It is conjectured that the only integrable metrics on the two-dimensional torus are Liouville metrics. In this paper, we study a deformative version of this conjecture: we consider integrable deformations of a non-flat Liouville metric in a conformal class and show that for a fairly large class of such deformations, the deformed metric is again Liouville. The principal idea of the argument is that the preservation of rational invariant tori in the foliation of the phase space forces a linear combination on the Fourier coefficients of the deformation to vanish. Showing that the resulting linear system is non-degenerate will then yield the claim. Since our method of proof immediately carries over to higher dimensional tori, we obtain analogous statements in this more general case. To put our results in perspective, we review existing results about integrable metrics on the torus.
{"title":"Deformational rigidity of integrable metrics on the torus","authors":"JOSCHA HENHEIK","doi":"10.1017/etds.2024.48","DOIUrl":"https://doi.org/10.1017/etds.2024.48","url":null,"abstract":"It is conjectured that the only integrable metrics on the two-dimensional torus are Liouville metrics. In this paper, we study a deformative version of this conjecture: we consider integrable deformations of a non-flat Liouville metric in a conformal class and show that for a fairly large class of such deformations, the deformed metric is again Liouville. The principal idea of the argument is that the preservation of rational invariant tori in the foliation of the phase space forces a linear combination on the Fourier coefficients of the deformation to vanish. Showing that the resulting linear system is non-degenerate will then yield the claim. Since our method of proof immediately carries over to higher dimensional tori, we obtain analogous statements in this more general case. To put our results in perspective, we review existing results about integrable metrics on the torus.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185313","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 proper, Gromov-hyperbolic metric space and a discrete, non-elementary, group of isometries, we define a natural subset of the limit set at infinity of the group called the ergodic limit set. The name is motivated by the fact that every ergodic measure which is invariant for the geodesic flow on the quotient metric space is concentrated on geodesics with endpoints belonging to the ergodic limit set. We refine the classical Bishop–Jones theorem proving that the packing dimension of the ergodic limit set coincides with the critical exponent of the group.
{"title":"Bishop–Jones’ theorem and the ergodic limit set","authors":"NICOLA CAVALLUCCI","doi":"10.1017/etds.2024.49","DOIUrl":"https://doi.org/10.1017/etds.2024.49","url":null,"abstract":"For a proper, Gromov-hyperbolic metric space and a discrete, non-elementary, group of isometries, we define a natural subset of the limit set at infinity of the group called the ergodic limit set. The name is motivated by the fact that every ergodic measure which is invariant for the geodesic flow on the quotient metric space is concentrated on geodesics with endpoints belonging to the ergodic limit set. We refine the classical Bishop–Jones theorem proving that the packing dimension of the ergodic limit set coincides with the critical exponent of the group.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224266","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 present a streamlined proof of a result essentially presented by the author in [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28(4) (2008), 1291–1322], namely that for every set $S = {s_1, s_2, ldots } subset mathbb {N}$ of zero Banach density and finite set A, there exists a minimal zero-entropy subshift $(X, sigma )$ so that for every sequence $u in A^{mathbb {Z}}$ , there is $x_u in X$ with $x_u(s_n) = u(n)$ for all $n in mathbb {N}$ . Informally, minimal deterministic sequences can achieve completely arbitrary behavior upon restriction to a set of zero Banach density. As a corollary, this provides counterexamples to the polynomial Sarnak conjecture reported by Eisner [A polynomial version of Sarnak’s conjecture. C. R. Math. Acad. Sci. Paris353(7) (2015), 569–572] which are significantly more general than some recently provided by Kanigowski, Lemańczyk and Radziwiłł [Prime number theorem for analytic skew products. Ann. of Math. (2)199 (2024), 591–705] and by Lian and Shi [A counter-example for polynomial version of Sarnak’s conjecture. Adv. Math.384 (2021), Paper no. 107765] and shows that no similar result can hold under only the assumptions of minimality and zero entropy.
