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A new dynamical proof of the Shmerkin–Wu theorem Shmerkin-Wu定理的一个新的动态证明
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2020-09-02 DOI: 10.3934/jmd.2022001
Tim Austin
{"title":"A new dynamical proof of the Shmerkin–Wu theorem","authors":"Tim Austin","doi":"10.3934/jmd.2022001","DOIUrl":"https://doi.org/10.3934/jmd.2022001","url":null,"abstract":"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\"M1\">begin{document}$ a < b $end{document}</tex-math></inline-formula> be multiplicatively independent integers, both at least <inline-formula><tex-math id=\"M2\">begin{document}$ 2 $end{document}</tex-math></inline-formula>. Let <inline-formula><tex-math id=\"M3\">begin{document}$ A,B $end{document}</tex-math></inline-formula> be closed subsets of <inline-formula><tex-math id=\"M4\">begin{document}$ [0,1] $end{document}</tex-math></inline-formula> that are forward invariant under multiplication by <inline-formula><tex-math id=\"M5\">begin{document}$ a $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M6\">begin{document}$ b $end{document}</tex-math></inline-formula> respectively, and let <inline-formula><tex-math id=\"M7\">begin{document}$ C : = Atimes B $end{document}</tex-math></inline-formula>. An old conjecture of Furstenberg asserted that any planar line <inline-formula><tex-math id=\"M8\">begin{document}$ L $end{document}</tex-math></inline-formula> not parallel to either axis must intersect <inline-formula><tex-math id=\"M9\">begin{document}$ C $end{document}</tex-math></inline-formula> in Hausdorff dimension at most <inline-formula><tex-math id=\"M10\">begin{document}$ max{dim C,1} - 1 $end{document}</tex-math></inline-formula>. Two recent works by Shmerkin and Wu have given two different proofs of this conjecture. This note provides a third proof. Like Wu's, it stays close to the ergodic theoretic machinery that Furstenberg introduced to study such questions, but it uses less substantial background from ergodic theory. The same method is also used to re-prove a recent result of Yu about certain sequences of sums.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44020894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
Horospherically invariant measures and finitely generated Kleinian groups 星象不变测度与有限生成Kleinian群
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2020-08-12 DOI: 10.3934/jmd.2021012
Or Landesberg
{"title":"Horospherically invariant measures and finitely generated Kleinian groups","authors":"Or Landesberg","doi":"10.3934/jmd.2021012","DOIUrl":"https://doi.org/10.3934/jmd.2021012","url":null,"abstract":"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\"M1\">begin{document}$ Gamma < {rm{PSL}}_2( mathbb{C}) $end{document}</tex-math></inline-formula> be a Zariski dense finitely generated Kleinian group. We show all Radon measures on <inline-formula><tex-math id=\"M2\">begin{document}$ {rm{PSL}}_2( mathbb{C}) / Gamma $end{document}</tex-math></inline-formula> which are ergodic and invariant under the action of the horospherical subgroup are either supported on a single closed horospherical orbit or quasi-invariant with respect to the geodesic frame flow and its centralizer. We do this by applying a result of Landesberg and Lindenstrauss [<xref ref-type=\"bibr\" rid=\"b18\">18</xref>] together with fundamental results in the theory of 3-manifolds, most notably the Tameness Theorem by Agol [<xref ref-type=\"bibr\" rid=\"b2\">2</xref>] and Calegari-Gabai [<xref ref-type=\"bibr\" rid=\"b10\">10</xref>].</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48785530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Computing the Rabinowitz Floer homology of tentacular hyperboloids 触手双曲面的Rabinowitz-Floer同调的计算
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2020-06-30 DOI: 10.3934/jmd.2021013
Alexander Fauck, W. Merry, J. Wi'sniewska
{"title":"Computing the Rabinowitz Floer homology of tentacular hyperboloids","authors":"Alexander Fauck, W. Merry, J. Wi'sniewska","doi":"10.3934/jmd.2021013","DOIUrl":"https://doi.org/10.3934/jmd.2021013","url":null,"abstract":"<p style='text-indent:20px;'>We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids <inline-formula><tex-math id=\"M1\">begin{document}$ Sigmasimeq S^{n+k-1}timesmathbb{R}^{n-k} $end{document}</tex-math></inline-formula>. Using an embedding of a compact sphere <inline-formula><tex-math id=\"M2\">begin{document}$ Sigma_0simeq S^{2k-1} $end{document}</tex-math></inline-formula> into the hypersurface <inline-formula><tex-math id=\"M3\">begin{document}$ Sigma $end{document}</tex-math></inline-formula>, we construct a chain map from the Floer complex of <inline-formula><tex-math id=\"M4\">begin{document}$ Sigma $end{document}</tex-math></inline-formula> to the Floer complex of <inline-formula><tex-math id=\"M5\">begin{document}$ Sigma_0 $end{document}</tex-math></inline-formula>. In contrast to the compact case, the Rabinowitz Floer homology groups of <inline-formula><tex-math id=\"M6\">begin{document}$ Sigma $end{document}</tex-math></inline-formula> are both non-zero and not equal to its singular homology. As a consequence, we deduce that the Weinstein Conjecture holds for any strongly tentacular deformation of such a hyperboloid.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46822839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Higher bifurcations for polynomial skew products 多项式偏积的高分岔
