[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86227-en":3,"doc-seo-86227-105":30,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86227,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Computable Ergodic Optimisation","Links between dynamical systems and computability have been actively studied. In the zero-temperature regime of ergodic optimisation, this work shows that for a computable potential, under several reasonable assumptions, the maximum ergodic average is a computable real and the set of maximising measures forms a Π1-computable compact set. In symbolic dynamics for finite-range interactions on shifts of finite type, it provides a finite-time algorithm to compute the maximum average and the maximising measures.","arXiv :2607 . 11404v1 [math .DS] 13 Jul 2026  \nComputable Ergodic Optimisation  \nGayral Léo, Hoyrup Mathieu  \nAbstract  \nLinks between physicals systems and computability properties have been an active field of investigation in recent years. Inspired by a previous work [GST25] in the context of positive temperature Gibbs measures, we prove here that in the context of zero-temperature ergodic optimisation, for a computable potential and provided with several reasonable assumptions, the maximum ergodic average is a computable real number, and the set of maximising measures is a Π1-computable compact set.  \nThen, in the more specific context of symbolic dynamics, with finite-range interactionson subshifts of finite type, we provide an explicit algorithm to compute both the maximum ergodic average and the set of maximising measures in finite time, with a matching code repository [Gay26] .  \n1 Introduction  \nErgodic optimisation can be interpreted as the study of the zero-temperature limit behaviours of a thermodynamic model, in statistical physics. These theoretical arguments help us reach a better understanding of what actually happens in a physical material at low temperatures, notably regarding the emergence of (quasi)crystalline (a)periodic structures. We redirect interested readers to Jenkinson’s relatively recent survey [Jen19] for more details and context on many typical results on ergodic optimisation, as we will here focus on a quite barebones general definition, to better highlight the seldom studied computational aspects.  \nWe start from a dynamical system (X, d, T ), where (X, d) is a compact metric space, and T : X ! X a continuous transformation. Let MT the set of T-invariant probability measures (i. e . such that µ 􀀎 T −1 = µ), compact in the weak-􀀃 topology. Given a continuous potential function φ : X ! R, we can thus define the maximum ergodic average β (φ) = maxµ∈MT R φ dµ among T-invariant probability measures, and the (convex and compact) set of such φ-maximising measures Mmax (φ) =􀀈µ 2 MT : R φ dµ = β (φ)􀀉 .  \nComputable analysis aims at providing a general framework and tools to computationally describe analytic objects from mathematics, such as real numbers or closed subsets for instance. For general considerations on computable analysis, we refer the readers to the classical textbook by Weihrauch [Wei00], as well as more recent works for more details on links with dynamical systems [Col20] and measure and probability theory [HR21] . Computability theory has permitted the characterisation of the computably realisable values of many conjugacy invariants [HM10; Mey11; JV15 ; Boy+15; Wes17 ; HS18 ; ENT23] of dynamical systems, providing practical tools to distinguish non-equivalent explicitely defined systems. Beyond these invariants, broader links between computability and dynamical systems represent an active area of research [BGZ11; BW24 ; CR24 ; GZ24] .  \nA notable fact, that will represent an unavoidable limitation for broad general results about ergodic optimisation, is that there exist some “simple” computable dynamical systems for which the invariant measures are known to be non-computable [GHR11; BGR12] (and that’s before taking into account any further information regarding the potential, for ergodic optimisation) .  \nA few existing results concern more specifically the interactions between statistical physics and computability, such as the computability of β !7 supµ∈MT h (µ)􀀀β R φ dµ (the pressure function) on some classes of one-dimensional subshifts [Spa08; Bur+22] and higher-dimensional full shifts [GST25], or of some thermodynamic invariants like the residual entropy [BW20] . The general framework for computable analysis, and the specific case of computable dynamical systems will be properly introduced in Section 2.  \nIn the context of ergodic optimisation, the space of potentials we consider (continuous, Lipschitz, Hölder, finite-range…) has a huge influence on the typical (i. e . generic) behav","cbCaihBrGpZbNIaN","https://ap.wps.com/l/cbCaihBrGpZbNIaN","pdf",216335,2,1,21,"English","en",105,"# Introduction\n## Background on Computable Analysis\n## Ergodic Optimisation and Invariant Measures\n## Computable Potentials and Complexity Goals\n## Main Approach and Section Roadmap\n# Background on Computabi","[{\"question\":\"What does the paper prove about the maximum ergodic average in zero-temperature ergodic optimisation?\",\"answer\":\"Under several reasonable assumptions for a computable potential, the maximum ergodic average is a computable real number.\"},{\"question\":\"What is the computability characterization of the set of maximising measures?\",\"answer\":\"The set of maximising measures is a Π1-computable compact set.\"},{\"question\":\"What additional results are provided for symbolic dynamics on shifts of finite type?\",\"answer\":\"For finite-range interactions on subshifts of finite type, the paper gives an explicit finite-time algorithm to compute both the maximum ergodic average and the maximising measures.\"}]",1784209644,53,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"computable-ergodic-optimisation","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/computable-ergodic-optimisation/86227/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-21","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the paper prove about the maximum ergodic average in zero-temperature ergodic optimisation?","Question",{"text":75,"@type":76},"Under several reasonable assumptions for a computable potential, the maximum ergodic average is a computable real number.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the computability characterization of the set of maximising measures?",{"text":80,"@type":76},"The set of maximising measures is a Π1-computable compact set.",{"name":82,"@type":73,"acceptedAnswer":83},"What additional results are provided for symbolic dynamics on shifts of finite type?",{"text":84,"@type":76},"For finite-range interactions on subshifts of finite type, the paper gives an explicit finite-time algorithm to compute both the maximum ergodic average and the maximising 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