Meeting Details

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Title: "Coexistence of Hyperbolic and Non-hyperbolic Behavior in Smooth Dynamical Systems" Ph.D. Thesis Defense Jianyu Chen, Adviser: Yakov Pesin, Penn State http:// We investigate the essential coexistence of hyperbolic and non-hyperbolic behavior in dynamical systems, which was numerically observed in classical mechanics but theoretically remains an important open problem. We present some recent devel- opments of such coexistence phenomenon in the category of smooth conservative systems in both discrete and continuous-time cases. In the discrete-time case, we show that there is a smooth volume preserving di eomorphism of a 4-dimensional compact smooth manifold, which is close to the identity map and has nonzero Lyapunov exponents on an open and dense subset of positive but not full volume while having zero Lyapunov exponents on its com- plement. Moreover, this subset consists of countably many connected components, on each of which the di eomorphism is isomorphic to a Bernoulli automorphism. We demonstrate an essential coexistence phenomenon in the continuous-time case by constructing a smooth volume preserving ow on a 5-dimensional compact smooth manifold that has nonzero Lyapunov exponents almost everywhere on an open and dense subset of positive but not full volume and is ergodic on this subset while having zero Lyapunov exponents on its complement. The latter is a union of 3-dimensional invariant submanifolds and on each of these submanifolds the ow is linear with Diophantine frequency vector. Thus the ow exhibits a KAM-type phenomenon.

Room Reservation Information

Room Number: MB106 04 / 23 / 2012 01:00pm - 03:00pm