Indecomposable continua for unbounded itineraries of exponential maps

TL;DR

This paper investigates the complex dynamics of exponential maps with specific unbounded itineraries, demonstrating the existence of indecomposable continua and analyzing their limit sets, extending previous bounded itinerary results.

Contribution

It introduces new classes of indecomposable continua for unbounded itineraries in exponential maps, generalizing prior bounded itinerary findings.

Findings

Existence of indecomposable continua for certain unbounded itineraries.

Unique point with omega-limit set containing the repelling fixed point.

Other points have omega-limit sets at infinity or the orbit of zero.

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.