We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk [A homology theory for étale groupoids. J. Reine Angew. Math.521 (2000), 25–46], with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for principal groupoids whose orbits are copies of an Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.
我们考虑的是 Crainic 和 Moerdijk 引入的 étale 子群的同调理论[A homology theory for étale groupoids.J. Reine Angew.Math.521 (2000)，25-46]，特别关注拓扑动力学系统中产生的群集。我们证明了群集乘积的库奈特公式，以及轨道为欧几里得空间副本的主群集的波恩卡列对偶类型结果。最后，我们举例说明了与自相似作用等零能群相关的系统的计算，并推广了伯克和普特南之前对代数数产生的类似孤子的系统进行的同调计算。对于后一种系统，我们证明了HK猜想，即使所得到的群集不是充裕的。
{"title":"Homology and K-theory of dynamical systems IV. Further structural results on groupoid homology","authors":"VALERIO PROIETTI, Makoto Yamashita","doi":"10.1017/etds.2024.37","DOIUrl":"https://doi.org/10.1017/etds.2024.37","url":null,"abstract":"\u0000 We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk [A homology theory for étale groupoids. J. Reine Angew. Math.521 (2000), 25–46], with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for principal groupoids whose orbits are copies of an Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140972261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this paper is to determine the asymptotic growth rate of the complexity function of cut-and-project sets in the non-abelian case. In the case of model sets of polytopal type in homogeneous two-step nilpotent Lie groups, we can establish that the complexity function asymptotically behaves like $r^{{mathrm {homdim}}(G) dim (H)}$ . Further, we generalize the concept of acceptance domains to locally compact second countable groups.
本文的目的是确定在非阿贝尔情况下切投集复杂性函数的渐近增长率。在同质两步零钾烈群中的多顶型模型集的情况下，我们可以确定复杂性函数渐近地表现为 $r^{{mathrm {homdim}}(G) dim (H)}$ 。此外，我们还将接受域的概念推广到局部紧凑的第二可数群。
{"title":"Complexity of non-abelian cut-and-project sets of polytopal type I: special homogeneous Lie groups","authors":"PETER KAISER","doi":"10.1017/etds.2024.38","DOIUrl":"https://doi.org/10.1017/etds.2024.38","url":null,"abstract":"The aim of this paper is to determine the asymptotic growth rate of the complexity function of cut-and-project sets in the non-abelian case. In the case of model sets of polytopal type in homogeneous two-step nilpotent Lie groups, we can establish that the complexity function asymptotically behaves like <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000385_inline1.png\"/> <jats:tex-math> $r^{{mathrm {homdim}}(G) dim (H)}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Further, we generalize the concept of acceptance domains to locally compact second countable groups.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930852","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define the co-spectral radius of inclusions ${mathcal S}leq {mathcal R}$ of discrete, probability- measure-preserving equivalence relations as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on $G/H$ for inclusion $Hleq G$ of groups. For the proof, we develop a more general version of the 2–3 method we used in another work on the growth of unimodular random rooted trees. We use this method to show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for hyperfinite relations, and discuss new critical exponents for percolation that can be defined using the co-spectral radius.
{"title":"Co-spectral radius for countable equivalence relations","authors":"MIKLÓS ABERT, MIKOLAJ FRACZYK, BENJAMIN HAYES","doi":"10.1017/etds.2024.32","DOIUrl":"https://doi.org/10.1017/etds.2024.32","url":null,"abstract":"We define the co-spectral radius of inclusions <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000324_inline1.png\"/> <jats:tex-math> ${mathcal S}leq {mathcal R}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of discrete, probability- measure-preserving equivalence relations as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000324_inline2.png\"/> <jats:tex-math> $G/H$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for inclusion <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000324_inline3.png\"/> <jats:tex-math> $Hleq G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of groups. For the proof, we develop a more general version of the 2–3 method we used in another work on the growth of unimodular random rooted trees. We use this method to show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for hyperfinite relations, and discuss new critical exponents for percolation that can be defined using the co-spectral radius.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that for $C^{1+theta }$ , $theta $ -bunched, dynamically coherent partially hyperbolic diffeomorphisms, the stable and unstable holonomies between center leaves are $C^1$ , and the derivative depends continuously on the points and on the map. Also for $C^{1+theta }$ , $theta $ -bunched partially hyperbolic diffeomorphisms, the derivative cocycle restricted to the center bundle has invariant continuous holonomies which depend continuously on the map. This generalizes previous results by Pugh, Shub, and Wilkinson; Burns and Wilkinson; Brown; Obata; Avila, Santamaria, and Viana; and Marin.
