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

[2607.18143] Non-locally connected spaces I: Coverings and branched coverings on metric spaces Skip to main content Search Submit Donate Log in Mathematics > Dynamical Systems arXiv:2607.18143 (math) [Submitted on 20 Jul 2026 ( v1 ), last revised 22 Aug 2026 (this version, v2)] Title: Non-locally connected spaces I: Coverings and branched coverings on metric spaces Authors: Jorge Iglesias , Aldo Portela , Álvaro Rovella , Juliana Xavier View PDF HTML (experimental) 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. Subjects: Dynamical Systems (math.DS) Cite as: arXiv:2607.18143 [math.DS] (or arXiv:2607.18143v2 [math.DS] for this version) https://doi.org/10.48550/arXiv.2607.18143 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Juliana Xavier [ view email ] [v1] Mon, 20 Jul 2026 16:42:40 UTC (59 KB) [v2] Sat, 22 Aug 2026 17:54:20 UTC (61 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DS < prev | next > new | recent | 2026-07 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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