We construct some positive entropy automorphisms of rational surfaces with no periodic curves. The surfaces in question, which we term tri-Coble surfaces, are blow-ups of the projective plane at 12 points which have contractions down to three different Coble surfaces. The automorphisms arise as compositions of lifts of Bertini involutions from certain degree 1 weak del Pezzo surfaces.
{"title":"Tri-Coble surfaces and their automorphisms","authors":"John Lesieutre","doi":"10.3934/JMD.2021008","DOIUrl":"https://doi.org/10.3934/JMD.2021008","url":null,"abstract":"We construct some positive entropy automorphisms of rational surfaces with no periodic curves. The surfaces in question, which we term tri-Coble surfaces, are blow-ups of the projective plane at 12 points which have contractions down to three different Coble surfaces. The automorphisms arise as compositions of lifts of Bertini involutions from certain degree 1 weak del Pezzo surfaces.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41589835","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}
For negative-torsion maps on the annulus we show that on every $mathcal{C}^1$ essential curve there is at least one point of zero torsion. As an outcome, we deduce that the Hausdorff dimension of the set of points of zero torsion is greater or equal 1. As a byproduct, we obtain a Birkhoff's-theorem-like result for $mathcal{C}^1$ essential curves in the framework of negative-torsion maps.
{"title":"On the set of points of zero torsion for negative-torsion maps of the annulus","authors":"Anna Florio","doi":"10.3934/jmd.2022017","DOIUrl":"https://doi.org/10.3934/jmd.2022017","url":null,"abstract":"For negative-torsion maps on the annulus we show that on every $mathcal{C}^1$ essential curve there is at least one point of zero torsion. As an outcome, we deduce that the Hausdorff dimension of the set of points of zero torsion is greater or equal 1. As a byproduct, we obtain a Birkhoff's-theorem-like result for $mathcal{C}^1$ essential curves in the framework of negative-torsion maps.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46385664","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}
We consider straight line flows on a translation surface that are minimal but not uniquely ergodic. We give bounds for the number of generic invariant probability measures.
我们考虑平移表面上的直线流是最小的，但不是唯一遍历的。我们给出了一般不变概率测度的个数的界限。
{"title":"Generic measures for translation surface flows","authors":"H. Masur","doi":"10.3934/jmd.2022014","DOIUrl":"https://doi.org/10.3934/jmd.2022014","url":null,"abstract":"<p style='text-indent:20px;'>We consider straight line flows on a translation surface that are minimal but not uniquely ergodic. We give bounds for the number of generic invariant probability measures.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44134621","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}
Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the begin{document}$ i^text{th} $end{document} and begin{document}$ (i+1)^text{th} $end{document} Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index begin{document}$ i $end{document}. We establish a similar result for sofic subshifts coming from word hyperbolic groups, in relation with Anosov representations of such groups. We discuss the case of finitely generated semigroups, and propose a notion of Anosov representation in this setting.
{"title":"Eigenvalue gaps for hyperbolic groups and semigroups","authors":"Fanny Kassel, R. Potrie","doi":"10.3934/jmd.2022008","DOIUrl":"https://doi.org/10.3934/jmd.2022008","url":null,"abstract":"<p style='text-indent:20px;'>Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the <inline-formula><tex-math id=\"M1\">begin{document}$ i^text{th} $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M2\">begin{document}$ (i+1)^text{th} $end{document}</tex-math></inline-formula> Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index <inline-formula><tex-math id=\"M3\">begin{document}$ i $end{document}</tex-math></inline-formula>. We establish a similar result for sofic subshifts coming from word hyperbolic groups, in relation with Anosov representations of such groups. We discuss the case of finitely generated semigroups, and propose a notion of Anosov representation in this setting.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45658136","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}
We review recent advances in the spectral approach to studying statistical properties of dynamical systems highlighting, in particular, the role played by Sebastien Gouezel.
