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Lyapunov exponents for random perturbations of some area-preserving maps including the standard map

Alex Blumenthal University of Maryland-College Park Jinxin Xue University of Chicago Lai-Sang Young Courant institute of mathematical sciences, New York University

Dynamical Systems mathscidoc:1703.11002

Distinguished Paper Award in 2017

Annals of Mathematics, 185, (1), 285-310, 2017.1
We consider a large class of 2D area-preserving diffeomorphisms that are not uniformly hyperbolic but have strong hyperbolicity properties on large regions of their phase spaces. A prime example is the standard map. Lower bounds for Lyapunov exponents of such systems are very hard to estimate, due to the potential switching of “stable” and “unstable” directions. This paper shows that with the addition of (very) small random perturbations, one obtains with relative ease Lyapunov exponents reflecting the geometry of the deterministic maps.
Lyapunov exponent, standard map.
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@inproceedings{alex2017lyapunov,
  title={Lyapunov exponents for random perturbations of some area-preserving maps including the standard map},
  author={Alex Blumenthal, Jinxin Xue, and Lai-Sang Young},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170314235950438109710},
  booktitle={Annals of Mathematics},
  volume={185},
  number={1},
  pages={285-310},
  year={2017},
}
Alex Blumenthal, Jinxin Xue, and Lai-Sang Young. Lyapunov exponents for random perturbations of some area-preserving maps including the standard map. 2017. Vol. 185. In Annals of Mathematics. pp.285-310. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170314235950438109710.
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