Abstract: In this talk, I will present and discuss the topological and geometric
structure of some dynamically defined subsets of $\mathbb C$ which
naturally appear in the dynamics of the family$f_\lambda(z)=\lambda\exp(z)$.
In particular, I shall discuss the structure of Julia sets (this leads
to study of so-called Cantor bouquets), radial Julia sets, and I shall
describe dynamically indecomposable continua. Also, I shall
discuss Hausdorff dimension of these sets.