  We study the dynamics of the exponential maps $E_\lambda: \mathbb{C}
\longrightarrow \mathbb{C}$ defined by $E_\lambda(z) = \lambda e^z$, where
$\lambda > \frac{1}{e}$. We prove that for itineraries of a certain form, the
set of all points sharing the given itinerary, together with the point at
infinity, is an indecomposable continuum in the Riemann sphere. These
itineraries contain infinitely many blocks of zeros whose lengths increase, and
they may be unbounded. We prove that in every such continuum, there exists
exactly one point whose $\omega$-limit set contains the repelling fixed point
of $E_\lambda$. For every other point, the $\omega$-limit set is equal either
to the point at infinity, or to the forward orbit of $0$ together with the
point at infinity. Thus, we generalize the results of R. Devaney and X. Jarque
concerning indecomposable continua for bounded itineraries.
