Indecomposable continua for unbounded itineraries of exponential maps

Radosław Opoka

doi:10.3934/dcds.2025080

Article Contents

2025, Volume 45, Issue 12: 4967-4997. Doi: 10.3934/dcds.2025080

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Indecomposable continua for unbounded itineraries of exponential maps

Radosław Opoka

University of Warsaw, Poland

Received: July 2024

Revised: March 2025

Early access: June 6, 2025

Published online: June 6, 2025

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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, its $ \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 from 2002 concerning indecomposable continua for bounded itineraries.

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Mathematics Subject Classification: Primary: 37F10, 37F20.

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Figure 1. The tail $ \omega_{s,\zeta} $ and its base $ \alpha_{s,\zeta} $

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Figure 2. Two circles which are $ \delta $-vertical at height $ y_1 $. The circle on the right is also $ \delta $-vertical at height $ y_2 $, while the circle on the left is not

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Figure 3. Rectangles $ V(a_n,b_{n+k},M_n) $, $ V(a_{n+1},b_{n+k+1},M_{n+1}) $ and part of image of $ V(a_n,b_{n+k},M_n) $

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Figure 4. The curve $ \nu_2 $ passes twice through $ V(a_n,b_{n+j},M_l) $

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Figure 5. The disks $ B_{j,n} $, $ j\in\{0,1,\ldots,\} $. Their radii are smaller than $ 1 $

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Figure 6. The set of $ A^+(n) $ and $ E_\lambda(A^+(n)) $

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Figure 7. The sets $ C_j $ and $ E_\lambda(A^+(n)) $ for $ n $ large enough

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