View AbstractAbstract: For smooth dynamical systems, hyperbolicity (strong expansion and contraction at infinitesimal level) is the main mechanism by which deterministic dynamical systems have chaotic asymptotic regimes. Uniformly hyperbolic systems, those exhibiting hyperbolicity on all of phase space or on the entirety of a stable attractor, are well-known to be asymptotically chaotic. The hyperbolicity mechanism is quite sensitive, however, in the sense that even for systems which are âmostlyâ hyperbolic except for a small exceptional set in phase space, the problem of determining the asymptotic regime (chaotic versus `ordered') is notoriously challenging.
In this talk I will discuss some of the inherent difficulties (coexistence phenomena, cone twisting) in studying âmostlyâ hyperbolic systems. Then, I will put forward the view that the addition of some IID randomness at each timestep has the effect of âunlockingâ hyperbolicity, greatly simplifying the study of these systems. I will discuss results on classes of 1D and 2D models which are âmostlyâ hyperbolic, including multimodal maps of the circle and the well-studied Chirikov standard map family on the torus.
This work is joint with Yun Yang, Lai-Sang Young and Jinxin Xue.