as well as the role of cohomology in dynamical systems analysis. Gain insights into how these methods are applied to investigate smooth coordinate changes in dynamical systems and their implications for understanding the behavior of larger centralizer groups. Learn about the connection between these techniques and the study of algebraic dynamical systems, particularly in the context of automorphisms and translations on homogeneous spaces. Examine how these methods contribute to the ongoing research on centralizer rigidity and the classification of dynamical systems with large centralizers.