Methods for Studying Abelian Actions and Centralizers - Lecture 1

Explore the intricacies of abelian actions and centralizers in dynamical systems through this comprehensive six-lecture mini-course. Delve into Smale's conjecture and its proof in C^1 topology by Bonatti-Crovisier-Wilkinson, then examine non-generic situations where smooth centralizers form larger groups. Investigate algebraic dynamical systems, defined by automorphisms and translations on homogeneous spaces. Learn about the progression of research showing that large centralizers for Anosov systems often indicate essentially algebraic nature. Cover topics including local and global rigidity for abelian actions, centralizer rigidity, invariant structures, linearization methods, cohomology, partially hyperbolic conservative dynamics, volume disintegration along foliations, transitive centralizers, fibered partially hyperbolic systems, and centralizer classification in accessible examples.