We obtain the following embedding theorem for symbolic dynamical systems. Let G be a countable amenable group with the comparison property. Let X be a strongly aperiodic subshift over G. Let Y be a strongly irreducible shift of finite type over G that has no global period, meaning that the shift action is faithful on Y. If the topological entropy of X is strictly less than that of Y and Y contains at least one factor of X, then X embeds into Y. This result partially extends the classical result of Krieger when $G = mathbb {Z}$ and the results of Lightwood when $G = mathbb {Z}^d$ for $d geq 2$. The proof relies on recent developments in the theory of tilings and quasi-tilings of amenable groups.
我们得到以下符号动力系统的嵌入定理。设 G 是具有比较性质的可数可调群。让 X 是 G 上的强无周期子移位。让 Y 是 G 上有限类型的强不可还原移位，它没有全局周期，即移位作用在 Y 上是忠实的。如果 X 的拓扑熵严格小于 Y 的拓扑熵，且 Y 至少包含 X 的一个因子，那么 X 嵌入 Y。这个结果部分地扩展了克里格在 $G = mathbb {Z}$ 时的经典结果，以及莱特伍德在 $G = mathbb {Z}^d$ 时对于 $d geq 2$ 的结果。证明依赖于可平分群的倾斜和准倾斜理论的最新发展。
{"title":"An embedding theorem for subshifts over amenable groups with the comparison property","authors":"ROBERT BLAND","doi":"10.1017/etds.2024.21","DOIUrl":"https://doi.org/10.1017/etds.2024.21","url":null,"abstract":"<p>We obtain the following embedding theorem for symbolic dynamical systems. Let <span>G</span> be a countable amenable group with the comparison property. Let <span>X</span> be a strongly aperiodic subshift over <span>G</span>. Let <span>Y</span> be a strongly irreducible shift of finite type over <span>G</span> that has no global period, meaning that the shift action is faithful on <span>Y</span>. If the topological entropy of <span>X</span> is strictly less than that of <span>Y</span> and <span>Y</span> contains at least one factor of <span>X</span>, then <span>X</span> embeds into <span>Y</span>. This result partially extends the classical result of Krieger when <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304130334345-0052:S014338572400021X:S014338572400021X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$G = mathbb {Z}$</span></span></img></span></span> and the results of Lightwood when <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304130334345-0052:S014338572400021X:S014338572400021X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$G = mathbb {Z}^d$</span></span></img></span></span> for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304130334345-0052:S014338572400021X:S014338572400021X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$d geq 2$</span></span></img></span></span>. The proof relies on recent developments in the theory of tilings and quasi-tilings of amenable groups.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"24 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034056","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}
PAULINA CECCHI BERNALES, MARÍA ISABEL CORTEZ, JAIME GÓMEZ
<p>Let <span>G</span> be a countable residually finite group (for instance, <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline1.png"><span data-mathjax-type="texmath"><span>${mathbb F}_2$</span></span></img></span></span>) and let <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline2.png"><span data-mathjax-type="texmath"><span>$overleftarrow {G}$</span></span></img></span></span> be a totally disconnected metric compactification of <span>G</span> equipped with the action of <span>G</span> by left multiplication. For every <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline3.png"><span data-mathjax-type="texmath"><span>$rgeq 1$</span></span></img></span></span>, we construct a Toeplitz <span>G</span>-subshift <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline4.png"><span data-mathjax-type="texmath"><span>$(X,sigma ,G)$</span></span></img></span></span>, which is an almost one-to-one extension of <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline5.png"><span