MATHEUS M. CASTRO, VINCENT P. H. GOVERSE, JEROEN S. W. LAMB, MARTIN RASMUSSEN
Abstract In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $mu $ on M where the transition kernel ${mathcal P}$ admits an eigenfunction $0leq eta in L^1(M,mu )$ . We find conditions on the transition densities of ${mathcal P}$ with respect to $mu $ which ensure that $eta (x) mu (mathrm {d} x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure $mu $ -almost surely. We apply this result to the random logistic map $X_{n+1} = omega _n X_n (1-X_n)$ absorbed at ${mathbb R} setminus [0,1],$ where $omega _n$ is an independent and identically distributed sequence of random variables uniformly distributed in $[a,b],$ for $1leq a <4$ and $b>4.$
摘要本文考虑M上吸收马尔可夫链$X_n$允许一个拟平稳测度$mu $，其中转移核${mathcal P}$允许一个本征函数$0leq eta in L^1(M,mu )$。我们找到了${mathcal P}$相对于$mu $的跃迁密度的条件，几乎可以肯定地保证$eta (x) mu (mathrm {d} x)$是$X_n$的拟遍历测度，并且Yaglom极限收敛于拟平稳测度$mu $。我们将这一结果应用于${mathbb R} setminus [0,1],$吸收的随机逻辑图$X_{n+1} = omega _n X_n (1-X_n)$，其中$omega _n$是一个独立的、同分布的随机变量序列，这些随机变量均匀分布在$1leq a <4$和$[a,b],$中 $b>4.$
{"title":"On the quasi-ergodicity of absorbing Markov chains with unbounded transition densities, including random logistic maps with escape","authors":"MATHEUS M. CASTRO, VINCENT P. H. GOVERSE, JEROEN S. W. LAMB, MARTIN RASMUSSEN","doi":"10.1017/etds.2023.69","DOIUrl":"https://doi.org/10.1017/etds.2023.69","url":null,"abstract":"Abstract In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $mu $ on M where the transition kernel ${mathcal P}$ admits an eigenfunction $0leq eta in L^1(M,mu )$ . We find conditions on the transition densities of ${mathcal P}$ with respect to $mu $ which ensure that $eta (x) mu (mathrm {d} x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure $mu $ -almost surely. We apply this result to the random logistic map $X_{n+1} = omega _n X_n (1-X_n)$ absorbed at ${mathbb R} setminus [0,1],$ where $omega _n$ is an independent and identically distributed sequence of random variables uniformly distributed in $[a,b],$ for $1leq a <4$ and $b>4.$","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135815922","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 present an elementary proof that the Rauzy gasket has Hausdorff dimension strictly smaller than two.
摘要给出了Rauzy垫片具有严格小于2的Hausdorff维数的初等证明。
{"title":"An elementary proof that the Rauzy gasket is fractal","authors":"MARK POLLICOTT, BENEDICT SEWELL","doi":"10.1017/etds.2023.66","DOIUrl":"https://doi.org/10.1017/etds.2023.66","url":null,"abstract":"Abstract We present an elementary proof that the Rauzy gasket has Hausdorff dimension strictly smaller than two.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135815868","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 characterize measure-theoretic sequence entropy pairs of continuous actions of abelian groups using mean sensitivity. This addresses an open question of Li and Yu [On mean sensitive tuples. J. Differential Equations 297 (2021), 175–200]. As a consequence of our results, we provide a simpler characterization of Kerr and Li’s independence sequence entropy pairs ( $mu $ -IN-pairs) when the measure is ergodic and the group is abelian.