{"title":"The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces","authors":"D. Dolgopyat","doi":"10.3934/jmd.2020014","DOIUrl":"https://doi.org/10.3934/jmd.2020014","url":null,"abstract":"We review recent advances in the spectral approach to studying statistical properties of dynamical systems highlighting, in particular, the role played by Sebastien Gouezel.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70084625","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}
We study joining rigidity in the class of von Neumann flows with one singularity. They are given by a smooth vector field begin{document}$ mathscr{X} $end{document} on begin{document}$ mathbb{T}^2setminus {a} $end{document} , where begin{document}$ mathscr{X} $end{document} is not defined at begin{document}$ ain mathbb{T}^2 $end{document} and begin{document}$ mathscr{X} $end{document} has one critical point which is a center. It follows that the phase space can be decomposed into a (topological disc) begin{document}$ D_mathscr{X} $end{document} and an ergodic component begin{document}$ E_mathscr{X} = mathbb{T}^2setminus D_mathscr{X} $end{document} . Let begin{document}$ omega_mathscr{X} $end{document} be the 1-form associated to begin{document}$ mathscr{X} $end{document} . We show that if begin{document}$ |int_{E_{mathscr{X}_1}}domega_{mathscr{X}_1}|neq |int_{E_{mathscr{X}_2}}domega_{mathscr{X}_2}| $end{document} , then the corresponding flows begin{document}$ (v_t^{mathscr{X}_1}) $end{document} and begin{document}$ (v_t^{mathscr{X}_2}) $end{document} are disjoint. It also follows that for every begin{document}$ mathscr{X} $end{document} there is a uniquely associated frequency begin{document}$ alpha = alpha_{mathscr{X}}in mathbb{T} $end{document} . We show that for a full measure set of begin{document}$ alphain mathbb{T} $end{document} the class of smooth time changes of begin{document}$ (v_t^mathscr{X_ alpha}) $end{document} is joining rigid, i.e., every two smooth time changes are either cohomologous or disjoint. This gives a natural class of flows for which the answer to [ 15 ,Problem 3] is positive.
We study joining rigidity in the class of von Neumann flows with one singularity. They are given by a smooth vector field begin{document}$ mathscr{X} $end{document} on begin{document}$ mathbb{T}^2setminus {a} $end{document} , where begin{document}$ mathscr{X} $end{document} is not defined at begin{document}$ ain mathbb{T}^2 $end{document} and begin{document}$ mathscr{X} $end{document} has one critical point which is a center. It follows that the phase space can be decomposed into a (topological disc) begin{document}$ D_mathscr{X} $end{document} and an ergodic component begin{document}$ E_mathscr{X} = mathbb{T}^2setminus D_mathscr{X} $end{document} . Let begin{document}$ omega_mathscr{X} $end{document} be the 1-form associated to begin{document}$ mathscr{X} $end{document} . We show that if begin{document}$ |int_{E_{mathscr{X}_1}}domega_{mathscr{X}_1}|neq |int_{E_{mathscr{X}_2}}domega_{mathscr{X}_2}| $end{document} , then the corresponding flows begin{document}$ (v_t^{mathscr{X}_1}) $end{document} and begin{document}$ (v_t^{mathscr{X}_2}) $end{document} are disjoint. It also follows that for every begin{document}$ mathscr{X} $end{document} there is a uniquely associated frequency begin{document}$ alpha = alpha_{mathscr{X}}in mathbb{T} $end{document} . We show that for a full measure set of begin{document}$ alphain mathbb{T} $end{document} the class of smooth time changes of begin{document}$ (v_t^mathscr{X_ alpha}) $end{document} is joining rigid, i.e., every two smooth time changes are either cohomologous or disjoint. This gives a natural class of flows for which the answer to [ 15 ,Problem 3] is positive.
