Shadowing and Hyperbolicity for Endomorphisms of Locally Compact Groups

[2606.27647] Shadowing and Hyperbolicity for Endomorphisms of Locally Compact Groups Skip to main content Search Submit Donate Log in Mathematics > Group Theory arXiv:2606.27647 (math) [Submitted on 26 Jun 2026 ( v1 ), last revised 27 Aug 2026 (this version, v3)] Title: Shadowing and Hyperbolicity for Endomorphisms of Locally Compact Groups Authors: Dekui Peng View PDF HTML (experimental) Abstract: We study shadowing, with respect to the left uniformity, for continuous endomorphisms of Lie groups and totally disconnected locally compact groups. For Lie groups, an endomorphism has shadowing if and only if its differential is hyperbolic, with zero eigenvalues allowed. This includes singular maps outside the classical theory of Anosov endomorphisms. As consequences, positively expansive Lie group endomorphisms are topologically expanding, while on connected semisimple Lie groups the shadowing endomorphisms are precisely the nilpotent ones. On compact connected Lie groups, the nonsingular case agrees with classical Anosov theory. In sharp contrast, every continuous endomorphism of an arbitrary totally disconnected locally compact group has shadowing, without compactness, metrizability or invertibility assumptions; the proof uses Willis' tidy-above decomposition. Consequently, in this category topological expansion and the topologically Anosov property reduce to positive expansiveness and expansiveness, respectively. We also discuss group shifts and revisit Aoki's dense-orbit compactness result without assuming metrizability. Subjects: Group Theory (math.GR) ; Dynamical Systems (math.DS) Cite as: arXiv:2606.27647 [math.GR] (or arXiv:2606.27647v3 [math.GR] for this version) https://doi.org/10.48550/arXiv.2606.27647 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Dekui Peng [ view email ] [v1] Fri, 26 Jun 2026 01:56:46 UTC (20 KB) [v2] Sun, 2 Aug 2026 02:56:48 UTC (23 KB) [v3] Thu, 27 Aug 2026 06:07:37 UTC (26 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.GR < prev | next > new | recent | 2026-06 Change to browse by: math math.DS 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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