我们对作者在 [Some counterexamples in topological dynamics.Ergod.Th. & Dynam.Sys.28(4)(2008),1291-1322]，即对于每一个集合 $S = {s_1, s_2, ldots }和有限集 A，存在一个最小零熵子移位 $(X, sigma )$，这样对于 A^{mathbb {Z}}$ 中的每一个序列 $u ，在 X$ 中存在 $x_u ，对于 mathbb {N}$ 中的所有 $n ，具有 $x_u(s_n) = u(n)$。非正式地讲，最小确定性序列在限制到零巴纳赫密度集合时可以实现完全任意的行为。作为推论，这为艾斯纳报告的多项式萨尔纳克猜想提供了反例 [A polynomial version of Sarnak's conjecture.C. R. Math.Acad.Sci. Paris353(7) (2015)，569-572] 所报告的猜想比卡尼戈夫斯基、莱曼奇克和拉齐维乌最近提供的一些猜想要宽泛得多 [Prime number theorem for analytic skew products.(2)199 (2024), 591-705] 以及 Lian 和 Shi [A counter-example for polynomial version of Sarnak's conjecture.Adv. Math.384 (2021), Paper no.107765]，并表明仅在最小性和零熵假设条件下，类似结果不可能成立。
{"title":"Minimal zero entropy subshifts can be unrestricted along any sparse set","authors":"RONNIE PAVLOV","doi":"10.1017/etds.2024.42","DOIUrl":"https://doi.org/10.1017/etds.2024.42","url":null,"abstract":"We present a streamlined proof of a result essentially presented by the author in [Some counterexamples in topological dynamics. <jats:italic>Ergod. Th. & Dynam. Sys.</jats:italic>28(4) (2008), 1291–1322], namely that for every set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000427_inline1.png\"/> <jats:tex-math> $S = {s_1, s_2, ldots } subset mathbb {N}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of zero Banach density and finite set <jats:italic>A</jats:italic>, there exists a minimal zero-entropy subshift <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000427_inline2.png\"/> <jats:tex-math> $(X, sigma )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> so that for every sequence <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000427_inline3.png\"/> <jats:tex-math> $u in A^{mathbb {Z}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, there is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000427_inline4.png\"/> <jats:tex-math> $x_u in X$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000427_inline5.png\"/> <jats:tex-math> $x_u(s_n) = u(n)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000427_inline6.png\"/> <jats:tex-math> $n in mathbb {N}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Informally, minimal deterministic sequences can achieve completely arbitrary behavior upon restriction to a set of zero Banach density. As a corollary, this provides counterexamples to the polynomial Sarnak conjecture reported by Eisner [A polynomial version of Sarnak’s conjecture. <jats:italic>C. R. Math. Acad. Sci. Paris</jats:italic>353(7) (2015), 569–572] which are significantly more general than some recently provided by Kanigowski, Lemańczyk and Radziwiłł [Prime number theorem for analytic skew products. <jats:italic>Ann. of Math. (2)</jats:italic>199 (2024), 591–705] and by Lian and Shi [A counter-example for polynomial version of Sarnak’s conjecture. <jats:italic>Adv. Math.</jats:italic>384 (2021), Paper no. 107765] and shows that no similar result can hold under only the assumptions of minimality and zero entropy.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224268","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 definition of subshifts of finite symbolic rank is motivated by the finite rank measure-preserving transformations which have been extensively studied in ergodic theory. In this paper, we study subshifts of finite symbolic rank as essentially minimal Cantor systems. We show that minimal subshifts of finite symbolic rank have finite topological rank, and conversely, every minimal Cantor system of finite topological rank is either an odometer or conjugate to a minimal subshift of finite symbolic rank. We characterize the class of all minimal Cantor systems conjugate to a rank- $1$ subshift and show that it is dense but not generic in the Polish space of all minimal Cantor systems. Within some different Polish coding spaces of subshifts, we also show that the rank-1 subshifts are dense but not generic. Finally, we study topological factors of minimal subshifts of finite symbolic rank. We show that every infinite odometer and every irrational rotation is the maximal equicontinuous factor of a minimal subshift of symbolic rank $2$ , and that a subshift factor of a minimal subshift of finite symbolic rank has finite symbolic rank.
