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Explore rigidity theorems in dynamical systems through this mathematical lecture that examines how orbit closures and invariant measures behave in systems with homogeneous structures. Delve into the asymptotic behavior of dynamical system orbits by analyzing orbit closures and invariant measures, particularly in systems with Lie group homogeneous structures where rigidity theorems demonstrate that ergodic invariant measures and orbit closures must be well-behaved and classifiable. Learn about groundbreaking joint research with Brown, Eskin, and Rodriguez Hertz that establishes rigidity results for general smooth dynamical systems exhibiting hyperbolicity properties. Discover the necessary assumptions required for these rigidity theorems and understand the homogeneous structures that naturally emerge in these mathematical frameworks, extending classical results beyond traditional homogeneous dynamics into broader mathematical territories.
