In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000051_inline1.png" /> <jats:tex-math> $[0,1]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000051_inline2.png" /> <jats:tex-math> $nu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We consider a class of non-square-integrable observables <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000051_inline3.png" /> <jats:tex-math> $phi $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, mostly of form <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000051_inline4.png" /> <jats:tex-math> $phi (x)=d(x,x_0)^{-{1}/{alpha }}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000051_inline5.png" /> <jats:tex-math> $x_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000051_inline6.png" /> <jats:tex-math> $alpha in (0,2)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. The two types of maps we concatenate are a class of piecewise <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000051_inline7.png" /> <jats:tex-math> $C^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000051_inline8.png" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants fo
{"title":"Stable laws for random dynamical systems","authors":"ROMAIN AIMINO, MATTHEW NICOL, ANDREW TÖRÖK","doi":"10.1017/etds.2024.5","DOIUrl":"https://doi.org/10.1017/etds.2024.5","url":null,"abstract":"In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline1.png\" /> <jats:tex-math> $[0,1]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline2.png\" /> <jats:tex-math> $nu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We consider a class of non-square-integrable observables <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline3.png\" /> <jats:tex-math> $phi $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, mostly of form <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline4.png\" /> <jats:tex-math> $phi (x)=d(x,x_0)^{-{1}/{alpha }}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline5.png\" /> <jats:tex-math> $x_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline6.png\" /> <jats:tex-math> $alpha in (0,2)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. The two types of maps we concatenate are a class of piecewise <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline7.png\" /> <jats:tex-math> $C^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000051_inline8.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants fo","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"20 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139771865","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}
Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000038_inline1.png" /> <jats:tex-math> $f: Mrightarrow M$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000038_inline2.png" /> <jats:tex-math> $C^{1+alpha }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> diffeomorphism on an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000038_inline3.png" /> <jats:tex-math> $m_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-dimensional compact smooth Riemannian manifold <jats:italic>M</jats:italic> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000038_inline4.png" /> <jats:tex-math> $mu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> a hyperbolic ergodic <jats:italic>f</jats:italic>-invariant probability measure. This paper obtains an upper bound for the stable (unstable) pointwise dimension of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000038_inline5.png" /> <jats:tex-math> $mu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, which is given by the unique solution of an equation involving the sub-additive measure-theoretic pressure. If <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000038_inline6.png" /> <jats:tex-math> $mu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a Sinai–Ruelle–Bowen (SRB) measure, then the Kaplan–Yorke conjecture is true under some additional conditions and the Lyapunov dimension of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000038_inline7.png" /> <jats:tex-math> $mu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> can be approximated gradually by the Hausdorff dimension of a sequence of hyperbolic sets <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000038_inline8.png" /> <jats:tex-math> ${Lambda _n}_{ngeq 1}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. The limit behaviour of the Carathéodory singular dimension of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385724000038_inline9.png" /> <jats:tex-math> $Lambda _n$ </jats:tex-math> </jats:alternatives> </jats:inl
{"title":"Dimension estimates and approximation in non-uniformly hyperbolic systems","authors":"JUAN WANG, YONGLUO CAO, YUN ZHAO","doi":"10.1017/etds.2024.3","DOIUrl":"https://doi.org/10.1017/etds.2024.3","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline1.png\" /> <jats:tex-math> $f: Mrightarrow M$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline2.png\" /> <jats:tex-math> $C^{1+alpha }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> diffeomorphism on an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline3.png\" /> <jats:tex-math> $m_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-dimensional compact smooth Riemannian manifold <jats:italic>M</jats:italic> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline4.png\" /> <jats:tex-math> $mu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> a hyperbolic ergodic <jats:italic>f</jats:italic>-invariant probability measure. This paper obtains an upper bound for the stable (unstable) pointwise dimension of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline5.png\" /> <jats:tex-math> $mu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, which is given by the unique solution of an equation involving the sub-additive measure-theoretic pressure. If <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline6.png\" /> <jats:tex-math> $mu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a Sinai–Ruelle–Bowen (SRB) measure, then the Kaplan–Yorke conjecture is