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

@article{Blumenthal2017LyapunovEF, title={Lyapunov exponents for random perturbations of some area-preserving maps including the standard map}, author={Alex Blumenthal and Jinxin Xue and Lai-Sang Young}, journal={Annals of Mathematics}, year={2017}, volume={185}, pages={285-310} }

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…

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## References

SHOWING 1-10 OF 54 REFERENCES

Genericity of zero Lyapunov exponents

- MathematicsErgodic Theory and Dynamical Systems
- 2002

We show that, for any compact surface, there is a residual (dense G_{\delta}) set of C^{1} area-preserving diffeomorphisms which either are Anosov or have zero Lyapunov exponents a.e. This result was…

On Stochastic Sea of the Standard Map

- Mathematics
- 2012

AbstractConsider a generic one-parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism. We show that for many parameters (residual subset in an open set approaching…

Stability of Lyapunov exponents

- MathematicsErgodic Theory and Dynamical Systems
- 1991

Abstract We consider small random perturbations of matrix cocycles over Lipschitz homeomorphisms of compact metric spaces. Lyapunov exponents are shown to be stable provided that our perturbations…

Extremal Lyapunov exponents: an invariance principle and applications

- Mathematics
- 2010

We propose a new approach to analyzing dynamical systems that combine hyperbolic and non-hyperbolic (“center”) behavior, e.g. partially hyperbolic diffeomorphisms. A number of applications illustrate…

Bounds on the Lyapunov Exponent via Crude Estimates on the Density of States

- Mathematics
- 2014

We study the Chirikov (standard) map at large coupling λ ≫ 1, and prove that the Lyapounov exponent of the associated Schrödinger operator is of order log λ except for a set of energies of measure…

Lyapunov exponents with multiplicity 1 for deterministic products of matrices

- MathematicsErgodic Theory and Dynamical Systems
- 2004

We exhibit an explicit criterion for the simplicity of the Lyapunov spectrum of linear cocycles, either locally constant or dominated, over hyperbolic (Axiom A) transformations. This criterion is…

Random Versus Deterministic Exponents in a Rich Family of Diffeomorphisms

- Mathematics
- 2002

We study, both numerically and theoretically, the relationship between the random Lyapunov exponent of a family of area preserving diffeomorphisms of the 2-sphere and the mean of the Lyapunov…

Elliptic isles in families of area-preserving maps

- MathematicsErgodic Theory and Dynamical Systems
- 2008

Abstract We prove that every one-parameter family of area-preserving maps unfolding a homoclinic tangency has a sequence of parameter intervals, approaching the bifurcation parameter, where the…

The Lyapunov exponents of generic volume-preserving and symplectic maps

- Mathematics
- 2005

We show that the integrated Lyapunov exponents of C1 volume-preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding Oseledets splitting is trivial…