L’un des principaux invariants dynamiques associes a un homeomorphisme de surface isotope a l’identite est son ensemble de rotation, decrivant les vitesses et directions asymptotiques moyennes selon lesquelles les points “tournent” autour de la surface sous l’action de l’homeomorphisme. Sur le tore en particulier, de nombreux resultats relient la forme ou la taille de l’ensemble de rotation a des proprietes dynamiques de l’homeomorphisme. Cette these a pour but de generaliser au cas des surfaces de genre ≥ 2 un certain nombre de resultats connus sur le tore pour les homeomorphismes ayant un “gros” ensemble de rotation : positivite de l’entropie, realisation de vecteurs de rotation par des points periodiques, deviations bornees, etc. L’outil principal utilise est la theorie de forcage de Le Calvez et Tal, reposant sur la construction d’un feuilletage transverse et l’etude des trajectoires des points relativement a ce feuilletage. Les deux premiers chapitres presentent des resultats preliminaires a ce cadre general. Au chapitre 3, nous menons une etude globale sur les cycles asymptotiques de points dont les trajectoires ont des directions homologiques qui s’intersectent. Nous montrons que cette situation suffit a assurer la positivite de l’entropie, ce qui permet d’aboutir a la generalisation de deux resultats connus sur le tore, les theoremes de Llibre-Mackay et de Franks. Enfin, au chapitre 4, nous montrons a l’aide de ce dernier resultat qu’un homeomorphisme dont l’ensemble de rotation contient 0 dans son interieur est a deviation bornee, generalisant encore une propriete connue sur le tore. Nous terminons avec diverses consequences de ce resultat.
{"title":"Sur les ensembles de rotation des homéomorphismes de surface en genre ≥ 2","authors":"Gabriel Lellouch","doi":"10.24033/msmf.486","DOIUrl":"https://doi.org/10.24033/msmf.486","url":null,"abstract":"L’un des principaux invariants dynamiques associes a un homeomorphisme de surface isotope a l’identite est son ensemble de rotation, decrivant les vitesses et directions asymptotiques moyennes selon lesquelles les points “tournent” autour de la surface sous l’action de l’homeomorphisme. Sur le tore en particulier, de nombreux resultats relient la forme ou la taille de l’ensemble de rotation a des proprietes dynamiques de l’homeomorphisme. Cette these a pour but de generaliser au cas des surfaces de genre ≥ 2 un certain nombre de resultats connus sur le tore pour les homeomorphismes ayant un “gros” ensemble de rotation : positivite de l’entropie, realisation de vecteurs de rotation par des points periodiques, deviations bornees, etc. L’outil principal utilise est la theorie de forcage de Le Calvez et Tal, reposant sur la construction d’un feuilletage transverse et l’etude des trajectoires des points relativement a ce feuilletage. Les deux premiers chapitres presentent des resultats preliminaires a ce cadre general. Au chapitre 3, nous menons une etude globale sur les cycles asymptotiques de points dont les trajectoires ont des directions homologiques qui s’intersectent. Nous montrons que cette situation suffit a assurer la positivite de l’entropie, ce qui permet d’aboutir a la generalisation de deux resultats connus sur le tore, les theoremes de Llibre-Mackay et de Franks. Enfin, au chapitre 4, nous montrons a l’aide de ce dernier resultat qu’un homeomorphisme dont l’ensemble de rotation contient 0 dans son interieur est a deviation bornee, generalisant encore une propriete connue sur le tore. Nous terminons avec diverses consequences de ce resultat.","PeriodicalId":55332,"journal":{"name":"Bulletin De La Societe Mathematique De France","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135805171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Curvature estimate and the ramification of the holomorphic maps over hypersurfaces on Riemann surfaces","authors":"Si DUC QUANG","doi":"10.24033/bsmf.2864","DOIUrl":"https://doi.org/10.24033/bsmf.2864","url":null,"abstract":"","PeriodicalId":55332,"journal":{"name":"Bulletin De La Societe Mathematique De France","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135503295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive a lower bound for all Levine-Tristram signatures of positive braid links, linear in terms of the first Betti number. As a consequence, we obtain upper and lower bounds on the ratio of fixed pairs of Levine-Tristram signature invariants, valid uniformly on all positive braid monoids.
{"title":"Signature spectrum of positive braids","authors":"Sebastian Baader","doi":"10.24033/bsmf.2865","DOIUrl":"https://doi.org/10.24033/bsmf.2865","url":null,"abstract":"We derive a lower bound for all Levine-Tristram signatures of positive braid links, linear in terms of the first Betti number. As a consequence, we obtain upper and lower bounds on the ratio of fixed pairs of Levine-Tristram signature invariants, valid uniformly on all positive braid monoids.","PeriodicalId":55332,"journal":{"name":"Bulletin De La Societe Mathematique De France","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135525613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work brings to light some partial emph{anabelian behaviours} of analytic annuli in the context of Berkovich geometry. More specifically, if $k$ is a valued non-archimedean complete field of mixed characteristic which is algebraically closed, and $mathcal{C}_1$, $mathcal{C}_2$ are two $k$-analytic annuli with isomorphic tempered fundamental group, we show that the lengths of $mathcal{C}_1$ and $mathcal{C}_2$ cannot be too far from each other. When they are finite, we show that the absolute value of their difference is bounded above with a bound depending only on the residual characteristic $p$.
