Ergodic Theory and Dynamical Systems最新文献

{"title":"Weighted topological pressure revisited","authors":"NIMA ALIBABAEI","doi":"10.1017/etds.2024.35","DOIUrl":"https://doi.org/10.1017/etds.2024.35","url":null,"abstract":"Feng and Huang [Variational principle for weighted topological pressure. <jats:italic>J. Math. Pures Appl. (9)</jats:italic>106 (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto [New approach to weighted topological entropy and pressure. <jats:italic>Ergod. Th. &amp; Dynam. Sys.</jats:italic>43 (2023), 1004–1034] redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto’s approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"100 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931000","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":"Poissonian pair correlation for directions in multi-dimensional affine lattices and escape of mass estimates for embedded horospheres","authors":"WOOYEON KIM, JENS MARKLOF","doi":"10.1017/etds.2024.31","DOIUrl":"https://doi.org/10.1017/etds.2024.31","url":null,"abstract":"We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for <jats:italic>any</jats:italic> irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author [The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. <jats:italic>Ann. of Math. (2)</jats:italic>172 (2010), 1949–2033], and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000312_inline1.png\"/> <jats:tex-math> $operatorname {SL}(d,mathbb {R})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-horospheres in the space of affine lattices.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"101 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800664","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":"Khintchine-type double recurrence in abelian groups","authors":"ETHAN ACKELSBERG","doi":"10.1017/etds.2024.29","DOIUrl":"https://doi.org/10.1017/etds.2024.29","url":null,"abstract":"We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline1.png\"/&gt; &lt;jats:tex-math&gt; $Gamma $ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is a countable discrete abelian group, &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline2.png\"/&gt; &lt;jats:tex-math&gt; $varphi , psi in mathrm {End}(Gamma )$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline3.png\"/&gt; &lt;jats:tex-math&gt; $psi - varphi $ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is an injective endomorphism with finite index image, then for any ergodic measure-preserving &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline4.png\"/&gt; &lt;jats:tex-math&gt; $Gamma $ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-system &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline5.png\"/&gt; &lt;jats:tex-math&gt; $( X, {mathcal {X}}, mu , (T_g)_{g in Gamma } )$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, any measurable set &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline6.png\"/&gt; &lt;jats:tex-math&gt; $A in {mathcal {X}}$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, and any &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline7.png\"/&gt; &lt;jats:tex-math&gt; ${varepsilon }&gt; 0$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, there is a syndetic set of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline8.png\"/&gt; &lt;jats:tex-math&gt; $g in Gamma$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; such that &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000294_inline9.png\"/&gt; &lt;jats:tex-math&gt; $mu ( A cap T_{varphi(g)}^{-1} A cap T_{psi(g)}^{-1} A ) &gt; mu(A)^3 - varepsilon$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. This generalizes the main results of Ackelsberg &lt;jats:italic&gt;et al&lt;/jats:italic&gt; [Khintchine-type recurrence for 3-point configurations. &lt;jats:italic","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"51 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800661","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":"Schmidt games and Cantor winning sets","authors":"DZMITRY BADZIAHIN, STEPHEN HARRAP, EREZ NESHARIM, DAVID SIMMONS","doi":"10.1017/etds.2024.23","DOIUrl":"https://doi.org/10.1017/etds.2024.23","url":null,"abstract":"Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of study. We survey the definitions of the most common variants and connections between them. A new game called the Cantor game is invented and helps with presenting a unifying framework. We prove surprising new results such as the coincidence of absolute winning and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000233_inline1.png\" /> <jats:tex-math> $1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> Cantor winning in metric spaces, and the fact that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000233_inline2.png\" /> <jats:tex-math> $1/2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> winning implies absolute winning for subsets of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000233_inline3.png\" /> <jats:tex-math> $mathbb {R}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We also suggest a prototypical example of a Cantor winning set to show the ubiquity of such sets in metric number theory and ergodic theory.","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"38 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140623891","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":"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 &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline1.png\" /&gt; &lt;jats:tex-math&gt; ${mathbf q}$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and a linear form &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline2.png\" /&gt; &lt;jats:tex-math&gt; ${mathbf l}$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; over the set of integral vectors, a problem initiated by Dani and Margulis [Orbit closures of generic unipotent flows on homogeneous spaces of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline3.png\" /&gt; &lt;jats:tex-math&gt; $mathrm{SL}_3(mathbb{R})$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. &lt;jats:italic&gt;Math. Ann.&lt;/jats:italic&gt;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 &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline4.png\" /&gt; &lt;jats:tex-math&gt; $n ge 5$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, then under the assumptions that for every &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline5.png\" /&gt; &lt;jats:tex-math&gt; $(alpha , beta ) in {mathbb {R}}^2 setminus { (0,0) }$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, the form &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline6.png\" /&gt; &lt;jats:tex-math&gt; $alpha {mathbf q} + beta {mathbf l}^2$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is irrational and that the signature of the restriction of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline7.png\" /&gt; &lt;jats:tex-math&gt; ${mathbf q}$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; to the kernel of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline8.png\" /&gt; &lt;jats:tex-math&gt; ${mathbf l}$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000300_inline9.png\" /&gt; &lt;jats:tex-math&gt; $(p, n-1-p)$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formu","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"49 1","pages":""},"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":"86 1","pages":""},"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":"214 1","pages":""},"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":"272 1","pages":""},"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":"67 1","pages":""},"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":"28 1","pages":""},"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}