{"title":"非均匀双曲映射的几乎确定不变性原理中的速率","authors":"C. Cuny, J. Dedecker, A. Korepanov, F. Merlevède","doi":"10.1007/s00220-024-05090-8","DOIUrl":null,"url":null,"abstract":"<div><p>We prove the almost sure invariance principle (ASIP) with close to optimal error rates for nonuniformly hyperbolic maps. We do not assume exponential contraction along stable leaves, therefore our result covers in particular slowly mixing invertible dynamical systems as Bunimovich flowers, billiards with flat points as in Chernov and Zhang (Stoch Dyn 5:535–553, 2005a, Nonlinearity 18:1527–1553, 2005b) and Wojtkowski’ (Commun Math Phys 126:507–533, 1990) system of two falling balls.For these examples, the ASIP is a new result, not covered by prior works for various reasons, notably because in absence of exponential contraction along stable leaves, it is challenging to employ the so-called Sinai’s trick (Sinai in Russ Math Surv 27:21–70, 1972; Bowen, Lecture Notes in Math vol. 470 (1975)) of reducing a nonuniformly hyperbolic system to a nonuniformly expanding one. Our strategy follows our previous papers on the ASIP for nonuniformly expanding maps, where we build a semiconjugacy to a specific renewal Markov shift and adapt the argument of Berkes et al. (Ann Probab 42:794–817, 2014). The main difference is that now the Markov shift is <i>two-sided</i>, the observables depend on the full trajectory, both the future and the past.</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"405 10","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00220-024-05090-8.pdf","citationCount":"0","resultStr":"{\"title\":\"Rates in Almost Sure Invariance Principle for Nonuniformly Hyperbolic Maps\",\"authors\":\"C. Cuny, J. Dedecker, A. Korepanov, F. Merlevède\",\"doi\":\"10.1007/s00220-024-05090-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove the almost sure invariance principle (ASIP) with close to optimal error rates for nonuniformly hyperbolic maps. We do not assume exponential contraction along stable leaves, therefore our result covers in particular slowly mixing invertible dynamical systems as Bunimovich flowers, billiards with flat points as in Chernov and Zhang (Stoch Dyn 5:535–553, 2005a, Nonlinearity 18:1527–1553, 2005b) and Wojtkowski’ (Commun Math Phys 126:507–533, 1990) system of two falling balls.For these examples, the ASIP is a new result, not covered by prior works for various reasons, notably because in absence of exponential contraction along stable leaves, it is challenging to employ the so-called Sinai’s trick (Sinai in Russ Math Surv 27:21–70, 1972; Bowen, Lecture Notes in Math vol. 470 (1975)) of reducing a nonuniformly hyperbolic system to a nonuniformly expanding one. Our strategy follows our previous papers on the ASIP for nonuniformly expanding maps, where we build a semiconjugacy to a specific renewal Markov shift and adapt the argument of Berkes et al. (Ann Probab 42:794–817, 2014). The main difference is that now the Markov shift is <i>two-sided</i>, the observables depend on the full trajectory, both the future and the past.</p></div>\",\"PeriodicalId\":522,\"journal\":{\"name\":\"Communications in Mathematical Physics\",\"volume\":\"405 10\",\"pages\":\"\"},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-09-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s00220-024-05090-8.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00220-024-05090-8\",\"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}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s00220-024-05090-8","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
摘要
我们以接近最优的误差率证明了非均匀双曲映射的几乎确定不变性原理(ASIP)。我们不假定沿稳定叶的指数收缩,因此我们的结果尤其涵盖了缓慢混合的可逆动力系统,如布尼莫维奇花、切尔诺夫和张 (Stoch Dyn 5:535-553, 2005a, Nonlinearity 18:1527-1553, 2005b) 中的带平点的台球,以及沃伊特科夫斯基 (Commun Math Phys 126:507-533, 1990) 的两落球系统。对于这些例子,ASIP 是一个新结果,由于种种原因而未被先前的工作所涵盖,主要是因为在没有沿稳定叶的指数收缩的情况下,采用所谓的西奈技巧(Sinai in Russ Math Surv 27:21-70, 1972; Bowen, Lecture Notes in Math vol. 470 (1975))将非均匀双曲系统还原为非均匀膨胀系统具有挑战性。我们的策略沿袭了我们之前关于非均匀扩展映射的 ASIP 的论文,在这些论文中,我们为特定的更新马尔可夫移动建立了一个半轭,并改编了 Berkes 等人(Ann Probab 42:794-817, 2014)的论证。主要区别在于现在的马尔可夫变换是双面的,观测值取决于整个轨迹,包括未来和过去。
Rates in Almost Sure Invariance Principle for Nonuniformly Hyperbolic Maps
We prove the almost sure invariance principle (ASIP) with close to optimal error rates for nonuniformly hyperbolic maps. We do not assume exponential contraction along stable leaves, therefore our result covers in particular slowly mixing invertible dynamical systems as Bunimovich flowers, billiards with flat points as in Chernov and Zhang (Stoch Dyn 5:535–553, 2005a, Nonlinearity 18:1527–1553, 2005b) and Wojtkowski’ (Commun Math Phys 126:507–533, 1990) system of two falling balls.For these examples, the ASIP is a new result, not covered by prior works for various reasons, notably because in absence of exponential contraction along stable leaves, it is challenging to employ the so-called Sinai’s trick (Sinai in Russ Math Surv 27:21–70, 1972; Bowen, Lecture Notes in Math vol. 470 (1975)) of reducing a nonuniformly hyperbolic system to a nonuniformly expanding one. Our strategy follows our previous papers on the ASIP for nonuniformly expanding maps, where we build a semiconjugacy to a specific renewal Markov shift and adapt the argument of Berkes et al. (Ann Probab 42:794–817, 2014). The main difference is that now the Markov shift is two-sided, the observables depend on the full trajectory, both the future and the past.
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.