data-mathjax-type="texmath"><span>$overleftarrow {G}$</span></span></img></span></span>, having <span>r</span> ergodic measures <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline6.png"><span data-mathjax-type="texmath"><span>$nu _1, ldots ,nu _r$</span></span></img></span></span> such that for every <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline7.png"><span data-mathjax-type="texmath"><span>$1leq ileq r$</span></span></img></span></span>, the measure-theoretic dynamical system <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline8.png"><span data-mathjax-type="texmath"><span>$(X,sigma ,G,nu _i)$</span></span></img></span></span> is isomorphic to <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_
让 G 是一个可数余有限群（例如，${mathbb F}_2$），并让 $overleftarrow {G}$ 是 G 的一个完全断开的度量紧凑化，配备有 G 的左乘作用。对于每一个 $rgeq 1$，我们构造一个托普利兹 G 子移位 $(X,sigma,G)$，它是 $overleftarrow {G}$ 的一个几乎一一对应的扩展，有 r 个遍历度量 $nu _1, ldots 、nu _r$，这样对于每1$leq ileq r$，度量理论动力系统$(X,sigma ,G,nu _i)$与赋予哈尔度量的$overleftarrow {G}$是同构的。我们提出的构造是通用的（适用于可驯化和不可驯化的残余有限群）；然而，我们指出了当作用群不可驯化时可能出现的差异和障碍。
{"title":"Invariant measures of Toeplitz subshifts on non-amenable groups","authors":"PAULINA CECCHI BERNALES, MARÍA ISABEL CORTEZ, JAIME GÓMEZ","doi":"10.1017/etds.2024.16","DOIUrl":"https://doi.org/10.1017/etds.2024.16","url":null,"abstract":"<p>Let <span>G</span> be a countable residually finite group (for instance, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb F}_2$</span></span></img></span></span>) and let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$overleftarrow {G}$</span></span></img></span></span> be a totally disconnected metric compactification of <span>G</span> equipped with the action of <span>G</span> by left multiplication. For every <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$rgeq 1$</span></span></img></span></span>, we construct a Toeplitz <span>G</span>-subshift <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$(X,sigma ,G)$</span></span></img></span></span>, which is an almost one-to-one extension of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$overleftarrow {G}$</span></span></img></span></span>, having <span>r</span> ergodic measures <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$nu _1, ldots ,nu _r$</span></span></img></span></span> such that for every <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$1leq ileq r$</span></span></img></span></span>, the measure-theoretic dynamical system <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$(X,sigma ,G,nu _i)$</span></span></img></span></span> is isomorphic to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240301125356903-0281:S0143385724000166:S0143385724000166_","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"13 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025330","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}
In this work we prove that every shift of finite type (SFT), sofic shift, and strongly irreducible shift on locally finite groups has strong dynamical properties. These properties include that every sofic shift is an SFT, every SFT is strongly irreducible, every strongly irreducible shift is an SFT, every SFT is entropy minimal, and every SFT has a unique measure of maximal entropy, among others. In addition, we show that if every SFT on a group is strongly irreducible, or if every sofic shift is an SFT, then the group must be locally finite, and this extends to all of the properties we explore. These results are collected in two main theorems which characterize the local finiteness of groups by purely dynamical properties. In pursuit of these results, we present a formal construction of free extension shifts on a group G, which takes a shift on a subgroup H of G, and naturally extends it to a shift on all of G.
