We show that the Hausdorff dimension of any slice of the graph of the Takagi function is bounded above by the Assouad dimension of the graph minus one, and that the bound is sharp. The result is deduced from a statement on more general self-affine sets, which is of independent interest. We also prove that Marstrand’s slicing theorem on the graph of the Takagi function extends to all slices if and only if the upper pointwise dimension of every projection of the length measure on the x-axis lifted to the graph is at least one.
我们证明，高木函数图的任何切片的豪斯多夫维度都受图的阿苏阿德维度减一的约束，而且这个约束是尖锐的。这个结果是从一个关于更一般的自阿芬集合的声明中推导出来的，这也是我们的兴趣所在。我们还证明了马斯特兰关于高木函数图的切片定理扩展到所有切片，当且仅当长度度量在 x 轴上的每一个投影抬升到图的上点维度至少为一。
{"title":"Slices of the Takagi function","authors":"ROOPE ANTTILA, BALÁZS BÁRÁNY, ANTTI KÄENMÄKI","doi":"10.1017/etds.2023.117","DOIUrl":"https://doi.org/10.1017/etds.2023.117","url":null,"abstract":"<p>We show that the Hausdorff dimension of any slice of the graph of the Takagi function is bounded above by the Assouad dimension of the graph minus one, and that the bound is sharp. The result is deduced from a statement on more general self-affine sets, which is of independent interest. We also prove that Marstrand’s slicing theorem on the graph of the Takagi function extends to all slices if and only if the upper pointwise dimension of every projection of the length measure on the <span>x</span>-axis lifted to the graph is at least one.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"37 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138817777","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}
For a post-critically finite branched covering of the sphere that is a subdivision map of a finite subdivision rule, we define non-expanding spines which determine the existence of a Levy cycle in a non-exhaustive semi-decidable algorithm. Especially when a finite subdivision rule has polynomial growth of edge subdivisions, the algorithm terminates very quickly, and the existence of a Levy cycle is equivalent to the existence of a Thurston obstruction. To show the equivalence between Levy and Thurston obstructions, we generalize the arcs intersecting obstruction theorem by Pilgrim and Tan [Combining rational maps and controlling obstructions. Ergod. Th. & Dynam. Sys.18(1) (1998), 221–245] to a graph intersecting obstruction theorem. As a corollary, we prove that for a pair of post-critically finite polynomials, if at least one polynomial has core entropy zero, then their mating has a Levy cycle if and only if the mating has a Thurston obstruction.
对于作为有限细分规则的细分映射的球面后关键有限分支覆盖，我们定义了非扩展刺，通过非穷举半可判定算法确定利维循环的存在。特别是当有限细分规则的边细分有多项式增长时，算法会很快终止，而李维循环的存在等同于瑟斯顿阻塞的存在。为了证明 Levy 阻碍和 Thurston 阻碍之间的等价性，我们推广了 Pilgrim 和 Tan [Combining rational maps and controlling obstructions.Ergod.Th. & Dynam.Sys.18(1)(1998),221-245]中的弧相交障碍定理。作为推论，我们证明对于一对后临界有限多项式，如果至少有一个多项式的核熵为零，那么它们的配对有一个列维循环，当且仅当配对有一个瑟斯顿障碍。
{"title":"Levy and Thurston obstructions of finite subdivision rules","authors":"INSUNG PARK","doi":"10.1017/etds.2023.115","DOIUrl":"https://doi.org/10.1017/etds.2023.115","url":null,"abstract":"For a post-critically finite branched covering of the sphere that is a subdivision map of a finite subdivision rule, we define non-expanding spines which determine the existence of a Levy cycle in a non-exhaustive semi-decidable algorithm. Especially when a finite subdivision rule has polynomial growth of edge subdivisions, the algorithm terminates very quickly, and the existence of a Levy cycle is equivalent to the existence of a Thurston obstruction. To show the equivalence between Levy and Thurston obstructions, we generalize the arcs intersecting obstruction theorem by Pilgrim and Tan [Combining rational maps and controlling obstructions. <jats:italic>Ergod. Th. & Dynam. Sys.</jats:italic>18(1) (1998), 221–245] to a graph intersecting obstruction theorem. As a corollary, we prove that for a pair of post-critically finite polynomials, if at least one polynomial has core entropy zero, then their mating has a Levy cycle if and only if the mating has a Thurston obstruction.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"70 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688204","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 study bracket words, which are a far-reaching generalization of Sturmian words, along Hardy field sequences, which are a far-reaching generalization of Piatetski-Shapiro sequences $lfloor n^c rfloor $. We show that sequences thus obtained are deterministic (that is, they have subexponential subword complexity) and satisfy Sarnak’s conjecture.