{"title":"On invariant holonomies between centers","authors":"RADU SAGHIN","doi":"10.1017/etds.2024.33","DOIUrl":"https://doi.org/10.1017/etds.2024.33","url":null,"abstract":"We prove that for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000336_inline1.png\"/> <jats:tex-math> $C^{1+theta }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000336_inline2.png\"/> <jats:tex-math> $theta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-bunched, dynamically coherent partially hyperbolic diffeomorphisms, the stable and unstable holonomies between center leaves are <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000336_inline3.png\"/> <jats:tex-math> $C^1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, and the derivative depends continuously on the points and on the map. Also for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000336_inline4.png\"/> <jats:tex-math> $C^{1+theta }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000336_inline5.png\"/> <jats:tex-math> $theta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-bunched partially hyperbolic diffeomorphisms, the derivative cocycle restricted to the center bundle has invariant continuous holonomies which depend continuously on the map. This generalizes previous results by Pugh, Shub, and Wilkinson; Burns and Wilkinson; Brown; Obata; Avila, Santamaria, and Viana; and Marin.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Feng and Huang [Variational principle for weighted topological pressure. J. Math. Pures Appl. (9)106 (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto [New approach to weighted topological entropy and pressure. Ergod. Th. & Dynam. Sys.43 (2023), 1004–1034] redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto’s approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.
Feng and Huang [Variational principle for weighted topological pressure.J. Math.Pures Appl. (9)106 (2016)，411-452] 引入了动态系统间因子映射的加权拓扑熵和压力，并建立了其变分原理。Tsukamoto [New approach to weighted topological entropy and pressure.Ergod.Th. & Dynam.Sys.43（2023），1004-1034] 对最简单情况下的这些不变式进行了完全不同的重新定义，并通过变分原理证明这两个定义是重合的。我们推广了塚本的方法，重新定义了更高维度的加权拓扑熵和压力，并证明了变分原理。我们的结果允许对仿射不变集的豪斯多夫维度进行基本计算，如自仿射海绵和驻留在任意维度欧几里得空间中的某些索菲克集。
{"title":"Weighted topological pressure revisited","authors":"NIMA ALIBABAEI","doi":"10.1017/etds.2024.35","DOIUrl":"https://doi.org/10.1017/etds.2024.35","url":null,"abstract":"Feng and Huang [Variational principle for weighted topological pressure. <jats:italic>J. Math. Pures Appl. (9)</jats:italic>106 (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto [New approach to weighted topological entropy and pressure. <jats:italic>Ergod. Th. & Dynam. Sys.</jats:italic>43 (2023), 1004–1034] redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto’s approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"ETS volume 44 issue 6 Cover and Back matter","authors":"","doi":"10.1017/etds.2023.88","DOIUrl":"https://doi.org/10.1017/etds.2023.88","url":null,"abstract":"","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141006450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"ETS volume 44 issue 6 Cover and Front matter","authors":"","doi":"10.1017/etds.2023.87","DOIUrl":"https://doi.org/10.1017/etds.2023.87","url":null,"abstract":"","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141007120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author [The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math. (2)172 (2010), 1949–2033], and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded $operatorname {SL}(d,mathbb {R})$ -horospheres in the space of affine lattices.
{"title":"Poissonian pair correlation for directions in multi-dimensional affine lattices and escape of mass estimates for embedded horospheres","authors":"WOOYEON KIM, JENS MARKLOF","doi":"10.1017/etds.2024.31","DOIUrl":"https://doi.org/10.1017/etds.2024.31","url":null,"abstract":"We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for <jats:italic>any</jats:italic> irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author [The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. <jats:italic>Ann. of Math. (2)</jats:italic>172 (2010), 1949–2033], and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000312_inline1.png\"/> <jats:tex-math> $operatorname {SL}(d,mathbb {R})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-horospheres in the space of affine lattices.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if $Gamma $ is a countable discrete abelian group, $varphi , psi in mathrm {End}(Gamma )$ , and $psi - varphi $ is an injective endomorphism with finite index image, then for any ergodic measure-preserving $Gamma $ -system $( X, {mathcal {X}}, mu , (T_g)_{g in Gamma } )$ , any measurable set $A in {mathcal {X}}$ , and any ${varepsilon }> 0$ , there is a syndetic set of $g in Gamma$ such that $mu ( A cap T_{varphi(g)}^{-1} A cap T_{psi(g)}^{-1} A ) > mu(A)^3 - varepsilon$ . This generalizes the main results of Ackelsberg et al [Khintchine-type recurrence for 3-point configurations.