{"title":"Rigidity of a class of smooth singular flows on begin{document}$ mathbb{T}^2 $end{document}","authors":"Changguang Dong, Adam Kanigowski","doi":"10.3934/jmd.2020002","DOIUrl":"https://doi.org/10.3934/jmd.2020002","url":null,"abstract":"We study joining rigidity in the class of von Neumann flows with one singularity. They are given by a smooth vector field begin{document}$ mathscr{X} $end{document} on begin{document}$ mathbb{T}^2setminus {a} $end{document} , where begin{document}$ mathscr{X} $end{document} is not defined at begin{document}$ ain mathbb{T}^2 $end{document} and begin{document}$ mathscr{X} $end{document} has one critical point which is a center. It follows that the phase space can be decomposed into a (topological disc) begin{document}$ D_mathscr{X} $end{document} and an ergodic component begin{document}$ E_mathscr{X} = mathbb{T}^2setminus D_mathscr{X} $end{document} . Let begin{document}$ omega_mathscr{X} $end{document} be the 1-form associated to begin{document}$ mathscr{X} $end{document} . We show that if begin{document}$ |int_{E_{mathscr{X}_1}}domega_{mathscr{X}_1}|neq |int_{E_{mathscr{X}_2}}domega_{mathscr{X}_2}| $end{document} , then the corresponding flows begin{document}$ (v_t^{mathscr{X}_1}) $end{document} and begin{document}$ (v_t^{mathscr{X}_2}) $end{document} are disjoint. It also follows that for every begin{document}$ mathscr{X} $end{document} there is a uniquely associated frequency begin{document}$ alpha = alpha_{mathscr{X}}in mathbb{T} $end{document} . We show that for a full measure set of begin{document}$ alphain mathbb{T} $end{document} the class of smooth time changes of begin{document}$ (v_t^mathscr{X_ alpha}) $end{document} is joining rigid, i.e., every two smooth time changes are either cohomologous or disjoint. This gives a natural class of flows for which the answer to [ 15 ,Problem 3] is positive.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":"16 1","pages":"37-57"},"PeriodicalIF":1.1,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70084554","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}
For a fixed frequency vector $omega in mathbb{R}^2 , setminus , lbrace 0 rbrace$ obeying $omega_1 omega_2 < 0$ we show the existence of Gevrey-smooth Hamiltonians, arbitrarily close to an integrable Kolmogorov non-degenerate analytic Hamiltonian, having a Lyapunov unstable elliptic equilibrium with frequency $omega$. In particular, the elliptic fixed points thus constructed will be KAM stable, i.e. accumulated by invariant tori whose Lebesgue density tend to one in the neighbourhood of the point and whose frequencies cover a set of positive measure. Similar examples for near-integrable Hamiltonians in action-angle coordinates in the neighbourhood of a Lagragian invariant torus with arbitrary rotation vector are also given in this work.
{"title":"Lyapunov instability in KAM stable Hamiltonians with two degrees of freedom","authors":"Frank Trujillo","doi":"10.3934/jmd.2023010","DOIUrl":"https://doi.org/10.3934/jmd.2023010","url":null,"abstract":"For a fixed frequency vector $omega in mathbb{R}^2 , setminus , lbrace 0 rbrace$ obeying $omega_1 omega_2 < 0$ we show the existence of Gevrey-smooth Hamiltonians, arbitrarily close to an integrable Kolmogorov non-degenerate analytic Hamiltonian, having a Lyapunov unstable elliptic equilibrium with frequency $omega$. In particular, the elliptic fixed points thus constructed will be KAM stable, i.e. accumulated by invariant tori whose Lebesgue density tend to one in the neighbourhood of the point and whose frequencies cover a set of positive measure. \u0000Similar examples for near-integrable Hamiltonians in action-angle coordinates in the neighbourhood of a Lagragian invariant torus with arbitrary rotation vector are also given in this work.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45013541","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}
We study the regularity of a conjugacy between an Anosov automorphism $L$ of a nilmanifold $N/Gamma$ and a volume-preserving, $C^1$-small perturbation $f$. We say that $L$ is locally Lyapunov spectrum rigid if this conjugacy is $C^{1+}$ whenever $f$ is $C^{1+}$ and has the same volume Lyapunov spectrum as $L$. For $L$ with simple spectrum, we show that local Lyapunov spectrum rigidity is equivalent to $L$ satisfying both an irreducibility condition and an ordering condition on its Lyapunov exponents.
{"title":"Local Lyapunov spectrum rigidity of nilmanifold automorphisms","authors":"Jonathan DeWitt","doi":"10.3934/JMD.2021003","DOIUrl":"https://doi.org/10.3934/JMD.2021003","url":null,"abstract":"We study the regularity of a conjugacy between an Anosov automorphism $L$ of a nilmanifold $N/Gamma$ and a volume-preserving, $C^1$-small perturbation $f$. We say that $L$ is locally Lyapunov spectrum rigid if this conjugacy is $C^{1+}$ whenever $f$ is $C^{1+}$ and has the same volume Lyapunov spectrum as $L$. For $L$ with simple spectrum, we show that local Lyapunov spectrum rigidity is equivalent to $L$ satisfying both an irreducibility condition and an ordering condition on its Lyapunov exponents.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41386825","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}
Let begin{document}$ f_c(z) = z^2+c $end{document} for begin{document}$ c in {mathbb C} $end{document}. We show there exists a uniform upper bound on the number of points in begin{document}$ {mathbb P}^1( {mathbb C}) $end{document} that can be preperiodic for both begin{document}$ f_{c_1} $end{document} and begin{document}$ f_{c_2} $end{document}, for any pair begin{document}$ c_1not = c_2 $end{document} in begin{document}$ {mathbb C} $end{document}. The proof combines arithmetic ingredients with complex-analytic: we estimate an adelic energy pairing when the parameters lie in begin{document}$ overline{mathbb{Q}} $end{document}, building on the quantitative arithmetic equidistribution theorem of Favre and Rivera-Letelier, and we use distortion theorems in complex analysis to control the size of the intersection of distinct Julia sets. The proofs are effective, and we provide explicit constants for each of the results.
