In this talk I will review and present results on stochastic properties of (generic) slow-fast partially hyperbolic local diffeomorphisms of the 2-torus.
- The past: In a work in collaboration with C. Liverani we showed a Local Central Limit Theorem which was instrumental in our proof of existence of finitely many SRB measures and exponential decay of correlation towards SRB measures, provided that the system admits statistical sinks and that it is mostly contracting (e.g. every SRB measure has negative center Lyapunov exponents).
- The present: In a work in progress with K. Fernando, we study mostly expanding systems. Such systems have paradoxical features (such as statistical sinks with positive Lyapunov exponents). Once again using the LCLT, we prove that in this situation we also have finitely many SRB measures and exponential decay of correlations with bounds similar to the mostly contracting case.
- The future: I will outline some future directions of this work, which will be done in collaboration with C. Liverani: in the sinkless case we now have a strategy to show exponential decay of correlation with explicit bounds. This builds upon our previous work and some earlier result of D. Dolgopyat.