Positive Lyapunov exponent by a random perturbation

Abstract

We study the effect of a random perturbation on a one-parameter family of dynamical systems whose behaviour in the absence of perturbation is ill-understood. We provide conditions under which the perturbed system is ergodic and admits a positive Lyapunov exponent, with an explicit lower bound, for a large and controlled set of parameter values.

Keywords:

• Lyapunov exponent
• random perturbation

AMS Subject Classfications:

• 37H15
• 70K60

Acknowledgements

The authors are indebted to Lai-Sang Young for stimulating discussions. Mikko Stenlund has received funding from the Academy of Finland.

Notes

Note

1. Alternatively, as is done in some of the cited references, one could consider the real quadratic family Q _a : x ↦ a + x ², .