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Applied and Computational Math Seminar

Lyapunov exponents for small random perturbations of predominantly hyperbolic volume-preserving diffeomorphisms, including the Standard Map 

Alex Blumenthal, University of Maryland

Location:  Hill 425
Date & time: Friday, 25 October 2019 at 12:00PM - 1:00PM

Abstract: An outstanding problem in smooth ergodic theory is the estimation from below of Lyapunov exponents for maps which exhibit hyperbolicity on a large but non-invariant subset of phase space, e.g. the Chirikov standard map or Henon map families. It is notoriously difficult to show that Lyapunov exponents actually reflect the predominant hyperbolicity in the system, due to cancellations caused by the switching of stable and unstable directions in those parts of phase space where hyperbolicity is violated. In this talk I will discuss the inherent difficulties of the above problem, and will discuss recent results when small random perturbations are introduced at every time-step. In this case, we show that for a large class of predominantly hyperbolic systems in two dimensions, the top Lypaunov exponent is large in proportion to the strength of the predominant hyperbolicity in the system. Our results apply to the standard map with large coefficient.

This work is joint with Lai-Sang Young and Jinxin Xue.

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