{"title":"Towards tempered anabelian behaviour of Berkovich annuli","authors":"Sylvain Gaulhiac","doi":"10.24033/bsmf.2862","DOIUrl":"https://doi.org/10.24033/bsmf.2862","url":null,"abstract":"This work brings to light some partial emph{anabelian behaviours} of analytic annuli in the context of Berkovich geometry. More specifically, if $k$ is a valued non-archimedean complete field of mixed characteristic which is algebraically closed, and $mathcal{C}_1$, $mathcal{C}_2$ are two $k$-analytic annuli with isomorphic tempered fundamental group, we show that the lengths of $mathcal{C}_1$ and $mathcal{C}_2$ cannot be too far from each other. When they are finite, we show that the absolute value of their difference is bounded above with a bound depending only on the residual characteristic $p$.","PeriodicalId":55332,"journal":{"name":"Bulletin De La Societe Mathematique De France","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135195809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define a general class of random systems of horizontal and vertical weighted broken lines on the quarter plane whose distribution are proved to be translation invariant. This invariance stems from a reversibility property of the model. This class of systems generalizes several classical processes of the same kind, such as Hammersley's broken line processes involved in Last Passage Percolation theory or such as the six-vertex model for some special sets of parameters. The novelty comes here from the introduction of a weight associated with each line. The lines are initially generated by spatially homogeneous weighted Poisson Point Process and their evolution (turn, split, crossing) are ruled by a Markovian dynamics which preserves Kirchhoff's node law for the line weights at each intersection. Among others, we derive some new explicit invariant measures for some bullet models as well as new reversible properties for some six-vertex models with an external electromagnetic field.
{"title":"Reversible Poisson-Kirchhoff Systems","authors":"Alexandre Boyer, Jérôme Casse, Nathanaël Enriquez, Arvind Singh","doi":"10.24033/bsmf.2863","DOIUrl":"https://doi.org/10.24033/bsmf.2863","url":null,"abstract":"We define a general class of random systems of horizontal and vertical weighted broken lines on the quarter plane whose distribution are proved to be translation invariant. This invariance stems from a reversibility property of the model. This class of systems generalizes several classical processes of the same kind, such as Hammersley's broken line processes involved in Last Passage Percolation theory or such as the six-vertex model for some special sets of parameters. The novelty comes here from the introduction of a weight associated with each line. The lines are initially generated by spatially homogeneous weighted Poisson Point Process and their evolution (turn, split, crossing) are ruled by a Markovian dynamics which preserves Kirchhoff's node law for the line weights at each intersection. Among others, we derive some new explicit invariant measures for some bullet models as well as new reversible properties for some six-vertex models with an external electromagnetic field.","PeriodicalId":55332,"journal":{"name":"Bulletin De La Societe Mathematique De France","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135195811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
— Given a finitely generated field extension K of the rational numbers and an abelian variety C over K, we consider the class of all abelian varieties over K which are isogenous (over K) to an abelian subvariety of a power of C. We show that there is a single, naturally constructed abelian variety C in the class whose ring of endomorphisms controls all isogenies in the class. Precisely, this means that if d is the discriminant of this ring then for any pair of isogenous abelian varieties in the class there exists an isogeny between them whose kernel has exponent at most d. Furthermore we prove that, for any element A in the class, the same number d governs several invariants attached to A such as the smallest degree of a polarisation on A, the discriminant of its ring of endomorphisms or the size of the invariant part of its geometric Brauer group. All these are bounded only in terms of d and the dimension of A. In the case where K is a number field we can go further and show that the period theorem applies to C in a natural way and gives an explicit bound for d in terms of the degree of K, the dimension of C and the stable Faltings height of C. This in turn yields explicit upper bounds for all the previous quantities related to isogenies, polarisations, endomorphisms, Brauer groups which significantly improve known results. MSC 2020 : 11G10 (14K02, 11R54, 14K15). Mots-clefs : variété abélienne, isogénie, ordre maximal, ordre principal, anneau de Lefschetz, module de Tate, polarisation, groupe de Brauer, réseau des périodes, théorème des périodes.