在这项工作中，我们证明了局部有限群上的每一个有限型移位（SFT）、sofic 移位和强不可还原移位都具有强动力学性质。这些性质包括：每个sofic shift 都是一个SFT，每个SFT 都是强不可还原的，每个强不可还原的 shift 都是一个SFT，每个SFT 都是熵最小的，每个SFT 都有一个唯一的最大熵量，等等。此外，我们还证明，如果一个群上的每一个 SFT 都是强不可还原的，或者每一个sofic shift 都是一个 SFT，那么这个群一定是局部有限的，这也扩展到了我们探索的所有性质。这些结果集合在两个主要定理中，它们通过纯粹的动力学性质描述了群的局部有限性。为了追寻这些结果，我们提出了群 G 上自由扩展位移的形式构造，它将 G 的一个子群 H 上的位移，自然地扩展为 G 全部上的位移。
{"title":"Shifts of finite type on locally finite groups","authors":"JADE RAYMOND","doi":"10.1017/etds.2024.14","DOIUrl":"https://doi.org/10.1017/etds.2024.14","url":null,"abstract":"In this work we prove that every shift of finite type (SFT), sofic shift, and strongly irreducible shift on locally finite groups has strong dynamical properties. These properties include that every sofic shift is an SFT, every SFT is strongly irreducible, every strongly irreducible shift is an SFT, every SFT is entropy minimal, and every SFT has a unique measure of maximal entropy, among others. In addition, we show that if every SFT on a group is strongly irreducible, or if every sofic shift is an SFT, then the group must be locally finite, and this extends to all of the properties we explore. These results are collected in two main theorems which characterize the local finiteness of groups by purely dynamical properties. In pursuit of these results, we present a formal construction of <jats:italic>free extension</jats:italic> shifts on a group <jats:italic>G</jats:italic>, which takes a shift on a subgroup <jats:italic>H</jats:italic> of <jats:italic>G</jats:italic>, and naturally extends it to a shift on all of <jats:italic>G</jats:italic>.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979059","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}
In this paper, we consider random iterations of polynomial maps $z^{2} + c_{n}$ , where $c_{n}$ are complex-valued independent random variables following the uniform distribution on the closed disk with center c and radius r. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter $c = -1$ , almost every random Julia set is totally disconnected with much smaller radial parameters r than expected. We also introduce several open questions worth discussing.
在本文中，我们考虑多项式映射 $z^{2} + c_{n}$ 的随机迭代。+ c_{n}$ ，其中 $c_{n}$ 是复值独立随机变量，在以 c 为圆心、r 为半径的封闭圆盘上服从均匀分布。首先，我们研究随机 Julia 集的（不）连通性。在这里，我们揭示了随机 Julia 集的分岔半径和连通性之间的关系。其次，我们研究了随机迭代的分岔，并给出了分岔参数的定量估计。特别是，我们证明了对于中心参数 $c = -1$ ，几乎每个随机 Julia 集都是完全断开的，其径向参数 r 比预期的要小得多。我们还介绍了几个值得讨论的开放问题。
{"title":"On the stochastic bifurcations regarding random iterations of polynomials of the form","authors":"TAKAYUKI WATANABE","doi":"10.1017/etds.2024.17","DOIUrl":"https://doi.org/10.1017/etds.2024.17","url":null,"abstract":"In this paper, we consider random iterations of polynomial maps <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000178_inline2.png\" /> <jats:tex-math> $z^{2} + c_{n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000178_inline3.png\" /> <jats:tex-math> $c_{n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are complex-valued independent random variables following the uniform distribution on the closed disk with center <jats:italic>c</jats:italic> and radius <jats:italic>r</jats:italic>. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000178_inline4.png\" /> <jats:tex-math> $c = -1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, almost every random Julia set is totally disconnected with much smaller radial parameters <jats:italic>r</jats:italic> than expected. We also introduce several open questions worth discussing.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139978846","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 BARROS, CHRISTIAN BONATTI, MARIA JOSÉ PACIFICO
We present a modified version of the well-known geometric Lorenz attractor. It consists of a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000087_inline1.png" /> <jats:tex-math> $C^1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> open set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000087_inline2.png" /> <jats:tex-math> ${mathcal O}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of vector fields in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000087_inline3.png" /> <jats:tex-math> ${mathbb R}^3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> having an attracting region <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000087_inline4.png" /> <jats:tex-math> ${mathcal U}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfying three properties. Namely, a unique singularity <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000087_inline5.png" /> <jats:tex-math> $sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>; a unique attractor <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000087_inline6.png" /> <jats:tex-math> $Lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> including the singular point and the maximal invariant in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000087_inline7.png" /> <jats:tex-math> ${mathcal U}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> has at most two chain recurrence classes, which are <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000087_inline8.png" /> <jats:tex-math> $Lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and (at most) one hyperbolic horseshoe. The horseshoe and the singular attractor have a collision along with the union of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000087_inline9.png" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> codimension <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000087_inline10.png" /> <jats:tex-math> $1$ </jats:t
