Indecomposable continua for unbounded itineraries of exponential maps

Abstract

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.