Methods for Studying Abelian Actions and Centralizers - Lecture 4

Explore the fourth lecture in a series on methods for studying abelian actions and centralizers in dynamical systems. Focus on partially hyperbolic conservative dynamics, examining the disintegration of volume along foliations and the relationship between large centralizers and pathological center foliations. Delve into the complexities of smooth coordinate changes in dynamical systems, building upon Smale's conjecture and its proof in C^1 topology by Bonatti-Crovisier-Wilkinson. Investigate non-generic situations where the smooth centralizer Z(f) of a dynamical system f is a larger group, particularly in the context of algebraic dynamical systems. Analyze the connection between large centralizers and the algebraic nature of Anosov systems, tracing the development from Katok-Spatzier's local results to more recent semi-local findings for partially hyperbolic systems.