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2020-06-26 DOI: 10.3934/jmd.2022003
M. Astorg, Fabrizio Bianchi
{"title":"Higher bifurcations for polynomial skew products","authors":"M. Astorg, Fabrizio Bianchi","doi":"10.3934/jmd.2022003","DOIUrl":"https://doi.org/10.3934/jmd.2022003","url":null,"abstract":"<p style='text-indent:20px;'>We continue our investigation of the parameter space of families of polynomial skew products. Assuming that the base polynomial has a Julia set not totally disconnected and is neither a Chebyshev nor a power map, we prove that, near any bifurcation parameter, one can find parameters where <inline-formula><tex-math id=\"M1\">begin{document}$ k $end{document}</tex-math></inline-formula> critical points bifurcate <i>independently</i>, with <inline-formula><tex-math id=\"M2\">begin{document}$ k $end{document}</tex-math></inline-formula> up to the dimension of the parameter space. This is a striking difference with respect to the one-dimensional case. The proof is based on a variant of the inclination lemma, applied to the postcritical set at a Misiurewicz parameter. By means of an analytical criterion for the non-vanishing of the self-intersections of the bifurcation current, we deduce the equality of the supports of the bifurcation current and the bifurcation measure for such families. Combined with results by Dujardin and Taflin, this also implies that the support of the bifurcation measure in these families has non-empty interior. As part of our proof we construct, in these families, subfamilies of codimension 1 where the bifurcation locus has non empty interior. This provides a new independent proof of the existence of holomorphic families of arbitrarily large dimension whose bifurcation locus has non empty interior. Finally, it shows that the Hausdorff dimension of the support of the bifurcation measure is maximal at any point of its support.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46403022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł, and Tao 关于Matomäki、Radziwiłł;和Tao的非周期乘法函数的Furstenberg系统
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2020-06-17 DOI: 10.3934/jmd.2021018
Aleksander Gomilko, M. Lemanczyk, T. Rue
{"title":"On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł, and Tao","authors":"Aleksander Gomilko, M. Lemanczyk, T. Rue","doi":"10.3934/jmd.2021018","DOIUrl":"https://doi.org/10.3934/jmd.2021018","url":null,"abstract":"<p style='text-indent:20px;'>It is shown that in a class of counterexamples to Elliott's conjecture by Matomäki, Radziwiłł, and Tao [<xref ref-type=\"bibr\" rid=\"b23\">23</xref>] the Chowla conjecture holds along a subsequence.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47482373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
On the relation between action and linking 论动作与衔接的关系
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2020-06-11 DOI: 10.3934/jmd.2021011
David Bechara Senior, Umberto L. Hryniewicz, Pedro A. S. Salomão
{"title":"On the relation between action and linking","authors":"David Bechara Senior, Umberto L. Hryniewicz, Pedro A. S. Salomão","doi":"10.3934/jmd.2021011","DOIUrl":"https://doi.org/10.3934/jmd.2021011","url":null,"abstract":"We introduce numerical invariants of contact forms in dimension three and use asymptotic cycles to estimate them. As a consequence, we prove a version for Anosov Reeb flows of results due to Hutchings and Weiler on mean actions of periodic points. The main tool is the Action-Linking Lemma, expressing the contact area of a surface bounded by periodic orbits as the Liouville average of the asymptotic intersection number of most trajectories with the surface.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43077860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
Cocycle superrigidity from higher rank lattices to $ {{rm{Out}}}{(F_N)} $ 从高阶格到${rm{Out}}}{（F_N）}的Cocycle超刚性$
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2020-05-15 DOI: 10.3934/jmd.2022010
Vincent Guirardel, Camille Horbez, Jean Lécureux