true under some additional conditions and the Lyapunov dimension of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline7.png\" /> <jats:tex-math> $mu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> can be approximated gradually by the Hausdorff dimension of a sequence of hyperbolic sets <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline8.png\" /> <jats:tex-math> ${Lambda _n}_{ngeq 1}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. The limit behaviour of the Carathéodory singular dimension of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000038_inline9.png\" /> <jats:tex-math> $Lambda _n$ </jats:tex-math> </jats:alternatives> </jats:inl","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"235 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139771871","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 show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose densities are regular and vary differentiably with respect to the dynamical system. The maps we consider are skew-products given by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723001219_inline1.png" /> <jats:tex-math> $T(x,y) = (E (x), C(x,y))$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:italic>E</jats:italic> is an expanding map of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723001219_inline2.png" /> <jats:tex-math> $mathbb {T}^u$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:italic>C</jats:italic> is a contracting map on each fiber. If <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723001219_inline3.png" /> <jats:tex-math> $inf |!det DT| inf | (D_yC)^{-1}| ^{-2s}>1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for some <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723001219_inline4.png" /> <jats:tex-math> ${s<r-(({u+d})/{2}+1)}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723001219_inline5.png" /> <jats:tex-math> $r geq 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:italic>T</jats:italic> satisfies a transversality condition between overlaps of iterates of <jats:italic>T</jats:italic> (a condition which we prove to be <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723001219_inline6.png" /> <jats:tex-math> $C^r$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-generic under mild assumptions), then the SRB measure <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723001219_inline7.png" /> <jats:tex-math> $mu _T$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:italic>T</jats:italic> is absolutely continuous and its density <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723001219_inline8.png" /> <jats:tex-math> $h_T$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> belongs to the Sobolev space <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S01433857230
我们证明了一类高维双曲内定形存在绝对连续的不变概率，这些概率的密度是有规律的，并且相对于动力系统是微分变化的。我们考虑的映射是由 $T(x,y) = (E (x), C(x,y))$ 给出的偏积，其中 E 是 $mathbb {T}^u$ 的扩张映射，C 是每个纤维上的收缩映射。如果 $inf |！det DT| inf | (D_yC)^{-1}| ^{-2s}>1$ for some ${s<；r-(({u+d})/{2}+1)}$，$r geq 2$，并且 T 满足 T 的迭代重叠之间的横向性条件（在温和的假设条件下，我们证明这个条件是$C^r$ -通用的）、那么 T 的 SRB 度量 $mu _T$ 是绝对连续的，其密度 $h_T$ 属于 Sobolev 空间 $H^s({mathbb {T}}^utimes {mathbb {R}}^d)$ 。当 $s> {u}/{2}$ 时，密度 $h_T$ 相对于 T 是可微分的也是有效的。对于接近几何势的热力学量，也证明了类似的结果。
{"title":"Regularity and linear response formula of the SRB measures for solenoidal attractors","authors":"CARLOS BOCKER, RICARDO BORTOLOTTI, ARMANDO CASTRO","doi":"10.1017/etds.2023.121","DOIUrl":"https://doi.org/10.1017/etds.2023.121","url":null,"abstract":"We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose densities are regular and vary differentiably with respect to the dynamical system. The maps we consider are skew-products given by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001219_inline1.png\" /> <jats:tex-math> $T(x,y) = (E (x), C(x,y))$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:italic>E</jats:italic> is an expanding map of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001219_inline2.png\" /> <jats:tex-math> $mathbb {T}^u$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:italic>C</jats:italic> is a contracting map on each fiber. If <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001219_inline3.png\" /> <jats:tex-math> $inf |!det DT| inf | (D_yC)^{-1}| ^{-2s}>1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for some <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001219_inline4.png\" /> <jats:tex-math> ${s<r-(({u+d})/{2}+1)}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001219_inline5.png\" /> <jats:tex-math> $r geq 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:italic>T</jats:italic> satisfies a transversality condition between overlaps of iterates of <jats:italic>T</jats:italic> (a condition which we prove to be <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001219_inline6.png\" /> <jats:tex-math> $C^r$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-generic under mild assumptions), then the SRB measure <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001219_inline7.png\" /> <jats:tex-math> $mu _T$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:italic>T</jats:italic> is absolutely continuous and its density <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385723001219_inline8.png\" /> <jats:tex-math> $h_T$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> belongs to the Sobolev space <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S01433857230","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"15 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139771801","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}
