We will talk about two results in the dynamics of $\lambda e^z$, which may still be generalized in the more general situation of non-autonomous systems (here it means that the real parameter $\lambda$ changes at every step). We will see that the set (naturally defined, and dynamically interesting) of points with so-called code $(0,0,\dots)$ has the topological structure of an indecomposable continuum and, moreover, it has the Hausdorff dimension equal to one. I will show the basic ideas of the proofs, and the problems that arise in the non-autonomous case.