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arXiv · 2607.18143

Non-locally connected spaces I: Coverings and branched coverings on metric spaces

Jorge Iglesias·Aldo Portela·Álvaro Rovella·Juliana Xavier
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Abstract

Whether a rational function can have an indecomposable continuum as its Julia set is a fundamental open problem in complex dynamics. We settle this negatively for one of the most classical examples of an indecomposable continuum: we prove that the Knaster continuum cannot be the Julia set of any rational function. To obtain this result, we develop a theory of coverings and branched coverings on general (non-locally connected) metric spaces. We introduce a new Euler characteristic and extend the Riemann--Hurwitz formula to a wider class of spaces, including continua for which the classical shape-theoretic invariants are trivial.

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Jorge Iglesias, Aldo Portela, Álvaro Rovella, Juliana Xavier. 2026-07-20. Non-locally connected spaces I: Coverings and branched coverings on metric spaces. https://arxiv.org/abs/2607.18143

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