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Explore chain recurrence classes of generic diffeomorphisms in this mathematics lecture from the Simons Semester on Dynamics series. Begin with a review of Conley's theory of Lyapunov functions separation and their role in chain recurrence classes, followed by examples of robustly non-hyperbolic diffeomorphisms. Examine how these examples lead to C∞-generic coexistence of uncountably many chain recurrence classes without periodic orbit (aperiodic classes) functioning as adding machines. Delve into properties of chain recurrence classes containing periodic points and their relationships with homoclinic classes, before investigating recent findings on C^1-locally generic coexistence of uncountably many aperiodic classes that are not adding machines. Master key concepts including viral chain properties of chain recurrence classes, flexible periodic points, and filtrating Markov partitions, while understanding how these elements combine to demonstrate generic coexistence of diverse dynamical behaviors in aperiodic classes.
