Pub Date : 2022-09-19DOI: 10.1007/s40598-022-00218-x
Trevor Clark, Kostiantyn Drach, Oleg Kozlovski, Sebastian van Strien
{"title":"Correction to: The Dynamics of Complex Box Mappings","authors":"Trevor Clark, Kostiantyn Drach, Oleg Kozlovski, Sebastian van Strien","doi":"10.1007/s40598-022-00218-x","DOIUrl":"10.1007/s40598-022-00218-x","url":null,"abstract":"","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-022-00218-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50497531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-16DOI: 10.1007/s40598-022-00217-y
Kiumars Kaveh, Peter Makhnatch
The paper concerns a result in linear algebra motivated by ideas from tropical geometry. Let A(t) be an (n times n) matrix whose entries are Laurent series in t. We show that, as (t rightarrow 0), the logarithms of singular values of A(t) approach the invariant factors of A(t). This leads us to suggest logarithms of singular values of an (n times n) complex matrix as an analog of the logarithm map on ((mathbb {C}^*)^n) for the matrix group ({text {GL}}(n, mathbb {C})).
{"title":"Invariant Factors as Limit of Singular Values of a Matrix","authors":"Kiumars Kaveh, Peter Makhnatch","doi":"10.1007/s40598-022-00217-y","DOIUrl":"10.1007/s40598-022-00217-y","url":null,"abstract":"<div><p>The paper concerns a result in linear algebra motivated by ideas from tropical geometry. Let <i>A</i>(<i>t</i>) be an <span>(n times n)</span> matrix whose entries are Laurent series in <i>t</i>. We show that, as <span>(t rightarrow 0)</span>, the logarithms of singular values of <i>A</i>(<i>t</i>) approach the invariant factors of <i>A</i>(<i>t</i>). This leads us to suggest logarithms of singular values of an <span>(n times n)</span> complex matrix as an analog of the logarithm map on <span>((mathbb {C}^*)^n)</span> for the matrix group <span>({text {GL}}(n, mathbb {C}))</span>.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-022-00217-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43034161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-12DOI: 10.1007/s40598-022-00213-2
Gil Bor, Connor Jackman
The Kepler orbits form a 3-parameter family of unparametrized plane curves, consisting of all conics sharing a focus at a fixed point. We study the geometry and symmetry properties of this family, as well as natural 2-parameter subfamilies, such as those of fixed energy or angular momentum. Our main result is that Kepler orbits is a ‘flat’ family, that is, the local diffeomorphisms of the plane preserving this family form a 7-dimensional local group, the maximum dimension possible for the symmetry group of a 3-parameter family of plane curves. These symmetries are different from the well-studied ‘hidden’ symmetries of the Kepler problem, acting on energy levels in the 4-dimensional phase space of the Kepler system. Each 2-parameter subfamily of Kepler orbits with fixed non-zero energy (Kepler ellipses or hyperbolas with fixed length of major axis) admits (mathrm { PSL}_2(mathbb {R})) as its (local) symmetry group, corresponding to one of the items of a classification due to Tresse (Détermination des invariants ponctuels de l’équation différentielle ordinaire du second ordre (y^{prime prime }= omega (x, y, y^{prime })), vol. 32, S. Hirzel, 1896) of 2-parameter families of plane curves admitting a 3-dimensional local group of symmetries. The 2-parameter subfamilies with zero energy (Kepler parabolas) or fixed non-zero angular momentum are flat (locally diffeomorphic to the family of straight lines). These results can be proved using techniques developed in the nineteenth century by Lie to determine ‘infinitesimal point symmetries’ of ODEs, but our proofs are much simpler, using a projective geometric model for the Kepler orbits (plane sections of a cone in projective 3-space). In this projective model, all symmetry groups act globally. Another advantage of the projective model is a duality between Kepler’s plane and Minkowski’s 3-space parametrizing the space of Kepler orbits. We use this duality to deduce several results on the Kepler system, old and new.