<p>Let <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline1.png"><span data-mathjax-type="texmath"><span>$(X,T)$</span></span></img></span></span> and <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline2.png"><span data-mathjax-type="texmath"><span>$(Y,S)$</span></span></img></span></span> be two topological dynamical systems, where <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline3.png"><span data-mathjax-type="texmath"><span>$(X,T)$</span></span></img></span></span> has the weak specification property. Let <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline4.png"><span data-mathjax-type="texmath"><span>$xi $</span></span></img></span></span> be an invariant measure on the product system <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline5.png"><span data-mathjax-type="texmath"><span>$(Xtimes Y, Ttimes S)$</span></span></img></span></span> with marginals <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline6.png"><span data-mathjax-type="texmath"><span>$mu $</span></span></img></span></span> on <span>X</span> and <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline7.png"><span data-mathjax-type="texmath"><span>$nu $</span></span></img></span></span> on <span>Y</span>, with <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline8.png"><span data-mathjax-type="texmath"><span>$mu $</span></span></img></span></span> ergodic. Let <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline9.png"><span data-mathjax-type="texmath"><span>$yin Y$</span></span></img></span></span> be quasi-generic for <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190
{"title":"Lifting generic points","authors":"TOMASZ DOWNAROWICZ, BENJAMIN WEISS","doi":"10.1017/etds.2023.119","DOIUrl":"https://doi.org/10.1017/etds.2023.119","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$(X,T)$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$(Y,S)$</span></span></img></span></span> be two topological dynamical systems, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$(X,T)$</span></span></img></span></span> has the weak specification property. Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$xi $</span></span></img></span></span> be an invariant measure on the product system <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$(Xtimes Y, Ttimes S)$</span></span></img></span></span> with marginals <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$mu $</span></span></img></span></span> on <span>X</span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$nu $</span></span></img></span></span> on <span>Y</span>, with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$mu $</span></span></img></span></span> ergodic. Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190:S0143385723001190_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$yin Y$</span></span></img></span></span> be quasi-generic for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203091301543-0170:S0143385723001190","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"9 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139688522","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 apply the Evans–Kishimoto intertwining argument to the classification of actions of discrete amenable groups into the normalizer of a full group of an ergodic transformation. Our proof does not depend on the types of ergodic transformations.
{"title":"Actions of discrete amenable groups into the normalizers of full groups of ergodic transformations","authors":"TOSHIHIKO MASUDA","doi":"10.1017/etds.2023.122","DOIUrl":"https://doi.org/10.1017/etds.2023.122","url":null,"abstract":"<p>We apply the Evans–Kishimoto intertwining argument to the classification of actions of discrete amenable groups into the normalizer of a full group of an ergodic transformation. Our proof does not depend on the types of ergodic transformations.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139688689","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 define the topological multiplicity of an invertible topological system $(X,T)$ as the minimal number k of real continuous functions $f_1,ldots , f_k$ such that the functions $f_icirc T^n$, $nin {mathbb {Z}}$, $1leq ileq k,$ span a dense linear vector space in the space of real continuous functions on X endowed with the supremum norm. We study some properties of topological systems with finite multiplicity. After giving some examples, we investigate the multiplicity of subshifts with linear growth complexity.
{"title":"Multiplicity of topological systems","authors":"DAVID BURGUET, RUXI SHI","doi":"10.1017/etds.2023.118","DOIUrl":"https://doi.org/10.1017/etds.2023.118","url":null,"abstract":"<p>We define the topological multiplicity of an invertible topological system <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203084901549-0722:S0143385723001189:S0143385723001189_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$(X,T)$</span></span></img></span></span> as the minimal number <span>k</span> of real continuous functions <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203084901549-0722:S0143385723001189:S0143385723001189_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$f_1,ldots , f_k$</span></span></img></span></span> such that the functions <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203084901549-0722:S0143385723001189:S0143385723001189_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$f_icirc T^n$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203084901549-0722:S0143385723001189:S0143385723001189_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$nin {mathbb {Z}}$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203084901549-0722:S0143385723001189:S0143385723001189_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$1leq ileq k,$</span></span></img></span></span> span a dense linear vector space in the space of real continuous functions on <span>X</span> endowed with the supremum norm. We study some properties of topological systems with finite multiplicity. After giving some examples, we investigate the multiplicity of subshifts with linear growth complexity.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"9 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139688710","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}
