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 $\tau _{r}(x,y)$ 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 \ $\mu$ at $y$, i.e. $$\underset{r\rightarrow 0}{\lim }\frac{\log \tau _{r}(x,y)}{-\log r}$$ 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.