{"title":"Bracket words along Hardy field sequences","authors":"JAKUB KONIECZNY, CLEMENS MÜLLNER","doi":"10.1017/etds.2023.112","DOIUrl":"https://doi.org/10.1017/etds.2023.112","url":null,"abstract":"<p>We study bracket words, which are a far-reaching generalization of Sturmian words, along Hardy field sequences, which are a far-reaching generalization of Piatetski-Shapiro sequences <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231213132148445-0557:S0143385723001128:S0143385723001128_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$lfloor n^c rfloor $</span></span></img></span></span>. We show that sequences thus obtained are deterministic (that is, they have subexponential subword complexity) and satisfy Sarnak’s conjecture.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"31 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138629710","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}
DIEGO ARMENTANO, GAUTAM CHINTA, SIDDHARTHA SAHI, MICHAEL SHUB
We consider orthogonally invariant probability measures on $operatorname {mathrm {GL}}_n(mathbb {R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices with the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu and Shub [On random and mean exponents for unitarily invariant probability measures on $operatorname {mathrm {GL}}_n(mathbb {C})$. Astérisque287 (2003), xvii, 1–18]. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.
我们考虑了$operatorname {mathrm {GL}}_n(mathbb {R})$上的正交不变概率度量，并比较了矩阵特征值对数的均值与随机矩阵乘积的李雅普诺夫指数，随机矩阵乘积是相对于度量独立抽取的。我们用后者给出了前者的下限。这些结果来自 Dedieu 和 Shub [On random and mean exponents for unitarily invariant probability measures on $operatorname {mathrm {GL}}_n(mathbb {C})$.Astérisque 287 (2003), xvii, 1-18].我们的处理方法的一个新特点是在证明我们的主要结果时使用了球面多项式理论。
{"title":"Random and mean Lyapunov exponents for","authors":"DIEGO ARMENTANO, GAUTAM CHINTA, SIDDHARTHA SAHI, MICHAEL SHUB","doi":"10.1017/etds.2023.106","DOIUrl":"https://doi.org/10.1017/etds.2023.106","url":null,"abstract":"<p>We consider orthogonally invariant probability measures on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207140347424-0756:S0143385723001062:S0143385723001062_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {mathrm {GL}}_n(mathbb {R})$</span></span></img></span></span> and compare the mean of the logs of the moduli of eigenvalues of the matrices with the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu and Shub [On random and mean exponents for unitarily invariant probability measures on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207140347424-0756:S0143385723001062:S0143385723001062_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {mathrm {GL}}_n(mathbb {C})$</span></span></img></span></span>. <span>Astérisque</span> <span>287</span> (2003), xvii, 1–18]. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"18 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138566110","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}
WEBERTY DOMINGOS, SÉBASTIEN FERENCZI, ALI MESSAOUDI, GLAUCO VALLE
In this work, we study ergodic and dynamical properties of symbolic dynamical system associated to substitutions on an infinite countable alphabet. Specifically, we consider shift dynamical systems associated to irreducible substitutions which have well-established properties in the case of finite alphabets. Based on dynamical properties of a countable integer matrix related to the substitution, we obtain results on existence and uniqueness of shift invariant measures.
{"title":"Invariant measures for substitutions on countable alphabets","authors":"WEBERTY DOMINGOS, SÉBASTIEN FERENCZI, ALI MESSAOUDI, GLAUCO VALLE","doi":"10.1017/etds.2023.113","DOIUrl":"https://doi.org/10.1017/etds.2023.113","url":null,"abstract":"<p>In this work, we study ergodic and dynamical properties of symbolic dynamical system associated to substitutions on an infinite countable alphabet. Specifically, we consider shift dynamical systems associated to irreducible substitutions which have well-established properties in the case of finite alphabets. Based on dynamical properties of a countable integer matrix related to the substitution, we obtain results on existence and uniqueness of shift invariant measures.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"8 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138566371","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 crossed products of fiberwise essentially minimal zero-dimensional dynamical systems, a class that includes systems in which all orbit closures are minimal, have isomorphic K-theory if and only if the dynamical systems are strong orbit equivalent. Under the additional assumption that the dynamical systems have no periodic points, this gives a classification theorem including isomorphism of the associated crossed product $C^*$-algebras as well. We additionally explore the K-theory of such crossed products and the Bratteli diagrams associated to the dynamical systems.