{"title":"Nouveaux théorèmes d’isogénie","authors":"Eric GAUDRON, Gaël RÉMOND","doi":"10.24033/msmf.484","DOIUrl":"https://doi.org/10.24033/msmf.484","url":null,"abstract":"— Given a finitely generated field extension K of the rational numbers and an abelian variety C over K, we consider the class of all abelian varieties over K which are isogenous (over K) to an abelian subvariety of a power of C. We show that there is a single, naturally constructed abelian variety C in the class whose ring of endomorphisms controls all isogenies in the class. Precisely, this means that if d is the discriminant of this ring then for any pair of isogenous abelian varieties in the class there exists an isogeny between them whose kernel has exponent at most d. Furthermore we prove that, for any element A in the class, the same number d governs several invariants attached to A such as the smallest degree of a polarisation on A, the discriminant of its ring of endomorphisms or the size of the invariant part of its geometric Brauer group. All these are bounded only in terms of d and the dimension of A. In the case where K is a number field we can go further and show that the period theorem applies to C in a natural way and gives an explicit bound for d in terms of the degree of K, the dimension of C and the stable Faltings height of C. This in turn yields explicit upper bounds for all the previous quantities related to isogenies, polarisations, endomorphisms, Brauer groups which significantly improve known results. MSC 2020 : 11G10 (14K02, 11R54, 14K15). Mots-clefs : variété abélienne, isogénie, ordre maximal, ordre principal, anneau de Lefschetz, module de Tate, polarisation, groupe de Brauer, réseau des périodes, théorème des périodes.","PeriodicalId":55332,"journal":{"name":"Bulletin De La Societe Mathematique De France","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135674628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a prime ideal $mathfrak{P}$ of the ring of integers of a number field $K$, we give a general definition of $mathfrak{P}$-adic continued fraction, which also includes classical definitions of continued fractions in the field of $p$--adic numbers. We give some necessary and sufficient conditions on $K$ ensuring that every $alphain K$ admits a finite $mathfrak{P}$-adic continued fraction expansion for all but finitely many $mathfrak{P}$, addressing a similar problem posed by Rosen in the archimedean setting.
对于数域$K$的整数环的素数理想$mathfrak{P}$，给出$mathfrak{P}$-进连续分数的一般定义，其中还包括$ P $-进数域内连分数的经典定义。我们给出了K$的一些充要条件，以保证K$中的每个$alpha对于除有限个$mathfrak{P}$以外的所有$mathfrak{P}$有一个有限的$mathfrak{P}$进进的连分式展开，解决了Rosen在阿基米德环境中提出的类似问题。
{"title":"On the finiteness of $mathfrak{P}$-adic continued fractions for number fields","authors":"Laura Capuano, Nadir Murru, Lea Terracini","doi":"10.24033/bsmf.2860","DOIUrl":"https://doi.org/10.24033/bsmf.2860","url":null,"abstract":"For a prime ideal $mathfrak{P}$ of the ring of integers of a number field $K$, we give a general definition of $mathfrak{P}$-adic continued fraction, which also includes classical definitions of continued fractions in the field of $p$--adic numbers. We give some necessary and sufficient conditions on $K$ ensuring that every $alphain K$ admits a finite $mathfrak{P}$-adic continued fraction expansion for all but finitely many $mathfrak{P}$, addressing a similar problem posed by Rosen in the archimedean setting.","PeriodicalId":55332,"journal":{"name":"Bulletin De La Societe Mathematique De France","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136178982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the classification of cubic planar Cremona maps","authors":"Alberto CALABRI, Nguyen THI NGOC GIAO","doi":"10.24033/bsmf.2857","DOIUrl":"https://doi.org/10.24033/bsmf.2857","url":null,"abstract":"","PeriodicalId":55332,"journal":{"name":"Bulletin De La Societe Mathematique De France","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136244536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish the finiteness of periodic points, that we called Geometric Dynamical Northcott Property, for regular polynomials automorphisms of the affine plane over a function field $mathbf{K}$ of characteristic zero, improving results of Ingram.
{"title":"The geometric dynamical Northcott property for regular polynomial automorphisms of the affine plane","authors":"Thomas Gauthier, Gabriel Vigny","doi":"10.24033/bsmf.2858","DOIUrl":"https://doi.org/10.24033/bsmf.2858","url":null,"abstract":"We establish the finiteness of periodic points, that we called Geometric Dynamical Northcott Property, for regular polynomials automorphisms of the affine plane over a function field $mathbf{K}$ of characteristic zero, improving results of Ingram. ","PeriodicalId":55332,"journal":{"name":"Bulletin De La Societe Mathematique De France","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136178980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we build the global determinant method of Salberger by Arakelov geometry explicitly. As an application, we study the dependence on the degree of the number of rational points of bounded height in plane curves. We will also explain why some constants will be more explicit if we admit the Generalized Riemann Hypothesis.
{"title":"On the global determinant method","authors":"Chunhui LIU","doi":"10.24033/bsmf.2859","DOIUrl":"https://doi.org/10.24033/bsmf.2859","url":null,"abstract":"In this paper, we build the global determinant method of Salberger by Arakelov geometry explicitly. As an application, we study the dependence on the degree of the number of rational points of bounded height in plane curves. We will also explain why some constants will be more explicit if we admit the Generalized Riemann Hypothesis.","PeriodicalId":55332,"journal":{"name":"Bulletin De La Societe Mathematique De France","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136244537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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