<p>Every Thurston map <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline1.png"><span data-mathjax-type="texmath"><span>$fcolon S^2rightarrow S^2$</span></span></img></span></span> on a <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline2.png"><span data-mathjax-type="texmath"><span>$2$</span></span></img></span></span>-sphere <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline3.png"><span data-mathjax-type="texmath"><span>$S^2$</span></span></img></span></span> induces a pull-back operation on Jordan curves <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline4.png"><span data-mathjax-type="texmath"><span>$alpha subset S^2smallsetminus {P_f}$</span></span></img></span></span>, where <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline5.png"><span data-mathjax-type="texmath"><span>${P_f}$</span></span></img></span></span> is the postcritical set of <span>f</span>. Here the isotopy class <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline6.png"><span data-mathjax-type="texmath"><span>$[f^{-1}(alpha )]$</span></span></img></span></span> (relative to <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline7.png"><span data-mathjax-type="texmath"><span>${P_f}$</span></span></img></span></span>) only depends on the isotopy class <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline8.png"><span data-mathjax-type="texmath"><span>$[alpha ]$</span></span></img></span></span>. We study this operation for Thurston maps with four postcritical points. In this case, a Thurston obstruction for the map <span>f</span> can be seen as a fixed point of the pull-back operation. We show that if a Thurston map <span>f</span> with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can ‘blow up’ suitable arcs in the underlying <span><span><img data-mimesubtype="png" data-type="" src="https:/
{"title":"Eliminating Thurston obstructions and controlling dynamics on curves","authors":"MARIO BONK, MIKHAIL HLUSHCHANKA, ANNINA ISELI","doi":"10.1017/etds.2023.114","DOIUrl":"https://doi.org/10.1017/etds.2023.114","url":null,"abstract":"<p>Every Thurston map <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$fcolon S^2rightarrow S^2$</span></span></img></span></span> on a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$2$</span></span></img></span></span>-sphere <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$S^2$</span></span></img></span></span> induces a pull-back operation on Jordan curves <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$alpha subset S^2smallsetminus {P_f}$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline5.png\"><span data-mathjax-type=\"texmath\"><span>${P_f}$</span></span></img></span></span> is the postcritical set of <span>f</span>. Here the isotopy class <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$[f^{-1}(alpha )]$</span></span></img></span></span> (relative to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline7.png\"><span data-mathjax-type=\"texmath\"><span>${P_f}$</span></span></img></span></span>) only depends on the isotopy class <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240116121843232-0091:S0143385723001141:S0143385723001141_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$[alpha ]$</span></span></img></span></span>. We study this operation for Thurston maps with four postcritical points. In this case, a Thurston obstruction for the map <span>f</span> can be seen as a fixed point of the pull-back operation. We show that if a Thurston map <span>f</span> with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can ‘blow up’ suitable arcs in the underlying <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https:/","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"24 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139481359","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 2 Cover and Back matter","authors":"","doi":"10.1017/etds.2023.80","DOIUrl":"https://doi.org/10.1017/etds.2023.80","url":null,"abstract":"","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"37 2","pages":"b1 - b2"},"PeriodicalIF":0.9,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139380041","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 prove a joint partial equidistribution result for common perpendiculars with given density on equidistributing equidistant hypersurfaces, towards a measure supported on truncated stable leaves. We recover a result of Marklof on the joint partial equidistribution of Farey fractions at a given density, and give several analogous arithmetic applications, including in Bruhat–Tits trees.
{"title":"Joint partial equidistribution of Farey rays in negatively curved manifolds and trees","authors":"JOUNI PARKKONEN, FRÉDÉRIC PAULIN","doi":"10.1017/etds.2023.116","DOIUrl":"https://doi.org/10.1017/etds.2023.116","url":null,"abstract":"<p>We prove a joint partial equidistribution result for common perpendiculars with given density on equidistributing equidistant hypersurfaces, towards a measure supported on truncated stable leaves. We recover a result of Marklof on the joint partial equidistribution of Farey fractions at a given density, and give several analogous arithmetic applications, including in Bruhat–Tits trees.</p>","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"8 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139396993","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 2 Cover and Front matter","authors":"","doi":"10.1017/etds.2023.79","DOIUrl":"https://doi.org/10.1017/etds.2023.79","url":null,"abstract":"","PeriodicalId":50504,"journal":{"name":"Ergodic Theory and Dynamical Systems","volume":"34 11","pages":"f1 - f2"},"PeriodicalIF":0.9,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139380099","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}
$(X,T)$ as the minimal number k of real continuous functions 
$f_1,ldots , f_k$ such that the functions 
$f_icirc T^n$, 
$nin {mathbb {Z}}$, 
$1leq ileq k,$ span a dense linear vector space in the space of real continuous functions on X endowed with the supremum norm. We study some properties of topological systems with finite multiplicity. After giving some examples, we investigate the multiplicity of subshifts with linear growth complexity.
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