我们证明，当且仅当动力学系统是强轨道等价物时，纤维本质上最小的零维动力学系统（包括所有轨道闭合都是最小的系统）的交叉积具有同构的 K 理论。在动力系统没有周期点的额外假设下，这给出了一个分类定理，包括相关交叉积$C^*$-代数的同构性。此外，我们还探讨了这种交叉积的 K 理论以及与动力系统相关的布拉泰利图。
{"title":"A dynamical classification for crossed products of fiberwise essentially minimal zero-dimensional dynamical systems","authors":"PAUL HERSTEDT","doi":"10.1017/etds.2023.104","DOIUrl":"https://doi.org/10.1017/etds.2023.104","url":null,"abstract":"<p>We prove that crossed products of fiberwise essentially minimal zero-dimensional dynamical systems, a class that includes systems in which all orbit closures are minimal, have isomorphic <span>K</span>-theory if and only if the dynamical systems are strong orbit equivalent. Under the additional assumption that the dynamical systems have no periodic points, this gives a classification theorem including isomorphism of the associated crossed product <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207142051786-0306:S0143385723001049:S0143385723001049_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$C^*$</span></span></img></span></span>-algebras as well. We additionally explore the <span>K</span>-theory of such crossed products and the Bratteli diagrams associated to the dynamical systems.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"56 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138566378","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 study equilibrium states for a class of non-uniformly expanding skew products, and show how a family of fiberwise transfer operators can be used to define the conditional measures along fibers of the product. We prove that the pushforward of the equilibrium state onto the base of the product is itself an equilibrium state for a Hölder potential defined via these fiberwise transfer operators.
{"title":"Equilibrium states for non-uniformly expanding skew products","authors":"GREGORY HEMENWAY","doi":"10.1017/etds.2023.111","DOIUrl":"https://doi.org/10.1017/etds.2023.111","url":null,"abstract":"<p>We study equilibrium states for a class of non-uniformly expanding skew products, and show how a family of fiberwise transfer operators can be used to define the conditional measures along fibers of the product. We prove that the pushforward of the equilibrium state onto the base of the product is itself an equilibrium state for a Hölder potential defined via these fiberwise transfer operators.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"40 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138566366","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 1 Cover and Back matter","authors":"","doi":"10.1017/etds.2023.78","DOIUrl":"https://doi.org/10.1017/etds.2023.78","url":null,"abstract":"","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"11 2","pages":"b1 - b2"},"PeriodicalIF":0.9,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138596693","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 1 Cover and Front matter","authors":"","doi":"10.1017/etds.2023.77","DOIUrl":"https://doi.org/10.1017/etds.2023.77","url":null,"abstract":"","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"49 1","pages":"f1 - f2"},"PeriodicalIF":0.9,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138596541","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}
Abstract We introduce a new method of constructing Birkhoff sections for pseudo-Anosov flows, which uses the connection between pseudo-Anosov flows and veering triangulations. This method allows for explicit constructions, as well as control over the Birkhoff section in terms of its Euler characteristic and the complexity of the boundary orbits. In particular, we show that any transitive pseudo-Anosov flow has a Birkhoff section with two boundary components.
{"title":"Constructing Birkhoff sections for pseudo-Anosov flows with controlled complexity","authors":"CHI CHEUK TSANG","doi":"10.1017/etds.2023.105","DOIUrl":"https://doi.org/10.1017/etds.2023.105","url":null,"abstract":"Abstract We introduce a new method of constructing Birkhoff sections for pseudo-Anosov flows, which uses the connection between pseudo-Anosov flows and veering triangulations. This method allows for explicit constructions, as well as control over the Birkhoff section in terms of its Euler characteristic and the complexity of the boundary orbits. In particular, we show that any transitive pseudo-Anosov flow has a Birkhoff section with two boundary components.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"32 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134954279","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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