{"title":"Inverse problems in multifractal analysis of measures","authors":"J. Barral","doi":"10.24033/ASENS.2274","DOIUrl":"https://doi.org/10.24033/ASENS.2274","url":null,"abstract":"","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"163 5 1","pages":"1457-1510"},"PeriodicalIF":1.9,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86670076","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}
H. Ammari, E. Bonnetier, Faouzi Triki, M. Vogelius
We consider a scalar elliptic equation for a composite medium consisting of homogeneous C^{1, α0} inclusions, 0< α0≤ 1, embedded in a constant matrix phase. When the inclusions are separated and are separated from the boundary, the solution has an integral representation, in terms of potential functions defined on the boundary of each inclusion. We study the system of integral equations satisfied by these potential functions as the distance between two inclusions tends to 0. We show that the potential functionsunctions converge in C^{0,α}, 0< α < α0 to limiting potential functions, with which one can represent the solution when the inclusions are touching. As a consequence, we obtain uniform C^{1,α} bounds on the solution, which are independent of the inter-inclusion distances.
{"title":"Elliptic estimates in composite media with smooth inclusions: an integral equation approach","authors":"H. Ammari, E. Bonnetier, Faouzi Triki, M. Vogelius","doi":"10.24033/ASENS.2249","DOIUrl":"https://doi.org/10.24033/ASENS.2249","url":null,"abstract":"We consider a scalar elliptic equation for a composite medium consisting of homogeneous C^{1, α0} inclusions, 0< α0≤ 1, embedded in a constant matrix phase. When the inclusions are separated and are separated from the boundary, the solution has an integral representation, in terms of potential functions defined on the boundary of each inclusion. We study the system of integral equations satisfied by these potential functions as the distance between two inclusions tends to 0. We show that the potential functionsunctions converge in C^{0,α}, 0< α < α0 to limiting potential functions, with which one can represent the solution when the inclusions are touching. As a consequence, we obtain uniform C^{1,α} bounds on the solution, which are independent of the inter-inclusion distances.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"24 1","pages":"453-495"},"PeriodicalIF":1.9,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75283445","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}
Mihalis Dafermos, I. Rodnianski, Yakov Shlapentokh-Rothman
We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|
对于一般亚极值情况下Kerr外背景下的标量波动方程，我们建立了一个确定的物理空间散射理论。特别地，我们证明了“散射态的存在唯一性”和“渐近完备性”所对应的结果，并进一步证明了将过去视界和过去零无穷辐射场映射到未来视界和未来零无穷辐射场的“散射矩阵”是一个有界算子。后者允许我们给出超辐射反射的时域理论。散射矩阵的有界性特别表明，与入射有限能量波包有关的解在过零无穷远处的最大放大是有界的。在频率方面，这对应于一种新的说法，即适当归一化的反射系数和透射系数是均匀有界的，与频率参数无关。我们进一步证明，超辐射反射确实放大了上述合适波包辐射到未来零无穷大的能量。这些结果充分利用了我们最近证明的一个改进[M]。Dafermos, I. Rodnianski和Y. Shlapentokh-Rothman, Kerr外时空上波动方程解的衰减:有界性的完全次极值情形[a]
{"title":"A scattering theory for the wave equation on kerr black hole exteriors","authors":"Mihalis Dafermos, I. Rodnianski, Yakov Shlapentokh-Rothman","doi":"10.24033/asens.2358","DOIUrl":"https://doi.org/10.24033/asens.2358","url":null,"abstract":"We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to \"existence and uniqueness of scattering states\" and \"asymptotic completeness\" and we show moreover that the resulting \"scattering matrix\" mapping radiation fields on the past horizon and past null infinity to radiation fields on the future horizon and future null infinity is a bounded operator. The latter allows us to give a time-domain theory of superradiant reflection. The boundedness of the scattering matrix shows in particular that the maximal amplification of solutions associated to ingoing finite-energy wave packets on past null infinity is bounded. On the frequency side, this corresponds to the novel statement that the suitably normalised reflection and transmission coefficients are uniformly bounded independently of the frequency parameters. We further complement this with a demonstration that superradiant reflection indeed amplifies the energy radiated to future null infinity of suitable wave-packets as above. The results make essential use of a refinement of our recent proof [M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case |a|<M, arXiv:1402.6034] of boundedness and decay for solutions of the Cauchy problem so as to apply in the class of solutions where only a degenerate energy is assumed finite. We show in contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerate energy solutions. This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shift instability when solving the wave equation backwards.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"89 1","pages":"371-486"},"PeriodicalIF":1.9,"publicationDate":"2014-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85637528","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 characterize n-rectifiable metric measure spaces as those spaces that admit a countable Borel decomposition so that each piece has positive and finite n-densities and one of the following : is an n-dimensional Lipschitz differentiability space ; has n-independent Alberti representations ; satisfies David's condition for an n-dimensional chart. The key tool is an iterative grid construction which allows us to show that the image of a ball with a high density of curves from the Alberti representations under a chart map contains a large portion of a uniformly large ball and hence satisfies David's condition. This allows us to apply modified versions of previously known ‘biLipschitz pieces' results on the charts.
