Abstract

This is joint work with Jacopo De Simoi and Kasun Fernando. We consider a class of sufficiently smooth, partially hyperbolic fast-slow systems on the 2-torus, obtained as size-ε perturbations of the trivial extension of a family of expanding circle maps. Such fast-slow systems satisfy an averaging principle: at time scale 1/ε, the slow component is approximated by an ODE. Assuming that this ODE has exactly one sink and that the Lyapunov exponent in the central direction is positive, we prove that there is a unique physical (SRB) measure. Moreover, we establish exponential decay of correlations, with explicit and nearly optimal bounds on the decay rate.
This result provides a ‘mostly expanding’ counterpart to the work of De Simoi–Liverani, who treated such systems in the ‘mostly contracting’ case (i.e., when the Lyapunov exponent in the central direction is negative).