{"title":"Upper, down, two-sided Lorenz attractor, collisions, merging, and switching","authors":"DIEGO BARROS, CHRISTIAN BONATTI, MARIA JOSÉ PACIFICO","doi":"10.1017/etds.2024.8","DOIUrl":"https://doi.org/10.1017/etds.2024.8","url":null,"abstract":"We present a modified version of the well-known geometric Lorenz attractor. It consists of a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline1.png\" /> <jats:tex-math> $C^1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> open set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline2.png\" /> <jats:tex-math> ${mathcal O}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of vector fields in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline3.png\" /> <jats:tex-math> ${mathbb R}^3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> having an attracting region <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline4.png\" /> <jats:tex-math> ${mathcal U}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfying three properties. Namely, a unique singularity <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline5.png\" /> <jats:tex-math> $sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>; a unique attractor <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline6.png\" /> <jats:tex-math> $Lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> including the singular point and the maximal invariant in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline7.png\" /> <jats:tex-math> ${mathcal U}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> has at most two chain recurrence classes, which are <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline8.png\" /> <jats:tex-math> $Lambda $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and (at most) one hyperbolic horseshoe. The horseshoe and the singular attractor have a collision along with the union of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline9.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> codimension <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000087_inline10.png\" /> <jats:tex-math> $1$ </jats:t","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"281 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139923275","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 find sufficient conditions for bounded density shifts to have a unique measure of maximal entropy. We also prove that every measure of maximal entropy of a bounded density shift is fully supported. As a consequence of this, we obtain that bounded density shifts are surjunctive.
{"title":"Measures of maximal entropy of bounded density shifts","authors":"FELIPE GARCÍA-RAMOS, RONNIE PAVLOV, CARLOS REYES","doi":"10.1017/etds.2024.6","DOIUrl":"https://doi.org/10.1017/etds.2024.6","url":null,"abstract":"We find sufficient conditions for bounded density shifts to have a unique measure of maximal entropy. We also prove that every measure of maximal entropy of a bounded density shift is fully supported. As a consequence of this, we obtain that bounded density shifts are surjunctive.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"41 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139923465","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}
Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000129_inline1.png" /> <jats:tex-math> $ G $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a connected semisimple real algebraic group and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000129_inline2.png" /> <jats:tex-math> $Gamma <G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a Zariski dense discrete subgroup. Let <jats:italic>N</jats:italic> denote a maximal horospherical subgroup of <jats:italic>G</jats:italic>, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000129_inline3.png" /> <jats:tex-math> $P=MAN$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> the minimal parabolic subgroup which is the normalizer of <jats:italic>N</jats:italic>. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000129_inline4.png" /> <jats:tex-math> $mathcal E$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the unique <jats:italic>P</jats:italic>-minimal subset of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000129_inline5.png" /> <jats:tex-math> $Gamma backslash G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000129_inline6.png" /> <jats:tex-math> $mathcal E_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000129_inline7.png" /> <jats:tex-math> $P^circ $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-minimal subset. We consider a notion of a horospherical limit point in the Furstenberg boundary <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000129_inline8.png" /> <jats:tex-math> $ G/P $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and show that the following are equivalent for any <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000129_inline9.png" /> <jats:tex-math> $[g]in mathcal E_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>: <jats:list list-type="number"> <jats:list-item> <jats:label>(1)</jats:label> <jats:inline-formula> <jats:alternatives> <jats:inline-gra
