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arXiv · 2608.16445

Dimension of the accumulation set of any hair for the exponential map

Joanna Horbaczewska·Radosław Opoka·Łukasz Pawelec
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Abstract

We study the dynamics of the exponential map on the complex plane. The set $\Lambda_{\mathbf{c}}$ of all points sharing a given itinerary $\mathbf{c}$ is non-empty if and only if $\mathbf{c}$ is an exponentially bounded itinerary. For such itineraries, $\Lambda_{\mathbf{c}}$ also contains a curve of escaping points, and hence its Hausdorff dimension is at least~$1$. We prove that for every exponentially bounded itinerary this dimension is in fact equal to~$1$. In comparison, for certain itineraries, the set $\Lambda_{\mathbf{c}}$ exhibits highly complicated topological structures, such as indecomposable continua.

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Joanna Horbaczewska, Radosław Opoka, Łukasz Pawelec. 2026-08-17. Dimension of the accumulation set of any hair for the exponential map. https://arxiv.org/abs/2608.16445

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