{"title":"Cocycle superrigidity from higher rank lattices to $ {{rm{Out}}}{(F_N)} $","authors":"Vincent Guirardel, Camille Horbez, Jean Lécureux","doi":"10.3934/jmd.2022010","DOIUrl":"https://doi.org/10.3934/jmd.2022010","url":null,"abstract":"<p style='text-indent:20px;'>We prove a rigidity result for cocycles from higher rank lattices to <inline-formula><tex-math id=\"M2\">begin{document}$ mathrm{Out}(F_N) $end{document}</tex-math></inline-formula> and more generally to the outer automorphism group of a torsion-free hyperbolic group. More precisely, let <inline-formula><tex-math id=\"M3\">begin{document}$ G $end{document}</tex-math></inline-formula> be either a product of connected higher rank simple algebraic groups over local fields, or a lattice in such a product. Let <inline-formula><tex-math id=\"M4\">begin{document}$ G curvearrowright X $end{document}</tex-math></inline-formula> be an ergodic measure-preserving action on a standard probability space, and let <inline-formula><tex-math id=\"M5\">begin{document}$ H $end{document}</tex-math></inline-formula> be a torsion-free hyperbolic group. We prove that every Borel cocycle <inline-formula><tex-math id=\"M6\">begin{document}$ Gtimes Xto mathrm{Out}(H) $end{document}</tex-math></inline-formula> is cohomologous to a cocycle with values in a finite subgroup of <inline-formula><tex-math id=\"M7\">begin{document}$ mathrm{Out}(H) $end{document}</tex-math></inline-formula>. This provides a dynamical version of theorems of Farb–Kaimanovich–Masur and Bridson–Wade asserting that every homomorphism from <inline-formula><tex-math id=\"M8\">begin{document}$ G $end{document}</tex-math></inline-formula> to either the mapping class group of a finite-type surface or the outer automorphism group of a free group, has finite image.</p><p style='text-indent:20px;'>The main new geometric tool is a barycenter map that associates to every triple of points in the boundary of the (relative) free factor graph a finite set of (relative) free splittings.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44865225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
A generic distal tower of arbitrary countable height over an arbitrary infinite ergodic system 在任意无限遍历系统上具有任意可数高度的一般远塔
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2020-05-14 DOI: 10.3934/jmd.2021015
E. Glasner, B. Weiss
{"title":"A generic distal tower of arbitrary countable height over an arbitrary infinite ergodic system","authors":"E. Glasner, B. Weiss","doi":"10.3934/jmd.2021015","DOIUrl":"https://doi.org/10.3934/jmd.2021015","url":null,"abstract":"<p style='text-indent:20px;'>We show the existence, over an arbitrary infinite ergodic <inline-formula><tex-math id=\"M1\">begin{document}$ mathbb{Z} $end{document}</tex-math></inline-formula>-dynamical system, of a generic ergodic relatively distal extension of arbitrary countable rank and arbitrary infinite compact extending groups (or more generally, infinite quotients of compact groups) in its canonical distal tower.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48081185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The orbital equivalence of Bernoulli actions and their Sinai factors 伯努利作用及其西奈因子的轨道等效性
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2020-05-06 DOI: 10.3934/JMD.2021005
Zemer Kosloff, Terry Soo
{"title":"The orbital equivalence of Bernoulli actions and their Sinai factors","authors":"Zemer Kosloff, Terry Soo","doi":"10.3934/JMD.2021005","DOIUrl":"https://doi.org/10.3934/JMD.2021005","url":null,"abstract":"Given a countable amenable group G and 0 < L < 1, we give an elementary construction of a type-III:L Bernoulli group action. In the case where G is the integers, we show that our nonsingular Bernoulli shifts have independent and identically distributed factors.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41473958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
Slow entropy of higher rank abelian unipotent actions 高阶阿贝尔无效动作的慢熵
IF 1.1 1区 数学
Journal of Modern Dynamics Pub Date : 2020-05-05 DOI: 10.3934/jmd.2022018
Adam Kanigowski, Philipp Kunde, Kurt Vinhage, Daren Wei
{"title":"Slow entropy of higher rank abelian unipotent actions","authors":"Adam Kanigowski, Philipp Kunde, Kurt Vinhage, Daren Wei","doi":"10.3934/jmd.2022018","DOIUrl":"https://doi.org/10.3934/jmd.2022018","url":null,"abstract":"We study slow entropy invariants for abelian unipotent actions $U$ on any finite volume homogeneous space $G/Gamma$. For every such action we show that the topological slow entropy can be computed directly from the dimension of a special decomposition of $operatorname{Lie}(G)$ induced by $operatorname{Lie}(U)$. Moreover, we are able to show that the metric slow entropy of the action coincides with its topological slow entropy. As a corollary, we obtain that the complexity of any abelian horocyclic action is only related to the dimension of $G$. This generalizes the rank one results from [A. Kanigowski, K. Vinhage, D. Wei, Commun. Math. Phys. 370 (2019), no. 2, 449-474.] to higher rank abelian actions.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42845446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
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