{"title":"Measure-theoretic sequence entropy pairs and mean sensitivity","authors":"FELIPE GARCÍA-RAMOS, VÍCTOR MUÑOZ-LÓPEZ","doi":"10.1017/etds.2023.65","DOIUrl":"https://doi.org/10.1017/etds.2023.65","url":null,"abstract":"Abstract We characterize measure-theoretic sequence entropy pairs of continuous actions of abelian groups using mean sensitivity. This addresses an open question of Li and Yu [On mean sensitive tuples. J. Differential Equations 297 (2021), 175–200]. As a consequence of our results, we provide a simpler characterization of Kerr and Li’s independence sequence entropy pairs ( $mu $ -IN-pairs) when the measure is ergodic and the group is abelian.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135015095","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 The construction of a spectral cocycle from the case of one-dimensional substitution flows [A. I. Bufetov and B. Solomyak. A spectral cocycle for substitution systems and translation flows. J. Anal. Math. 141 (1) (2020), 165–205] is extended to the setting of pseudo-self-similar tilings in ${mathbb R}^d$ , allowing expanding similarities with rotations. The pointwise upper Lyapunov exponent of this cocycle is used to bound the local dimension of spectral measures of deformed tilings. The deformations are considered, following the work of Treviño [Quantitative weak mixing for random substitution tilings. Israel J. Math. , to appear], in the simpler, non-random setting. We review some of the results of Treviño in this special case and illustrate them on concrete examples.
{"title":"Spectral cocycle for substitution tilings","authors":"BORIS SOLOMYAK, RODRIGO TREVIÑO","doi":"10.1017/etds.2023.64","DOIUrl":"https://doi.org/10.1017/etds.2023.64","url":null,"abstract":"Abstract The construction of a spectral cocycle from the case of one-dimensional substitution flows [A. I. Bufetov and B. Solomyak. A spectral cocycle for substitution systems and translation flows. J. Anal. Math. 141 (1) (2020), 165–205] is extended to the setting of pseudo-self-similar tilings in ${mathbb R}^d$ , allowing expanding similarities with rotations. The pointwise upper Lyapunov exponent of this cocycle is used to bound the local dimension of spectral measures of deformed tilings. The deformations are considered, following the work of Treviño [Quantitative weak mixing for random substitution tilings. Israel J. Math. , to appear], in the simpler, non-random setting. We review some of the results of Treviño in this special case and illustrate them on concrete examples.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135154072","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 prove that any strongly mixing action of a countable abelian group on a probability space has higher-order mixing properties. This is achieved via the utilization of $mathcal R$ -limits, a notion of convergence which is based on the classical Ramsey theorem. $mathcal R$ -limits are intrinsically connected with a new combinatorial notion of largeness which is similar to but has stronger properties than the classical notions of uniform density one and IP $^*$ . While the main goal of this paper is to establish a universal property of strongly mixing actions of countable abelian groups, our results, when applied to ${mathbb {Z}}$ -actions, offer a new way of dealing with strongly mixing transformations. In particular, we obtain several new characterizations of strong mixing for ${mathbb {Z}}$ -actions, including a result which can be viewed as the analogue of the weak mixing of all orders property established by Furstenberg in the course of his proof of Szemerédi’s theorem. We also demonstrate the versatility of $mathcal R$ -limits by obtaining new characterizations of higher-order weak and mild mixing for actions of countable abelian groups.
{"title":"Strongly mixing systems are almost strongly mixing of all orders","authors":"Vitaly Bergelson, Rigoberto Zelada","doi":"10.1017/etds.2023.63","DOIUrl":"https://doi.org/10.1017/etds.2023.63","url":null,"abstract":"Abstract We prove that any strongly mixing action of a countable abelian group on a probability space has higher-order mixing properties. This is achieved via the utilization of $mathcal R$ -limits, a notion of convergence which is based on the classical Ramsey theorem. $mathcal R$ -limits are intrinsically connected with a new combinatorial notion of largeness which is similar to but has stronger properties than the classical notions of uniform density one and IP $^*$ . While the main goal of this paper is to establish a universal property of strongly mixing actions of countable abelian groups, our results, when applied to ${mathbb {Z}}$ -actions, offer a new way of dealing with strongly mixing transformations. In particular, we obtain several new characterizations of strong mixing for ${mathbb {Z}}$ -actions, including a result which can be viewed as the analogue of the weak mixing of all orders property established by Furstenberg in the course of his proof of Szemerédi’s theorem. We also demonstrate the versatility of $mathcal R$ -limits by obtaining new characterizations of higher-order weak and mild mixing for actions of countable abelian groups.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134989635","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 43 issue 10 Cover and Back matter","authors":"","doi":"10.1017/etds.2022.104","DOIUrl":"https://doi.org/10.1017/etds.2022.104","url":null,"abstract":"","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":" ","pages":"b1 - b2"},"PeriodicalIF":0.9,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45584553","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 43 issue 10 Cover and Front matter","authors":"","doi":"10.1017/etds.2022.103","DOIUrl":"https://doi.org/10.1017/etds.2022.103","url":null,"abstract":"","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":" ","pages":"f1 - f2"},"PeriodicalIF":0.9,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45275010","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}