开普勒轨道形成了一个非三参数平面曲线族,由所有在固定点共享焦点的圆锥组成。我们研究了这个族的几何和对称性质,以及自然的双参数亚族,例如固定能量或角动量的亚族。我们的主要结果是开普勒轨道是一个“平坦”族,即保持该族的平面的局部微分同胚形成一个7维局部群,这是平面曲线的3参数族的对称群可能的最大维数。这些对称性不同于研究充分的开普勒问题的“隐藏”对称性,它们作用于开普勒系统的4维相空间中的能级。具有固定非零能量的开普勒轨道的每个2-参数子族(长轴长度固定的开普勒椭圆或双曲线)都承认(mathrm{PSL}_2(mathbb{R}))为其(局部)对称群,对应于由Tresse(Détermination des不变量ponctuels de l’équation différentielle ordinaire du second ordre (y^{primeprime}=omega(x,y,y^})),第32卷,S.Hirzel,1896)引起的平面曲线的2-参数族的分类的一个项目,其允许三维局部对称性组。具有零能量(开普勒抛物面)或固定非零角动量的2-参数亚族是平坦的(与直线族局部微分同胚)。这些结果可以使用李在19世纪开发的技术来证明,以确定常微分方程的“无穷小点对称性”,但我们的证明要简单得多,使用开普勒轨道的投影几何模型(投影3-空间中圆锥的平面截面)。在这个投影模型中,所有对称群都是全局作用的。投影模型的另一个优点是开普勒平面和闵可夫斯基3空间之间的对偶性,参数化了开普勒轨道的空间。我们利用这种对偶性来推导开普勒系统的几个结果,无论是旧的还是新的。
{"title":"Revisiting Kepler: New Symmetries of an Old Problem","authors":"Gil Bor, Connor Jackman","doi":"10.1007/s40598-022-00213-2","DOIUrl":"10.1007/s40598-022-00213-2","url":null,"abstract":"<div><p>The <i>Kepler orbits</i> form a 3-parameter family of <i>unparametrized</i> plane curves, consisting of all conics sharing a focus at a fixed point. We study the geometry and symmetry properties of this family, as well as natural 2-parameter subfamilies, such as those of fixed energy or angular momentum. Our main result is that Kepler orbits is a ‘flat’ family, that is, the local diffeomorphisms of the plane preserving this family form a 7-dimensional local group, the maximum dimension possible for the symmetry group of a 3-parameter family of plane curves. These symmetries are different from the well-studied ‘hidden’ symmetries of the Kepler problem, acting on energy levels in the 4-dimensional phase space of the Kepler system. Each 2-parameter subfamily of Kepler orbits with fixed non-zero energy (Kepler ellipses or hyperbolas with fixed length of major axis) admits <span>(mathrm { PSL}_2(mathbb {R}))</span> as its (local) symmetry group, corresponding to one of the items of a classification due to Tresse (Détermination des invariants ponctuels de l’équation différentielle ordinaire du second ordre <span>(y^{prime prime }= omega (x, y, y^{prime }))</span>, vol. 32, S. Hirzel, 1896) of 2-parameter families of plane curves admitting a 3-dimensional local group of symmetries. The 2-parameter subfamilies with zero energy (Kepler parabolas) or fixed non-zero angular momentum are flat (locally diffeomorphic to the family of straight lines). These results can be proved using techniques developed in the nineteenth century by Lie to determine ‘infinitesimal point symmetries’ of ODEs, but our proofs are much simpler, using a projective geometric model for the Kepler orbits (plane sections of a cone in projective 3-space). In this projective model, all symmetry groups act globally. Another advantage of the projective model is a duality between Kepler’s plane and Minkowski’s 3-space parametrizing the space of Kepler orbits. We use this duality to deduce several results on the Kepler system, old and new.\u0000</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-022-00213-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43059024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-29DOI: 10.1007/s40598-022-00216-z
Jayadev S. Athreya, David Aulicino, Harry Richman
Motivated by the problem of counting finite BPS webs, we count certain immersed metric graphs, tripods, on the flat torus. Classical Euclidean geometry turns this into a lattice point counting problem in ({mathbb {C}}^2), and we give an asymptotic counting result using lattice point counting techniques.
{"title":"Counting Tripods on the Torus","authors":"Jayadev S. Athreya, David Aulicino, Harry Richman","doi":"10.1007/s40598-022-00216-z","DOIUrl":"10.1007/s40598-022-00216-z","url":null,"abstract":"<div><p>Motivated by the problem of counting finite BPS webs, we count certain immersed metric graphs, <i>tripods</i>, on the flat torus. Classical Euclidean geometry turns this into a lattice point counting problem in <span>({mathbb {C}}^2)</span>, and we give an asymptotic counting result using lattice point counting techniques.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41344607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-24DOI: 10.1007/s40598-022-00215-0
Alexei A. Mailybaev, Artem Raibekas
We propose a simple model for the phenomenon of Eulerian spontaneous stochasticity in turbulence. This model is solved rigorously, proving that infinitesimal small-scale noise in otherwise a deterministic multi-scale system yields a large-scale stochastic process with Markovian properties. Our model shares intriguing properties with open problems of modern mathematical theory of turbulence, such as non-uniqueness of the inviscid limit, existence of wild weak solutions and explosive effect of random perturbations. Thereby, it proposes rigorous, often counterintuitive answers to these questions. Besides its theoretical value, our model opens new ways for the experimental verification of spontaneous stochasticity, and suggests new applications beyond fluid dynamics.
