Ergodic Theory and Dynamical Systems最新文献

{"title":"Asymptotic distribution for pairs of linear and quadratic forms at integral vectors","authors":"JIYOUNG HAN, SEONHEE LIM, KEIVAN MALLAHI-KARAI","doi":"10.1017/etds.2024.30","DOIUrl":"https://doi.org/10.1017/etds.2024.30","url":null,"abstract":"We study the joint distribution of values of a pair consisting of a quadratic form <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline1.png\" /> <jats:tex-math> ${mathbf q}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and a linear form <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline2.png\" /> <jats:tex-math> ${mathbf l}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> over the set of integral vectors, a problem initiated by Dani and Margulis [Orbit closures of generic unipotent flows on homogeneous spaces of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline3.png\" /> <jats:tex-math> $mathrm{SL}_3(mathbb{R})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. <jats:italic>Math. Ann.</jats:italic>286 (1990), 101–128]. In the spirit of the celebrated theorem of Eskin, Margulis and Mozes on the quantitative version of the Oppenheim conjecture, we show that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline4.png\" /> <jats:tex-math> $n ge 5$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then under the assumptions that for every <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline5.png\" /> <jats:tex-math> $(alpha , beta ) in {mathbb {R}}^2 setminus { (0,0) }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, the form <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline6.png\" /> <jats:tex-math> $alpha {mathbf q} + beta {mathbf l}^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is irrational and that the signature of the restriction of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline7.png\" /> <jats:tex-math> ${mathbf q}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> to the kernel of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline8.png\" /> <jats:tex-math> ${mathbf l}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline9.png\" /> <jats:tex-math> $(p, n-1-p)$ </jats:tex-math> </jats:alternatives> </jats:inline-formu","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140609087","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":"Similarities and differences between specification and non-uniform specification","authors":"WANSHAN LIN, XUETING TIAN, CHENWEI YU","doi":"10.1017/etds.2024.28","DOIUrl":"https://doi.org/10.1017/etds.2024.28","url":null,"abstract":"Pavlov [<jats:italic>Adv. Math.</jats:italic>295 (2016), 250–270; <jats:italic>Nonlinearity</jats:italic>32 (2019), 2441–2466] studied the measures of maximal entropy for dynamical systems with weak versions of specification property and found the existence of intrinsic ergodicity would be influenced by the assumptions of the gap functions. Inspired by these, in this article, we study the dynamical systems with non-uniform specification property. We give some basic properties these systems have and give an assumption for the gap functions to ensure the systems have the following five properties: CO-measures are dense in invariant measures; for every non-empty compact connected subset of invariant measures, its saturated set is dense in the total space; ergodic measures are residual in invariant measures; ergodic measures are connected; and entropy-dense. In addition, we will give examples to show the assumption is optimal.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140589962","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":"Non-existence of a universal zero-entropy system via generic actions of almost complete growth","authors":"GEORGII VEPREV","doi":"10.1017/etds.2024.24","DOIUrl":"https://doi.org/10.1017/etds.2024.24","url":null,"abstract":"We prove that a generic probability measure-preserving (p.m.p.) action of a countable amenable group <jats:italic>G</jats:italic> has scaling entropy that cannot be dominated by a given rate of growth. As a corollary, we obtain that there does not exist a topological action of <jats:italic>G</jats:italic> for which the set of ergodic invariant measures coincides with the set of all ergodic p.m.p. <jats:italic>G</jats:italic>-systems of entropy zero. We also prove that a generic action of a residually finite amenable group has scaling entropy that cannot be bounded from below by a given sequence. In addition, we show an example of an amenable group that has such a lower bound for every free p.m.p. action.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140589934","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":"On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms","authors":"FRANK TRUJILLO","doi":"10.1017/etds.2024.25","DOIUrl":"https://doi.org/10.1017/etds.2024.25","url":null,"abstract":"We show that, generically, the unique invariant measure of a sufficiently regular piecewise smooth circle homeomorphism with irrational rotation number and zero mean nonlinearity (e.g. piecewise linear) has zero Hausdorff dimension. To encode this generic condition, we consider piecewise smooth homeomorphisms as <jats:italic>generalized interval exchange transformations</jats:italic> (GIETs) of the interval and rely on the notion of <jats:italic>combinatorial rotation number</jats:italic> for GIETs, which can be seen as an extension of the classical notion of rotation number for circle homeomorphisms to the GIET setting.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140589963","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":"Substreetutions and more on trees","authors":"ALEXANDRE BARAVIERA, RENAUD LEPLAIDEUR","doi":"10.1017/etds.2023.108","DOIUrl":"https://doi.org/10.1017/etds.2023.108","url":null,"abstract":"We define a notion of substitution on colored binary trees that we call substreetution. We show that a point fixed by a substreetution may (or not) be almost periodic, and thus the closure of the orbit under the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001086_inline1.png\" /> <jats:tex-math> $mathbb {F}_{2}^{+}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-action may (or not) be minimal. We study one special example: we show that it belongs to the minimal case and that the number of preimages in the minimal set increases just exponentially fast, whereas it could be expected a super-exponential growth. We also give examples of periodic trees without invariant measures on their orbit. We use our construction to get quasi-periodic colored tilings of the hyperbolic disk.