{"title":"Subshifts of finite symbolic rank","authors":"SU GAO, RUIWEN LI","doi":"10.1017/etds.2024.45","DOIUrl":"https://doi.org/10.1017/etds.2024.45","url":null,"abstract":"The definition of subshifts of finite symbolic rank is motivated by the finite rank measure-preserving transformations which have been extensively studied in ergodic theory. In this paper, we study subshifts of finite symbolic rank as essentially minimal Cantor systems. We show that minimal subshifts of finite symbolic rank have finite topological rank, and conversely, every minimal Cantor system of finite topological rank is either an odometer or conjugate to a minimal subshift of finite symbolic rank. We characterize the class of all minimal Cantor systems conjugate to a rank-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000452_inline1.png\"/> <jats:tex-math> $1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> subshift and show that it is dense but not generic in the Polish space of all minimal Cantor systems. Within some different Polish coding spaces of subshifts, we also show that the rank-1 subshifts are dense but not generic. Finally, we study topological factors of minimal subshifts of finite symbolic rank. We show that every infinite odometer and every irrational rotation is the maximal equicontinuous factor of a minimal subshift of symbolic rank <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000452_inline2.png\"/> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, and that a subshift factor of a minimal subshift of finite symbolic rank has finite symbolic rank.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224273","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 7 Cover and Back matter","authors":"","doi":"10.1017/etds.2023.90","DOIUrl":"https://doi.org/10.1017/etds.2023.90","url":null,"abstract":"","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141715072","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 7 Cover and Front matter","authors":"","doi":"10.1017/etds.2023.89","DOIUrl":"https://doi.org/10.1017/etds.2023.89","url":null,"abstract":"","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141695213","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}
Two asymptotic configurations on a full $mathbb {Z}^d$ -shift are indistinguishable if, for every finite pattern, the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of $mathbb {Z}^d$ . We prove that indistinguishable asymptotic pairs satisfying a ‘flip condition’ are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together, the two results provide a generalization to $mathbb {Z}^d$ of the characterization of Sturmian sequences by their factor complexity $n+1$ . Many open questions are raised by the current work and are listed in the introduction.
{"title":"Indistinguishable asymptotic pairs and multidimensional Sturmian configurations","authors":"SEBASTIÁN BARBIERI, SÉBASTIEN LABBÉ","doi":"10.1017/etds.2024.39","DOIUrl":"https://doi.org/10.1017/etds.2024.39","url":null,"abstract":"Two asymptotic configurations on a full <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000397_inline1.png\"/> <jats:tex-math> $mathbb {Z}^d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-shift are indistinguishable if, for every finite pattern, the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000397_inline2.png\"/> <jats:tex-math> $mathbb {Z}^d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We prove that indistinguishable asymptotic pairs satisfying a ‘flip condition’ are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together, the two results provide a generalization to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000397_inline3.png\"/> <jats:tex-math> $mathbb {Z}^d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of the characterization of Sturmian sequences by their factor complexity <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000397_inline4.png\"/> <jats:tex-math> $n+1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Many open questions are raised by the current work and are listed in the introduction.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195472","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 study transitivity of partially hyperbolic endomorphisms of the two torus whose action in the first homology group has two integer eigenvalues of moduli greater than one. We prove that if the Jacobian is everywhere greater than the modulus of the largest eigenvalue, then the map is robustly transitive. For this, we introduce Blichfedt’s theorem as a tool for extracting dynamical information from the action of a map in homology. We also treat the case of specially partially hyperbolic endomorphisms, for which we obtain a complete dichotomy: either the map is transitive and conjugated to its linear part, or its unstable foliation must contain an annulus which may either be wandering or periodic.
{"title":"Partially hyperbolic endomorphisms with expanding linear part","authors":"MARTIN ANDERSSON, WAGNER RANTER","doi":"10.1017/etds.2024.36","DOIUrl":"https://doi.org/10.1017/etds.2024.36","url":null,"abstract":"In this paper, we study transitivity of partially hyperbolic endomorphisms of the two torus whose action in the first homology group has two integer eigenvalues of moduli greater than one. We prove that if the Jacobian is everywhere greater than the modulus of the largest eigenvalue, then the map is robustly transitive. For this, we introduce Blichfedt’s theorem as a tool for extracting dynamical information from the action of a map in homology. We also treat the case of specially partially hyperbolic endomorphisms, for which we obtain a complete dichotomy: either the map is transitive and conjugated to its linear part, or its unstable foliation must contain an annulus which may either be wandering or periodic.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141173051","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 main theorem of this paper establishes a uniform syndeticity result concerning the multiple recurrence of measure-preserving actions on probability spaces. More precisely, for any integers $d,lgeq 1$ and any $varepsilon> 0$ , we prove the existence of $delta>0$ and $Kgeq 1$ (dependent only on d, l, and $varepsilon $ ) such that the following holds: Consider a solvable group $Gamma $ of derived length l, a probability space $(X, mu )$ , and d pairwise commuting measure-preserving $Gamma $ -actions $T_1, ldots , T_d$ on