{"title":"Spontaneously Stochastic Arnold’s Cat","authors":"Alexei A. Mailybaev, Artem Raibekas","doi":"10.1007/s40598-022-00215-0","DOIUrl":"10.1007/s40598-022-00215-0","url":null,"abstract":"<div><p>We propose a simple model for the phenomenon of Eulerian spontaneous stochasticity in turbulence. This model is solved rigorously, proving that infinitesimal small-scale noise in otherwise a deterministic multi-scale system yields a large-scale stochastic process with Markovian properties. Our model shares intriguing properties with open problems of modern mathematical theory of turbulence, such as non-uniqueness of the inviscid limit, existence of wild weak solutions and explosive effect of random perturbations. Thereby, it proposes rigorous, often counterintuitive answers to these questions. Besides its theoretical value, our model opens new ways for the experimental verification of spontaneous stochasticity, and suggests new applications beyond fluid dynamics.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-022-00215-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46583415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-01DOI: 10.1007/s40598-022-00209-y
Trevor Clark, Kostiantyn Drach, Oleg Kozlovski, Sebastian van Strien
{"title":"Correction to: The Dynamics of Complex Box Mappings","authors":"Trevor Clark, Kostiantyn Drach, Oleg Kozlovski, Sebastian van Strien","doi":"10.1007/s40598-022-00209-y","DOIUrl":"10.1007/s40598-022-00209-y","url":null,"abstract":"","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-022-00209-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46369231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-22DOI: 10.1007/s40598-022-00211-4
Araceli Bonifant, Chad Estabrooks, Thomas Sharland
We describe a topological relationship between slices of the parameter space of cubic maps. In the paper [9], Milnor defined the curves (mathcal {S}_{p}) as the set of all cubic polynomials with a marked critical point of period p. In this paper, we will describe a relationship between the boundaries of the connectedness loci in the curves (mathcal {S}_{1}) and (mathcal {S}_{2}).
{"title":"Relations Between Escape Regions in the Parameter Space of Cubic Polynomials","authors":"Araceli Bonifant, Chad Estabrooks, Thomas Sharland","doi":"10.1007/s40598-022-00211-4","DOIUrl":"10.1007/s40598-022-00211-4","url":null,"abstract":"<div><p>We describe a topological relationship between slices of the parameter space of cubic maps. In the paper [9], Milnor defined the curves <span>(mathcal {S}_{p})</span> as the set of all cubic polynomials with a marked critical point of period <i>p</i>. In this paper, we will describe a relationship between the boundaries of the connectedness loci in the curves <span>(mathcal {S}_{1})</span> and <span>(mathcal {S}_{2})</span>.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44482491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-18DOI: 10.1007/s40598-022-00210-5
Kohei Kikuta, Genki Ouchi
We study the categorical entropy and counterexamples to Gromov–Yomdin type conjecture via homological mirror symmetry of K3 surfaces established by Sheridan–Smith. We introduce asymptotic invariants of quasi-endofunctors of dg categories, called the Hochschild entropy. It is proved that the categorical entropy is lower bounded by the Hochschild entropy. Furthermore, motivated by Thurston’s classical result, we prove the existence of a symplectic Torelli mapping class of positive categorical entropy. We also consider relations to the Floer-theoretic entropy.
{"title":"Hochschild Entropy and Categorical Entropy","authors":"Kohei Kikuta, Genki Ouchi","doi":"10.1007/s40598-022-00210-5","DOIUrl":"10.1007/s40598-022-00210-5","url":null,"abstract":"<div><p>We study the categorical entropy and counterexamples to Gromov–Yomdin type conjecture via homological mirror symmetry of K3 surfaces established by Sheridan–Smith. We introduce asymptotic invariants of quasi-endofunctors of dg categories, called the Hochschild entropy. It is proved that the categorical entropy is lower bounded by the Hochschild entropy. Furthermore, motivated by Thurston’s classical result, we prove the existence of a symplectic Torelli mapping class of positive categorical entropy. We also consider relations to the Floer-theoretic entropy.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46139264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-12DOI: 10.1007/s40598-022-00212-3
Boris Tsvelikhovskiy
We show that there are infinitely many nonisomorphic quandle structures on any topogical space X of positive dimension. In particular, we disprove Conjecture 5.2 in Cheng et al. (Topology Appl 248:64–74, 2018), asserting that there are no nontrivial quandle structures on the closed unit interval [0, 1].
{"title":"Nontrivial Topological Quandles","authors":"Boris Tsvelikhovskiy","doi":"10.1007/s40598-022-00212-3","DOIUrl":"10.1007/s40598-022-00212-3","url":null,"abstract":"<div><p>We show that there are infinitely many nonisomorphic quandle structures on any topogical space <i>X</i> of positive dimension. In particular, we disprove Conjecture 5.2 in Cheng et al. (Topology Appl 248:64–74, 2018), asserting that there are no nontrivial quandle structures on the closed unit interval [0, 1].</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48402695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-05DOI: 10.1007/s40598-022-00206-1
Grigory Kondyrev, Artem Prikhodko
We show how the formalism of 2-traces can be applied in the setting of derived algebraic geometry to obtain a generalization of the holomorphic Atiyah–Bott formula to the case when an endomorphism is replaced by a correspondence.
{"title":"Holomorphic Atiyah–Bott Formula for Correspondences","authors":"Grigory Kondyrev, Artem Prikhodko","doi":"10.1007/s40598-022-00206-1","DOIUrl":"10.1007/s40598-022-00206-1","url":null,"abstract":"<div><p>We show how the formalism of 2-traces can be applied in the setting of derived algebraic geometry to obtain a generalization of the holomorphic Atiyah–Bott formula to the case when an endomorphism is replaced by a correspondence.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42881435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}