Universal dense-open windows for generic conservative surface semigroups and graph-directed local systems

Universal dense-open windows for generic conservative surface semigroups and graph-directed local systems | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Universal dense-open windows for generic conservative surface semigroups and graph-directed local systems Yitzchak Shmalo This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9222043/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 5 You are reading this latest preprint version Abstract We prove a universal dense-open window theorem for generic conservative pairs on a closed orientable surface: there exists a residual set of area-preserving pairs $(g_0,g_1)$ such that for \emph{every} choice of dense open sets $D_0,D_1$, the restrictions $g_0|_{D_0}$ and $g_1|_{D_1}$ already generate an open-set transitive pseudosemigroup admitting a residual set of dense admissible branches. The residual set of pairs does not depend on the windows. We then prove a deterministic propagation theorem for graph-directed local systems: once a distinguished core atom carries a transitive pair and every other atom is connected to and from the core through dense open transport windows, dense admissible branches propagate throughout the network. Combining these results with a fibered genericity argument yields a universal parameter theorem for conservative graph-directed systems equipped with two loop words having private single-use edges, removing both the dependence on prescribed windows and any simplicity requirement on the loop words. Finally, we also state a conditional corollary for Hamiltonian window networks realized by Poincaré and scattering maps. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 07 Apr, 2026 Reviewers invited by journal 27 Mar, 2026 Editor assigned by journal 27 Mar, 2026 Submission checks completed at journal 26 Mar, 2026 First submitted to journal 25 Mar, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9222043","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":613733970,"identity":"4e63248e-87bb-4f48-95ed-68e75f04c4b6","order_by":0,"name":"Yitzchak Shmalo","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCUlEQVRIiWNgGAWjYDACZiBmbGAwYGDgAbIKGORAggce4NXCjKzFgMEYrCWBkDXIWhIbQIL4tBgc5z/84ecOO2MG/rNHN/wwsEmfH3b4IdAWOzndBhxaDjMzGPaeSTZjkMhLu9ljkJa78XaaAVBLsrHZAdxakhnbmG0YJHjMbvAYHM7dODsBpOVA4jY8Wg4zttXbMPCfMbv5x+BwuuHs9A+EtDA2M7YdNmNgyDG7DbQlQV46B78tkoeZjRl7244bs0kAtcgYpBlukM4pOJBggNsvfOcPPv7ws63asB/ksDcVNvLys9M3f/hQYSeHS4sCTJwN7lSwiAF25SAg30BYZBSMglEwCkY6AADYNl7ZPQ0rxwAAAABJRU5ErkJggg==","orcid":"","institution":"Hebrew University of Jerusalem","correspondingAuthor":true,"prefix":"","firstName":"Yitzchak","middleName":"","lastName":"Shmalo","suffix":""}],"badges":[],"createdAt":"2026-03-25 10:54:18","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9222043/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9222043/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105715707,"identity":"a529ec63-d2ba-4fd9-80df-19f88ce7280f","added_by":"auto","created_at":"2026-03-30 08:44:46","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1923801,"visible":true,"origin":"","legend":"","description":"","filename":"transitivedynamics.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9222043/v1_covered_141a554a-6da9-406f-b006-da7f557a6b09.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Universal dense-open windows for generic conservative surface semigroups and graph-directed local systems","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"qualitative-theory-of-dynamical-systems","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"qtds","sideBox":"Learn more about [Qualitative Theory of Dynamical Systems](http://link.springer.com/journal/12306)","snPcode":"12346","submissionUrl":"https://submission.nature.com/new-submission/12346/3","title":"Qualitative Theory of Dynamical Systems","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-9222043/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9222043/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe prove a universal dense-open window theorem for generic conservative pairs on a closed orientable surface: there exists a residual set of area-preserving pairs $(g_0,g_1)$ such that for \\emph{every} choice of dense open sets $D_0,D_1$, the restrictions $g_0|_{D_0}$ and $g_1|_{D_1}$ already generate an open-set transitive pseudosemigroup admitting a residual set of dense admissible branches. 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