{"title":"Uniform syndeticity in multiple recurrence","authors":"ASGAR JAMNESHAN, MINGHAO PAN","doi":"10.1017/etds.2024.40","DOIUrl":"https://doi.org/10.1017/etds.2024.40","url":null,"abstract":"The main theorem of this paper establishes a uniform syndeticity result concerning the multiple recurrence of measure-preserving actions on probability spaces. More precisely, for any integers <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000403_inline1.png\"/> <jats:tex-math> $d,lgeq 1$ </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=\"S0143385724000403_inline2.png\"/> <jats:tex-math> $varepsilon> 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we prove the existence of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000403_inline3.png\"/> <jats:tex-math> $delta>0$ </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=\"S0143385724000403_inline4.png\"/> <jats:tex-math> $Kgeq 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> (dependent only on <jats:italic>d</jats:italic>, <jats:italic>l</jats:italic>, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000403_inline5.png\"/> <jats:tex-math> $varepsilon $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>) such that the following holds: Consider a solvable group <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000403_inline6.png\"/> <jats:tex-math> $Gamma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of derived length <jats:italic>l</jats:italic>, a probability space <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000403_inline7.png\"/> <jats:tex-math> $(X, mu )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:italic>d</jats:italic> pairwise commuting measure-preserving <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000403_inline8.png\"/> <jats:tex-math> $Gamma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-actions <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000403_inline9.png\"/> <jats:tex-math> $T_1, ldots , T_d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141172888","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}
Bedford and Smillie [A symbolic characterization of the horseshoe locus in the Hénon family. Ergod. Th. & Dynam. Sys.37(5) (2017), 1389–1412] classified the dynamics of the Hénon map $f_{a, b} : (x, y)mapsto (x^2-a-by, x)$ defined on $mathbb {R}^2$ in terms of a symbolic dynamics when $(a, b)$ is close to the boundary of the horseshoe locus. The purpose of the current article is to generalize their results for all $bne 0$ (including the case $b < 0$ as well). The method of the proof is first to regard $f_{a, b}$ as a complex dynamical system in $mathbb {C}^2$ and second to introduce the new Markov-like partition in $mathbb {R}^2$ constructed by us [On parameter loci of the Hénon family. Comm. Math. Phys.361(2) (2018),
{"title":"Symbolic dynamics for Hénon maps near the boundary of the horseshoe locus","authors":"Yuki Hironaka, Yutaka Ishii","doi":"10.1017/etds.2024.34","DOIUrl":"https://doi.org/10.1017/etds.2024.34","url":null,"abstract":"\u0000\t <jats:p>Bedford and Smillie [A symbolic characterization of the horseshoe locus in the Hénon family. <jats:italic>Ergod. Th. & Dynam. Sys.</jats:italic><jats:bold>37</jats:bold>(5) (2017), 1389–1412] classified the dynamics of the Hénon map <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=\"S0143385724000348_inline1.png\"/>\u0000\t\t<jats:tex-math>\u0000$f_{a, b} : (x, y)mapsto (x^2-a-by, x)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> defined 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=\"S0143385724000348_inline2.png\"/>\u0000\t\t<jats:tex-math>\u0000$mathbb {R}^2$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> in terms of a symbolic dynamics when <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=\"S0143385724000348_inline3.png\"/>\u0000\t\t<jats:tex-math>\u0000$(a, b)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> is close to the boundary of the horseshoe locus. The purpose of the current article is to generalize their results for all <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=\"S0143385724000348_inline4.png\"/>\u0000\t\t<jats:tex-math>\u0000$bne 0$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> (including the case <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=\"S0143385724000348_inline5.png\"/>\u0000\t\t<jats:tex-math>\u0000$b < 0$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> as well). The method of the proof is first to regard <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=\"S0143385724000348_inline6.png\"/>\u0000\t\t<jats:tex-math>\u0000$f_{a, b}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> as a complex dynamical system in <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=\"S0143385724000348_inline7.png\"/>\u0000\t\t<jats:tex-math>\u0000$mathbb {C}^2$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> and second to introduce the new Markov-like partition in <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=\"S0143385724000348_inline8.png\"/>\u0000\t\t<jats:tex-math>\u0000$mathbb {R}^2$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> constructed by us [On parameter loci of the Hénon family. <jats:italic>Comm. Math. Phys.</jats:italic><jats:bold>361</jats:bold>(2) (2018), ","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141106142","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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