A logarithm law for nonautonomous systems fastly converging to equilibrium and mean field coupled systems

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