[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85171-en":3,"doc-seo-85171-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85171,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Computable Functions as Reeb Flows","A coorientable contact 3-manifold supports contact forms whose Reeb flows encode computation. For any computable partial function f from natural numbers to natural numbers, a contact structure and an appropriate defining contact form are constructed so that a chosen Poincaré section produces a partially defined first return map. The return map restricted to an embedded copy of N computes f, aligning dynamical return-time behavior with computability.","arXiv :2607 . 10022v1 [math . SG] 10 Jul 2026  \nCOMPUTABLE FUNCTIONS AS REEB FLOWS  \nKAI CIELIEBAK, ´ANGEL GONZ´ALEZ-PRIETO, AND EVA MIRANDA  \nAbstract. We prove that, given any contact 3-manifold and any computable function f : N 99K N, there exists a defining contact form and a Poincar´e section of its Reeb flow whose partially defined return map computes f.  \n1. Introduction  \nThe realization of computation within continuous dynamical systems is a very active research topic at the crossroads of logic, topology, and analysis. Despite its seemingly simple nature, computation is known to exhibit highly complex dynamics, including chaos, mixing, and undecidability. For this reason, the ability to simulate computability within a family of continuous dynamical systems provides strong evidence that very intricate dynamics can arise.  \nThe most celebrated model of computation is given by the notion of a Turing machine. This isan abstraction of a computer consisting of a simple read-write head inspecting a two-sided tape of symbols. At each step of the computation, the head reads one symbol and, according to the symbol read and its internal state, replaces it with another symbol, transitions to another internal state, and shifts either to the right or to the left along the tape in order to continue the read-write process. This procedure continues until the machine reaches one of its halting states, leaving the result of the computation written on the tape. Turing machines were conceived as a way to formalize the notion that a partially defined function f : N 99K N can be effectively computed by a mechanical algorithm. Such a function f is said to be computable if there exists a Turing machine M such that, when we write any n ∈ N on a tape in binary and run M on it, then M halts if and only if f (n) is defined, in which case the result of the computation is f(n) written in binary on the tape. From this perspective, M is the “algorithm” computing f, the piece of code that is executed to calculate f mechanically. Despite their apparent dependence on Turing machines, computable functions are first-class citizens in computability theory. As Kleene proved in [12], they can be fully characterized as partial recursive functions through a set of axioms independent of the notion of Turing machines. In fact, in line with the Church–Turing thesis, computable functions turn out to be models of computation fully equivalent to Turing machines. In this spirit, there exist computable functions with very complex behaviour, such as universal functions, capable of simulating any other computable function, or undecidable functions, for which there exists no algorithm able to determine their domain of definition.  \nThe three authors are supported by the Bilateral AEI–DFG project Celestial Mechanics, Hydrodynamics, and Turing Machines / Himmelsmechanik, Hydrodynamik und Turing-Maschinen (AQUACELL), with reference codes PCI2024-155042-2 and PCI2024-155062-2, and Deutsche Forschungsgemeinschaft (DFG) Projektnummer 541525489 . AGP has also been supported by projects PID2024-156578NB-I00 and PID2021-124440NB-I00, funded by MICIU/AEI/10.13039/501100011033 and by FEDER/EU. EM is supported by the Catalan Institution for Research and Advanced Studies through an ICREA Academia Prize 2021 . She is also supported by the Spanish State Research Agency through the Severo Ochoa and Mar´ıa de Maeztu Program for Centers and Units of Excellence in R&D, project CEX2020- 001084-M, and by grant PID2023-146936NB-I00, funded by MICIU/AEI/10.13039/501100011033 and by ERDF/EU.  \n2 KAI CIELIEBAK, ´ ´ANGEL GONZALEZ-PRIETO, AND EVA MIRANDA  \nPartially defined functions also arise naturally in continuous dynamical systems. Consider a manifold M with a smooth vector field X and a Poincar´e section D ⊂ M, i.e. , an embedded 2-disk transverse to X . This defines a partially defined first return map Φ : D 99K D as follows: if the forward X-orbit of x ∈ D hits D again we set Φ(x) to be th","cbCaifdfNpA3mTV9","https://ap.wps.com/l/cbCaifdfNpA3mTV9","pdf",1214295,1,18,"English","en",105,"# Introduction\n# Partially defined functions in dynamical systems\n## Poincaré first return maps\n## Contact geometry and Reeb dynamics\n# Main result and proof strategy","[{\"question\":\"What does the main theorem guarantee about computable functions?\",\"answer\":\"For any computable partial function f, there exists a coorientable contact 3-manifold setup with a defining contact form and a Poincaré disk such that the Reeb flow’s partially defined first return map restricted to an embedded N computes f.\"},{\"question\":\"How are partially defined functions represented in continuous dynamical systems?\",\"answer\":\"They arise as first return maps on a Poincaré section: points whose forward orbit returns to the section determine the map value, while points whose orbit never returns make the map undefined.\"},{\"question\":\"What role does contact geometry play in the construction?\",\"answer\":\"A contact form determines a contact structure and a Reeb vector field; the Reeb flow then provides the dynamical system whose transverse Poincaré section yields the computable return map.\"}]",1784201522,45,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"computable-functions-as-reeb-flows","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/computable-functions-as-reeb-flows/85171/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What does the main theorem guarantee about computable functions?","Question",{"text":74,"@type":75},"For any computable partial function f, there exists a coorientable contact 3-manifold setup with a defining contact form and a Poincaré disk such that the Reeb flow’s partially defined first return map restricted to an embedded N computes f.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How are partially defined functions represented in continuous dynamical systems?",{"text":79,"@type":75},"They arise as first return maps on a Poincaré section: points whose forward orbit returns to the section determine the map value, while points whose orbit never returns make the map undefined.",{"name":81,"@type":72,"acceptedAnswer":82},"What role does contact geometry play in the construction?",{"text":83,"@type":75},"A contact form determines a contact structure and a Reeb vector field; the Reeb flow then provides the dynamical system whose transverse Poincaré section yields the computable return map.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]