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140589959","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":"Density of mode-locking property for quasi-periodically forced Arnold circle maps","authors":"JIAN WANG, ZHIYUAN ZHANG","doi":"10.1017/etds.2024.27","DOIUrl":"https://doi.org/10.1017/etds.2024.27","url":null,"abstract":"We show that the mode-locking region of the family of quasi-periodically forced Arnold circle maps with a topologically generic forcing function is dense. This gives a rigorous verification of certain numerical observations in [M. Ding, C. Grebogi and E. Ott. Evolution of attractors in quasiperiodically forced systems: from quasiperiodic to strange nonchaotic to chaotic. <jats:italic>Phys. Rev. A</jats:italic> 39(5) (1989), 2593–2598] for such forcing functions. More generally, under some general conditions on the base map, we show the density of the mode-locking property among dynamically forced maps (defined in [Z. Zhang. On topological genericity of the mode-locking phenomenon. <jats:italic>Math. Ann.</jats:italic> 376 (2020), 707–72]) equipped with a topology that is much stronger than the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000270_inline1.png\" /> <jats:tex-math> $C^0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> topology, compatible with smooth fiber maps. For quasi-periodic base maps, our result generalizes the main results in [A. Avila, J. Bochi and D. Damanik. Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. <jats:italic>Duke Math. J.</jats:italic>146 (2009), 253–280], [J. Wang, Q. Zhou and T. Jäger. Genericity of mode-locking for quasiperiodically forced circle maps. <jats:italic>Adv. Math.</jats:italic>348 (2019), 353–377] and Zhang (2020).","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140589952","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":"Entropy, virtual Abelianness and Shannon orbit equivalence","authors":"DAVID KERR, HANFENG LI","doi":"10.1017/etds.2024.26","DOIUrl":"https://doi.org/10.1017/etds.2024.26","url":null,"abstract":"<p>We prove that if two free probability-measure-preserving (p.m.p.) <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240329070043690-0459:S0143385724000269:S0143385724000269_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb Z}$</span></span></img></span></span>-actions are Shannon orbit equivalent, then they have the same entropy. The argument also applies more generally to yield the same conclusion for free p.m.p. actions of finitely generated virtually Abelian groups. Together with the isomorphism theorems of Ornstein and Ornstein–Weiss and the entropy invariance results of Austin and Kerr–Li in the non-virtually-cyclic setting, this shows that two Bernoulli actions of any non-locally-finite countably infinite amenable group are Shannon orbit equivalent if and only if they are measure conjugate. We also show, at the opposite end of the stochastic spectrum, that every <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240329070043690-0459:S0143385724000269:S0143385724000269_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb Z}$</span></span></img></span></span>-odometer is Shannon orbit equivalent to the universal <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240329070043690-0459:S0143385724000269:S0143385724000269_inline3.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb Z}$</span></span></img></span></span>-odometer.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140589961","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":"Scale recurrence lemma and dimension formula for Cantor sets in the complex plane","authors":"CARLOS GUSTAVO T. DE A. MOREIRA, ALEX MAURICIO ZAMUDIO ESPINOSA","doi":"10.1017/etds.2024.15","DOIUrl":"https://doi.org/10.1017/etds.2024.15","url":null,"abstract":"We prove a multidimensional conformal version of the scale recurrence lemma of Moreira and Yoccoz [Stable intersections of regular Cantor sets with large Hausdorff dimensions. <jats:italic>Ann. of Math. (2)</jats:italic>154(1) (2001), 45–96] for Cantor sets in the complex plane. We then use this new recurrence lemma, together with Moreira’s ideas in [Geometric properties of images of Cartesian products of regular Cantor sets by differentiable real maps. <jats:italic>Math. Z.</jats:italic>303 (2023), 3], to prove that under the right hypothesis for the Cantor sets <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000154_inline1.png\" /> <jats:tex-math> $K_1,ldots ,K_n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and the function <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000154_inline2.png\" /> <jats:tex-math> $h:mathbb {C}^{n}to mathbb {R}^{l}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, the following formula holds: <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000154_eqnu1.png\" /> <jats:tex-math> $$ begin{align*}HD(h(K_1times K_2 times cdotstimes K_n))=min {l,HD(K_1)+cdots+HD(K_n)}.end{align*} $$ </jats:tex-math> </jats:alternatives> </jats:disp-formula>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300750","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":"Rigidity of pressures of Hölder potentials and the fitting of analytic functions through them","authors":"LIANGANG MA, MARK POLLICOTT","doi":"10.1017/etds.2024.9","DOIUrl":"https://doi.org/10.1017/etds.2024.9","url":null,"abstract":"<p>The first part of this work is devoted to the study of higher derivatives of pressure functions of Hölder potentials on shift spaces with finitely many symbols. By describing the derivatives of pressure functions via the central limit theorem for the associated random processes, we discover some rigid relationships between derivatives of various orders. The rigidity imposes obstructions on fitting candidate convex analytic functions by pressure functions of Hölder potentials globally, which answers a question of Kucherenko and Quas. In the second part of the work, we consider fitting candidate analytic germs by pressure functions of locally constant potentials. We prove that all 1-level candidate germs can be realised by pressures of some locally constant potentials, as long as the number of symbols in the symbolic set is large enough. There are also some results on fitting 2-level germs by pressures of locally constant potentials obtained in the work.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140151558","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":"Approximate homomorphisms and sofic approximations of orbit equivalence relations","authors":"BEN HAYES, SRIVATSAV KUNNAWALKAM ELAYAVALLI","doi":"10.1017/etds.2024.22","DOIUrl":"https://doi.org/10.1017/etds.2024.22","url":null,"abstract":"We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence relation is uniquely determined by the invariant random subgroup of the approximate homomorphisms. We record applications of this result to recover various known stability and conjugacy characterizations for almost homomorphisms of amenable groups.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140151529","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}