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The paper proves that µ is exponentially mixing of all orders for every d.s.h.-type observable built from plurisubharmonic functions, even when the observables are not assumed bounded. It also establishes the central limit theorem for all plurisubharmonic functions with respect to µ. Results extend beyond Hénon maps to the broader class of Hénon-Sibony maps on C^k.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/mixing-and-clt-for-henon-sibony-maps-plurisubharmonic-observables/268084/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/mixing-and-clt-for-henon-sibony-maps-plurisubharmonic-observables/268084.png","ImageObject",300,407,{"name":92,"@type":93},"McGucket","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-17","2026-09-14",true,{"@type":102,"interactionType":103,"userInteractionCount":44},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the equilibrium measure µ used in the paper?","Question",{"text":112,"@type":113},"µ is the unique measure of maximal entropy (equilibrium measure) associated to the Hénon/Sibony dynamical system, constructed in the pluripotential framework.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What does exponential mixing of all orders mean here?",{"text":117,"@type":113},"It provides bounds on deviations of k-point correlations from the product of integrals, uniformly over time indices with exponential control in the minimal gaps.",{"name":119,"@type":110,"acceptedAnswer":120},"How is the central limit theorem (CLT) obtained in this setting?",{"text":121,"@type":113},"The paper proves that all plurisubharmonic observables satisfy a CLT with respect to µ, complementing the established exponential mixing properties.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},268084,1790404922,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":44,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":46,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":31,"preview_page_count":4},1236954412713,"https://us-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","arXiv :2407 . 15418v3 [math .CV] 9 Jul 2025  \nMIXING AND CLT FOR HÉNON-SIBONY MAPS: PLURISUBHARMONIC  \nOBSERVABLES  \nMARCO VERGAMINI AND HAO WU  \nABSTRACT. Let f be a complex Hénon map and µ its unique measure of maximal entropy.  \nWe prove that µ is exponentially mixing of all orders for all (not necessarily bounded) plurisubharmonic observables, and that all plurisubharmonic functions satisfy the central limit theorem with respect to µ . Our results hold more generally for every Hénon-Sibony map on Ck.  \nMathematics Subject Classification 2020: 37F80, 32H50, 32U05, 60F05  \nKeywords: Hénon map, equilibrium measure, exponential mixing, central limit theorem  \n1. INTRODUCTION  \nHénon maps are among the most studied dynamical systems. They were introduced in the real setting by Michel Hénon [32] . Since the 1980s, the study of these maps in the complex setting has seen a surge in activity, following the development of pluripotential theory techniques, see for instance the works of Bedford-Lyubich-Smillie [2, 3, 4, 5] and Fornaess-Sibony [28, 29, 34] . In particular, a unique measure of maximal entropy µ (also called equilibrium measure) for these systems was introduced by Sibony, as the intersection of two canonical positive closed currents, and the study of the ergodic properties of µ has quickly become a central problem in the domain. It is shown in [3] that µ is Bernoulli. In particular, it is mixing of all orders. On the other hand, the control of the speed of mixing is a challenging problem, and is usually obtained only under strong hyperbolicity assumptions on the systems, see for instance [33] .  \nIn [8], Bianchi-Dinh proved the exponential mixing of all orders and the central limit theorem for smooth (and, by the interpolation techniques, Hölder continuous) observables for Hénon maps on C2 and, more generally, for all so called Hénon-Sibony maps on Ck [12, 24, 25] . This solved a long-standing question in the domain. De Thélin-Vigny [14] recently generalized this result for observables in a special subspace of bounded d.s.h. functions (their result is valid in the more general setting of generic birational maps [1, 13]) . The goal of this paper is to obtain these strong statistical properties for all d.s.h. observables. We recall that, roughly speaking, d.s.h. functions are the elements of the vector space spanned by (non necessarily bounded) plurisubharmonic (p.s.h. for short) functions, see [19] and Section 2 for precise definitions. Since their introduction, d.s.h. functions have been proven to be a fundamental tool in the study of holomorphic dynamical systems; they are naturally adapted to the underlying complex structure, and invariant by iteration of holomorphic maps, something that is not true for the usual “real” regularity conditions. We refer to [7, 17, 21, 22, 30, 31] for examples of applications.  \nLet us recall the following definition.  \nMIXING AND CLT FOR HÉNON-SIBONY MAPS: PLURISUBHARMONIC OBSERVABLES 2  \nDefinition 1.1. Let (X, f) be a dynamical system and ν an f-invariant measure. Let (E , ∥·∥E ) be a normed space of real functions on X with ∥·∥Lq(ν) ≲q ∥·∥E for all 1 ≤ q \u003C +∞ . We say that ν is exponentially mixing of all orders for observables in E if, for every κ ∈ N∗ , there exist constants Cκ > 0 and 0 \u003C θκ \u003C 1 such that, for all φ0 ,   ,φκ in E and integers 0 =: n0 ≤ n 1 ≤ ... ≤ nκ , we have  \nκ κ  \n􀀌 Z φ0 (φ1 ◦ fn1 )···(φκ ◦ fnκ )dν −YZ φj dν 􀀌 ≤ Cκ θminκ0≤j≤κ−1(nj+1−nj) Y ∥φj ∥E .  \nj=0 j=0  \nThe following is our first main result, which completely settles the question of the exponential mixing for d.s.h. observables.  \nTheorem 1.2. Let f be a Hénon-Sibony map and µ its equilibrium measure. Then, µ is exponentially mixing of all orders for all observables in DSH(Pk ). More precisely: there exists 0 \u003C θ \u003C 1 such that for every integers κ ∈ N∗, 0 = n0 ≤ n 1 ≤ ... ≤ nκ and for every φ0 ,φ 1 ,   ,φκ ∈ DSH(Pk ), we have  \nκ κ  \n􀀌 Z φ0 (φ1 ◦ fn1 )···(φκ ◦ fnκ )dµ −YZ φj dµ􀀌 ≤ Cκ θmin0≤j≤κ−1(nj+1−nj) ","cbCaitNQuHJq215J","https://ap.wps.com/l/cbCaitNQuHJq215J","pdf",464839,"English","# Introduction\n## Background on Hénon and Hénon-Sibony maps\n## Equilibrium measure and mixing properties\n## Main goal and role of d.s.h. observables\n# Definition and main results\n## Exponential mixing of all orders (Definition 1.1)\n## Exponential mixing for Hénon-Sibony maps (Theorem 1.2)","[{\"question\":\"What is the equilibrium measure µ used in the paper?\",\"answer\":\"µ is the unique measure of maximal entropy (equilibrium measure) associated to the Hénon/Sibony dynamical system, constructed in the pluripotential framework.\"},{\"question\":\"What does exponential mixing of all orders mean here?\",\"answer\":\"It provides bounds on deviations of k-point correlations from the product of integrals, uniformly over time indices with exponential control in the minimal gaps.\"},{\"question\":\"How is the central limit theorem (CLT) obtained in this setting?\",\"answer\":\"The paper proves that all plurisubharmonic observables satisfy a CLT with respect to µ, complementing the established exponential mixing properties.\"}]","MIXING AND CLT FOR HÉNON-SIBONY MAPS: PLURISUBHARMONIC OBSERVABLES | PDF",1789428325]