A logarithm law for non-autonomous systems rapidly converging to equilibrium and mean field coupled systems

We prove that if a non-autonomous system has in a certain sense a fast convergence to equilibrium (faster than any power law behavior), then the time τr(x,y) needed for a typical point x to enter for the first time in a ball B(y,r) centered at y, with small radius r, scales as the local dimension of the equilibrium measure μ at y, i.e., limr→0log⁡τr(x,y)−log⁡r=dμ(y). We then apply the general result to concrete systems of different kinds, showing such a logarithm law for asymptotically autonomous solenoidal maps and mean field coupled expanding maps.