For more information about this meeting, contact Becky Halpenny.
| Title: | "Coexistence of Hyperbolic and Non-hyperbolic Behavior in Smooth Dynamical Systems" |
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| Seminar: | Ph.D. Thesis Defense |
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| Speaker: | Jianyu Chen, Adviser: Yakov Pesin, Penn State |
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| Abstract Link: | http:// |
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| Abstract: |
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| 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 |
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| Date: | 04 / 23 / 2012 |
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| Time: | 01:00pm - 03:00pm |
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