{"title":"On denseness of horospheres in higher rank homogeneous spaces","authors":"OR LANDESBERG, HEE OH","doi":"10.1017/etds.2024.12","DOIUrl":"https://doi.org/10.1017/etds.2024.12","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline1.png\" /> <jats:tex-math> $ G $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a connected semisimple real algebraic group and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline2.png\" /> <jats:tex-math> $Gamma <G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a Zariski dense discrete subgroup. Let <jats:italic>N</jats:italic> denote a maximal horospherical subgroup of <jats:italic>G</jats:italic>, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline3.png\" /> <jats:tex-math> $P=MAN$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> the minimal parabolic subgroup which is the normalizer of <jats:italic>N</jats:italic>. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline4.png\" /> <jats:tex-math> $mathcal E$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the unique <jats:italic>P</jats:italic>-minimal subset of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline5.png\" /> <jats:tex-math> $Gamma backslash G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline6.png\" /> <jats:tex-math> $mathcal E_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline7.png\" /> <jats:tex-math> $P^circ $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-minimal subset. We consider a notion of a horospherical limit point in the Furstenberg boundary <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline8.png\" /> <jats:tex-math> $ G/P $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and show that the following are equivalent for any <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000129_inline9.png\" /> <jats:tex-math> $[g]in mathcal E_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>: <jats:list list-type=\"number\"> <jats:list-item> <jats:label>(1)</jats:label> <jats:inline-formula> <jats:alternatives> <jats:inline-gra","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139909838","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}
IZTOK BANIČ, RENE GRIL ROGINA, JUDY KENNEDY, VAN NALL
We introduce the notions of returns and well-aligned sets for closed relations on compact metric spaces and then use them to obtain non-trivial sufficient conditions for such a relation to have non-zero entropy. In addition, we give a characterization of finite relations with non-zero entropy in terms of Li–Yorke and DC2 chaos.
{"title":"Sufficient conditions for non-zero entropy of closed relations","authors":"IZTOK BANIČ, RENE GRIL ROGINA, JUDY KENNEDY, VAN NALL","doi":"10.1017/etds.2024.11","DOIUrl":"https://doi.org/10.1017/etds.2024.11","url":null,"abstract":"We introduce the notions of returns and well-aligned sets for closed relations on compact metric spaces and then use them to obtain non-trivial sufficient conditions for such a relation to have non-zero entropy. In addition, we give a characterization of finite relations with non-zero entropy in terms of Li–Yorke and DC2 chaos.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"5 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139766847","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}
TULLIO CECCHERINI-SILBERSTEIN, MICHEL COORNAERT, XUAN KIEN PHUNG
Let G be a group and let V be an algebraic variety over an algebraically closed field K. Let A denote the set of K-points of V. We introduce algebraic sofic subshifts ${Sigma subset A^G}$ and study endomorphisms $tau colon Sigma to Sigma $ . We generalize several results for dynamical invariant sets and nilpotency of $tau $ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that $tau $ is nilpotent if and only if its limit set, that is, the intersection of the images of its iterates, is a singleton. If moreover G is infinite, finitely generated and $Sigma $ is topologically mixing, we show that $tau $ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.
让 G 是一个群，让 V 是一个代数封闭域 K 上的代数簇，让 A 表示 V 的 K 点集合。我们引入代数的 sofic 子转移 ${Sigma subset A^G}$ 并研究 $tau colon Sigma to Sigma $ 的内同构。我们对有限字母蜂窝自动机中众所周知的动力学不变集和 $tau $ 的无势性的几个结果进行了归纳。在温和的假设条件下，我们证明当且仅当 $tau $ 的极限集（即其迭代的图像的交集）是单子时，它才是无穷的。此外，如果 G 是无限的、有限生成的，并且 $Sigma $ 是拓扑混合的，那么我们证明，只有当其极限集由周期性配置组成，并且具有有限的字母值集时，$tau $ 才是无穷的。
