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Let be a compact smooth manifold and a transitive Anosov diffeomorphism. If is an invariant Borel probability measure on that is absolutely continuous with respect to volume, then the Hopf argument can be used to show that is ergodic. In … Continue reading
An important issue in hyperbolic dynamics is that of absolute continuity. Suppose some neighbourhood of a smooth manifold is foliated by a collection of smooth submanifolds , where is some indexing set. (Here “smooth” may mean , or , or … Continue reading
The stable manifold theorem is one of the most important in the theory of nonlinear ODEs and dynamical systems. Unfortunately, some of the standard introductory texts (Hirsch–Smale, Perko) either do not give a proof, or do not motivate the proof, … Continue reading