This is a corrigendum to the paper ‘A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems’ [3]. Theorem 7.1 in that paper is incorrect as stated, and the error originates with Proposition 7.5, part (iii), which was incorrectly quoted from a paper by Bergelson, Host, and Kra [1]. Consequently, this invalidates the proof of Theorem 4.2, which was used in the proofs of the main results in [3]. In this corrigendum we fix the problem by establishing a slightly weaker version of Theorem 7.1 (see §2 below) and use it to give a new proof of Theorem 4.2 (see §3 below). This ensures that all main results in [3] remain correct. We thank Zhengxing Lian and Jiahao Qiu for bringing this mistake to our attention.
{"title":"A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems – CORRIGENDUM","authors":"Joel Moreira, F. Richter","doi":"10.1017/etds.2023.61","DOIUrl":"https://doi.org/10.1017/etds.2023.61","url":null,"abstract":"\u0000 This is a corrigendum to the paper ‘A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems’ [3]. Theorem 7.1 in that paper is incorrect as stated, and the error originates with Proposition 7.5, part (iii), which was incorrectly quoted from a paper by Bergelson, Host, and Kra [1]. Consequently, this invalidates the proof of Theorem 4.2, which was used in the proofs of the main results in [3]. In this corrigendum we fix the problem by establishing a slightly weaker version of Theorem 7.1 (see §2 below) and use it to give a new proof of Theorem 4.2 (see §3 below). This ensures that all main results in [3] remain correct. We thank Zhengxing Lian and Jiahao Qiu for bringing this mistake to our attention.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42940197","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":"and empirical measures, the irregular set and entropy","authors":"S. Usuki","doi":"10.1017/etds.2023.60","DOIUrl":"https://doi.org/10.1017/etds.2023.60","url":null,"abstract":"\u0000\t <jats:p>For integers <jats:italic>a</jats:italic> and <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723000603_inline3.png\" />\u0000\t\t<jats:tex-math>\u0000$bgeq 2$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, let <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723000603_inline4.png\" />\u0000\t\t<jats:tex-math>\u0000$T_a$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> and <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723000603_inline5.png\" />\u0000\t\t<jats:tex-math>\u0000$T_b$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> be multiplication by <jats:italic>a</jats:italic> and <jats:italic>b</jats:italic> on <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723000603_inline6.png\" />\u0000\t\t<jats:tex-math>\u0000$mathbb {T}=mathbb {R}/mathbb {Z}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>. The action on <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723000603_inline7.png\" />\u0000\t\t<jats:tex-math>\u0000$mathbb {T}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> by <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723000603_inline8.png\" />\u0000\t\t<jats:tex-math>\u0000$T_a$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> and <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723000603_inline9.png\" />\u0000\t\t<jats:tex-math>\u0000$T_b$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> is called <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723000603_inline10.png\" />\u0000\t\t<jats:tex-math>\u0000$times a,times b$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> action and it is known that, if <jats:italic>a</jats:italic> and <jats:italic>b</jats:italic> are multiplicatively independent, then the only <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723000603_inline11.png\" />\u0000\t\t<jats:tex-math>\u0000$times a,times b$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> invariant and ergodic measure with positive entropy of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inl","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42121803","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 43 issue 9 Cover and Front matter","authors":"","doi":"10.1017/etds.2022.101","DOIUrl":"https://doi.org/10.1017/etds.2022.101","url":null,"abstract":"","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":" ","pages":"f1 - f2"},"PeriodicalIF":0.9,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46375869","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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