{"title":"Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts","authors":"TULLIO CECCHERINI-SILBERSTEIN, MICHEL COORNAERT, XUAN KIEN PHUNG","doi":"10.1017/etds.2023.120","DOIUrl":"https://doi.org/10.1017/etds.2023.120","url":null,"abstract":"Let <jats:italic>G</jats:italic> be a group and let <jats:italic>V</jats:italic> be an algebraic variety over an algebraically closed field <jats:italic>K</jats:italic>. Let <jats:italic>A</jats:italic> denote the set of <jats:italic>K</jats:italic>-points of <jats:italic>V</jats:italic>. We introduce algebraic sofic subshifts <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline1.png\" /> <jats:tex-math> ${Sigma subset A^G}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and study endomorphisms <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline2.png\" /> <jats:tex-math> $tau colon Sigma to Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We generalize several results for dynamical invariant sets and nilpotency of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline3.png\" /> <jats:tex-math> $tau $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline4.png\" /> <jats:tex-math> $tau $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is nilpotent if and only if its limit set, that is, the intersection of the images of its iterates, is a singleton. If moreover <jats:italic>G</jats:italic> is infinite, finitely generated and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline5.png\" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is topologically mixing, we show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001207_inline6.png\" /> <jats:tex-math> $tau $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"313 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139767237","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}
Let $Ksubset {mathbb {R}}^d$ be a self-similar set generated by an iterated function system ${varphi _i}_{i=1}^m$ satisfying the strong separation condition and let f be a contracting similitude with $f(K)subseteq K$ . We show that $f(K)$ is relatively open in K if all $varphi _i$ share a common contraction ratio and orthogonal part. We also provide a counterexample when the orthogonal parts are allowed to vary. This partially answers a question of Elekes, Keleti and Máthé [Ergod. Th. & Dynam. Sys.30 (2010), 399–440]. As a byproduct of our argument, when $d=1$ and K admits two homogeneous generating iterated function systems satisfying the strong separation condition but with contraction ratios of opposite signs, we show that K is symmetric. This partially answers a question of Feng and Wang [Adv. Math.222 (2009), 1964–1981].
让 $Ksubset {mathbb {R}}^d$ 是由满足强分离条件的迭代函数系统 ${varphi _i}_{i=1}^m$ 生成的自相似集合，并让 f 是具有 $f(K)subseteq K$ 的收缩相似。我们证明，如果所有 $varphi _i$ 都有一个共同的收缩比和正交部分，那么 $f(K)$ 在 K 中是相对开放的。当允许正交部分变化时，我们还提供了一个反例。这部分回答了埃莱克斯、凯莱蒂和马特的一个问题[Ergod.作为我们论证的副产品，当 $d=1$ 且 K 包含两个满足强分离条件但收缩比符号相反的同质生成迭代函数系统时，我们证明 K 是对称的。这部分回答了冯和王的一个问题[Adv. Math.222 (2009), 1964-1981]。
{"title":"On a self-embedding problem for self-similar sets","authors":"JIAN-CI XIAO","doi":"10.1017/etds.2024.2","DOIUrl":"https://doi.org/10.1017/etds.2024.2","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline1.png\" /> <jats:tex-math> $Ksubset {mathbb {R}}^d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a self-similar set generated by an iterated function system <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline2.png\" /> <jats:tex-math> ${varphi _i}_{i=1}^m$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfying the strong separation condition and let <jats:italic>f</jats:italic> be a contracting similitude with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline3.png\" /> <jats:tex-math> $f(K)subseteq K$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline4.png\" /> <jats:tex-math> $f(K)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is relatively open in <jats:italic>K</jats:italic> if all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline5.png\" /> <jats:tex-math> $varphi _i$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> share a common contraction ratio and orthogonal part. We also provide a counterexample when the orthogonal parts are allowed to vary. This partially answers a question of Elekes, Keleti and Máthé [<jats:italic>Ergod. Th. & Dynam. Sys.</jats:italic>30 (2010), 399–440]. As a byproduct of our argument, when <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000026_inline6.png\" /> <jats:tex-math> $d=1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:italic>K</jats:italic> admits two homogeneous generating iterated function systems satisfying the strong separation condition but with contraction ratios of opposite signs, we show that <jats:italic>K</jats:italic> is symmetric. This partially answers a question of Feng and Wang [<jats:italic>Adv. Math.</jats:italic>222 (2009), 1964–1981].","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"31 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139767086","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}
$G = mathbb {Z}$ and the results of Lightwood when 
$G = mathbb {Z}^d$ for 
$d geq 2$. The proof relies on recent developments in the theory of tilings and quasi-tilings of amenable groups.
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