Let begin{document}$ f_c(z) = z^2+c $end{document} for begin{document}$ c in {mathbb C} $end{document}. We show there exists a uniform upper bound on the number of points in begin{document}$ {mathbb P}^1( {mathbb C}) $end{document} that can be preperiodic for both begin{document}$ f_{c_1} $end{document} and begin{document}$ f_{c_2} $end{document}, for any pair begin{document}$ c_1not = c_2 $end{document} in begin{document}$ {mathbb C} $end{document}. The proof combines arithmetic ingredients with complex-analytic: we estimate an adelic energy pairing when the parameters lie in begin{document}$ overline{mathbb{Q}} $end{document}, building on the quantitative arithmetic equidistribution theorem of Favre and Rivera-Letelier, and we use distortion theorems in complex analysis to control the size of the intersection of distinct Julia sets. The proofs are effective, and we provide explicit constants for each of the results.
{"title":"Common preperiodic points for quadratic polynomials","authors":"Laura Demarco, Holly Krieger, Hexi Ye","doi":"10.3934/jmd.2022012","DOIUrl":"https://doi.org/10.3934/jmd.2022012","url":null,"abstract":"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\"M1\">begin{document}$ f_c(z) = z^2+c $end{document}</tex-math></inline-formula> for <inline-formula><tex-math id=\"M2\">begin{document}$ c in {mathbb C} $end{document}</tex-math></inline-formula>. We show there exists a uniform upper bound on the number of points in <inline-formula><tex-math id=\"M3\">begin{document}$ {mathbb P}^1( {mathbb C}) $end{document}</tex-math></inline-formula> that can be preperiodic for both <inline-formula><tex-math id=\"M4\">begin{document}$ f_{c_1} $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M5\">begin{document}$ f_{c_2} $end{document}</tex-math></inline-formula>, for any pair <inline-formula><tex-math id=\"M6\">begin{document}$ c_1not = c_2 $end{document}</tex-math></inline-formula> in <inline-formula><tex-math id=\"M7\">begin{document}$ {mathbb C} $end{document}</tex-math></inline-formula>. The proof combines arithmetic ingredients with complex-analytic: we estimate an adelic energy pairing when the parameters lie in <inline-formula><tex-math id=\"M8\">begin{document}$ overline{mathbb{Q}} $end{document}</tex-math></inline-formula>, building on the quantitative arithmetic equidistribution theorem of Favre and Rivera-Letelier, and we use distortion theorems in complex analysis to control the size of the intersection of distinct Julia sets. The proofs are effective, and we provide explicit constants for each of the results.</p>","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48391614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the entropy function on the moduli space of real quadratic rational maps is not monotonic by exhibiting a continuum of disconnected level sets. This entropy behavior is in stark contrast with the case of polynomial maps, and establishes a conjecture on the failure of monotonicity for bimodal real quadratic rational maps of shape begin{document}$ (+-+) $end{document} which was posed in [ 10 ] based on experimental evidence.
We prove that the entropy function on the moduli space of real quadratic rational maps is not monotonic by exhibiting a continuum of disconnected level sets. This entropy behavior is in stark contrast with the case of polynomial maps, and establishes a conjecture on the failure of monotonicity for bimodal real quadratic rational maps of shape begin{document}$ (+-+) $end{document} which was posed in [ 10 ] based on experimental evidence.
{"title":"On the non-monotonicity of entropy for a class of real quadratic rational maps","authors":"Khashayar Filom, K. Pilgrim","doi":"10.3934/JMD.2020008","DOIUrl":"https://doi.org/10.3934/JMD.2020008","url":null,"abstract":"We prove that the entropy function on the moduli space of real quadratic rational maps is not monotonic by exhibiting a continuum of disconnected level sets. This entropy behavior is in stark contrast with the case of polynomial maps, and establishes a conjecture on the failure of monotonicity for bimodal real quadratic rational maps of shape begin{document}$ (+-+) $end{document} which was posed in [ 10 ] based on experimental evidence.","PeriodicalId":51087,"journal":{"name":"Journal of Modern Dynamics","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43207775","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}
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