{"title":"Persistent Non-statistical Dynamics in One-Dimensional Maps","authors":"Douglas Coates, Stefano Luzzatto","doi":"10.1007/s00220-024-04957-0","DOIUrl":null,"url":null,"abstract":"<p>We study a class <span>\\(\\widehat{{\\mathfrak {F}}}\\)</span> of one-dimensional full branch maps introduced in Coates et al. (Commun Math Phys 402(2):1845–1878, 2023), admitting two indifferent fixed points as well as critical points and/or singularities with unbounded derivative. We show that <span>\\(\\widehat{{\\mathfrak {F}}}\\)</span> can be partitioned into 3 pairwise disjoint subfamilies <span>\\(\\widehat{{\\mathfrak {F}}} = {\\mathfrak {F}} \\cup {\\mathfrak {F}}_\\pm \\cup {\\mathfrak {F}}_*\\)</span> such that all <span>\\(g \\in {\\mathfrak {F}}\\)</span> have a unique physical measure equivalent to Lebesgue, all <span>\\(g \\in {\\mathfrak {F}}_{\\pm }\\)</span> have a physical measure which is a Dirac-<span>\\(\\delta \\)</span> measure on one of the (repelling) fixed points, and all <span>\\(g \\in {\\mathfrak {F}}_{*}\\)</span> are non-statistical and in particular have no physical measure. Moreover we show that these subfamilies are <i>intermingled</i>: they can all be approximated by maps in the other subfamilies in natural topologies.</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s00220-024-04957-0","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
We study a class \(\widehat{{\mathfrak {F}}}\) of one-dimensional full branch maps introduced in Coates et al. (Commun Math Phys 402(2):1845–1878, 2023), admitting two indifferent fixed points as well as critical points and/or singularities with unbounded derivative. We show that \(\widehat{{\mathfrak {F}}}\) can be partitioned into 3 pairwise disjoint subfamilies \(\widehat{{\mathfrak {F}}} = {\mathfrak {F}} \cup {\mathfrak {F}}_\pm \cup {\mathfrak {F}}_*\) such that all \(g \in {\mathfrak {F}}\) have a unique physical measure equivalent to Lebesgue, all \(g \in {\mathfrak {F}}_{\pm }\) have a physical measure which is a Dirac-\(\delta \) measure on one of the (repelling) fixed points, and all \(g \in {\mathfrak {F}}_{*}\) are non-statistical and in particular have no physical measure. Moreover we show that these subfamilies are intermingled: they can all be approximated by maps in the other subfamilies in natural topologies.
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.