Explore the intricate relationship between topology and dynamics in this lecture from the Simons Semester on Dynamics series. Delve into situations where topological considerations, such as actions on homology or the fundamental group, reveal crucial insights about a system's dynamics. Examine how certain dynamics persist under large perturbations within the same homotopy class. Investigate model systems with well-understood yet complex dynamics, and learn how their behaviors are preserved under homotopy. Discover the essential role of hyperbolicity at various scales in these phenomena. Cover key topics including the fixed point index, Lefschetz formula, Nielsen Theory, Franks and Handel homotopy stability theorems, pseudoAnosov maps, and their applications to braid type partial orders, rotation sets, and fluid mixing. Progress through a comprehensive syllabus that includes introductory concepts, steering protocols, index theory, universal cover examples, and fat homotopy, culminating in the main theorem of Nielsen Theory. Read more