我们证明了可数离散无边群的成对内定态的欣钦钦型递推定理。作为主结果的一个特例，如果 $Gamma $ 是一个可数离散无边群， $varphi , psi in mathrm {End}(Gamma )$ 、并且 $psi - varphi $ 是一个具有有限索引映像的注入式内形变，那么对于任何保全遍历度量的 $Gamma $ 系统 $( X, {mathcal {X}}, mu , (T_g)_{g in Gamma } )$, 任何可度量集合 $g in Gamma } )$, 任何可度量集合 $g - varphi $ 是一个具有有限索引映像的注入式内形变。)$，{mathcal {X}}$中的任意可测集$A，以及任意${varepsilon }>；0$, there is a syndetic set of $g in Gamma$ such that $mu ( A cap T_{varphi(g)}^{-1} A cap T_{psi(g)}^{-1} A ) > mu(A)^3 - varepsilon$ .这概括了 Ackelsberg 等人[Khintchine-type recurrence for 3-point configurations.Forum Math.Sigma10 (2022), Paper no. e107] 并基本上回答了该论文中的一个未决问题 [Question 1.12; Khintchine-type recurrence for 3-point configurations.论坛数学。Sigma10 (2022), Paper no.］对于$Gamma = {mathbb {Z}}^d$ 群，结果适用于由差值为非奇异值的矩阵给出的成对内定态。证明的关键要素是(1) 与 Bergelson 和 Shalom 共同获得的最新结果[Khintchine-type recurrence for 3-point configurations.Forum Math.Sigma10 (2022), Paper no. e107]说，相关的遍历平均值由一个与准阿芬系数（或康泽-勒格朗系数）密切相关的特征因子控制；(2) 一个扩展技巧，以还原到具有良好离散谱（关于 $varphi $ 和 $psi $）的系统；(3) 描述与具有良好离散谱的旋转系统上的准阿芬环相关的麦基群。
{"title":"Khintchine-type double recurrence in abelian groups","authors":"ETHAN ACKELSBERG","doi":"10.1017/etds.2024.29","DOIUrl":"https://doi.org/10.1017/etds.2024.29","url":null,"abstract":"We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline1.png\"/> <jats:tex-math> $Gamma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a countable discrete abelian group, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline2.png\"/> <jats:tex-math> $varphi , psi in mathrm {End}(Gamma )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline3.png\"/> <jats:tex-math> $psi - varphi $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is an injective endomorphism with finite index image, then for any ergodic measure-preserving <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline4.png\"/> <jats:tex-math> $Gamma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-system <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline5.png\"/> <jats:tex-math> $( X, {mathcal {X}}, mu , (T_g)_{g in Gamma } )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, any measurable set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline6.png\"/> <jats:tex-math> $A in {mathcal {X}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, and any <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline7.png\"/> <jats:tex-math> ${varepsilon }> 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, there is a syndetic set of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline8.png\"/> <jats:tex-math> $g in Gamma$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline9.png\"/> <jats:tex-math> $mu ( A cap T_{varphi(g)}^{-1} A cap T_{psi(g)}^{-1} A ) > mu(A)^3 - varepsilon$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. This generalizes the main results of Ackelsberg <jats:italic>et al</jats:italic> [Khintchine-type recurrence for 3-point configurations. <jats:italic","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
DZMITRY BADZIAHIN, STEPHEN HARRAP, EREZ NESHARIM, DAVID SIMMONS
Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of study. We survey the definitions of the most common variants and connections between them. A new game called the Cantor game is invented and helps with presenting a unifying framework. We prove surprising new results such as the coincidence of absolute winning and $1$ Cantor winning in metric spaces, and the fact that $1/2$ winning implies absolute winning for subsets of $mathbb {R}$ . We also suggest a prototypical example of a Cantor winning set to show the ubiquity of such sets in metric number theory and ergodic theory.
{"title":"Schmidt games and Cantor winning sets","authors":"DZMITRY BADZIAHIN, STEPHEN HARRAP, EREZ NESHARIM, DAVID SIMMONS","doi":"10.1017/etds.2024.23","DOIUrl":"https://doi.org/10.1017/etds.2024.23","url":null,"abstract":"Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of study. We survey the definitions of the most common variants and connections between them. A new game called the Cantor game is invented and helps with presenting a unifying framework. We prove surprising new results such as the coincidence of absolute winning and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000233_inline1.png\" /> <jats:tex-math> $1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> Cantor winning in metric spaces, and the fact that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000233_inline2.png\" /> <jats:tex-math> $1/2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> winning implies absolute winning for subsets of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000233_inline3.png\" /> <jats:tex-math> $mathbb {R}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We also suggest a prototypical example of a Cantor winning set to show the ubiquity of such sets in metric number theory and ergodic theory.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140623891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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