{"title":"Characterizations of rectifiable metric measure spaces","authors":"David Bate, Sean Li","doi":"10.24033/asens.2314","DOIUrl":"https://doi.org/10.24033/asens.2314","url":null,"abstract":"We characterize n-rectifiable metric measure spaces as those spaces that admit a countable Borel decomposition so that each piece has positive and finite n-densities and one of the following : is an n-dimensional Lipschitz differentiability space ; has n-independent Alberti representations ; satisfies David's condition for an n-dimensional chart. The key tool is an iterative grid construction which allows us to show that the image of a ball with a high density of curves from the Alberti representations under a chart map contains a large portion of a uniformly large ball and hence satisfies David's condition. This allows us to apply modified versions of previously known ‘biLipschitz pieces' results on the charts.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"2 1","pages":"1-37"},"PeriodicalIF":1.9,"publicationDate":"2014-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74245551","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}
In our previous article, we proved that symplectic homeomorphisms preserving a coisotropic submanifold C, preserve its characteristic foliation as well. As a consequence, such symplectic homeomorphisms descend to the reduction of the coisotropic C. In this article we show that these reduced homeomorphisms continue to exhibit certain symplectic properties. In particular, in the specific setting where the symplectic manifold is a torus and the coisotropic is a standard subtorus, we prove that the reduced homeomorphism preserves spectral invariants and hence the spectral capacity.
{"title":"Reduction of symplectic homeomorphisms","authors":"Vincent Humilière, R. Leclercq, Sobhan Seyfaddini","doi":"10.24033/asens.2292","DOIUrl":"https://doi.org/10.24033/asens.2292","url":null,"abstract":"In our previous article, we proved that symplectic homeomorphisms preserving a coisotropic submanifold C, preserve its characteristic foliation as well. As a consequence, such symplectic homeomorphisms descend to the reduction of the coisotropic C. In this article we show that these reduced homeomorphisms continue to exhibit certain symplectic properties. In particular, in the specific setting where the symplectic manifold is a torus and the coisotropic is a standard subtorus, we prove that the reduced homeomorphism preserves spectral invariants and hence the spectral capacity.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"518 1","pages":"633-668"},"PeriodicalIF":1.9,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77164737","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}
The main purpose of this paper is to provide a description of the fundamental group of a symplectic manifold in terms of Floer theoretic objects. As an application, we show that when counted with a suitable notion of multiplicity, non degenerate Hamiltonian diffeomorphisms have enough fixed points to generate the fundamental group.
{"title":"A Floer fundamental group","authors":"J. Barraud","doi":"10.24033/asens.2366","DOIUrl":"https://doi.org/10.24033/asens.2366","url":null,"abstract":"The main purpose of this paper is to provide a description of the fundamental group of a symplectic manifold in terms of Floer theoretic objects. As an application, we show that when counted with a suitable notion of multiplicity, non degenerate Hamiltonian diffeomorphisms have enough fixed points to generate the fundamental group.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"62 1","pages":"773-809"},"PeriodicalIF":1.9,"publicationDate":"2014-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90413911","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}
In this article, we construct etale realization functors defined on the categories DAet(X, Λ) of etale motives (without transfers) over a scheme X. Our construction is natural and relies on a relative rigidity theorem a la Suslin-Voevodsky that we will establish first. Then, we show that these realization functors are compatible with Grothendieck operations and the "nearby cycles" functors. Along the way, we prove a number of properties concerning etale motives.
{"title":"La réalisation étale et les opérations de Grothendieck","authors":"J. Ayoub","doi":"10.24033/ASENS.2210","DOIUrl":"https://doi.org/10.24033/ASENS.2210","url":null,"abstract":"In this article, we construct etale realization functors defined on the categories DAet(X, Λ) of etale motives (without transfers) over a scheme X. Our construction is natural and relies on a relative rigidity theorem a la Suslin-Voevodsky that we will establish first. Then, we show that these realization functors are compatible with Grothendieck operations and the \"nearby cycles\" functors. Along the way, we prove a number of properties concerning etale motives.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"7 1","pages":"1-145"},"PeriodicalIF":1.9,"publicationDate":"2014-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74766802","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}
This article is the first in a series devoted to studying generalised Gross-KudlaSchoen diagonal cycles in the product of three Kuga-Sato varieties and the Euler system properties of the associated Selmer classes, with special emphasis on their application to the Birch–Swinnerton-Dyer conjecture and the theory of Stark-Heegner points. The basis for the entire study is a p-adic formula of Gross-Zagier type which relates the images of these diagonal cycles under the p-adic Abel-Jacobi map to special values of certain p-adic Lfunctions attached to the Garrett-Rankin triple convolution of three Hida families of modular forms. The main goal of this article is to describe and prove this formula. Cet article est le premier d’une serie consacree aux cycles de Gross-Kudla-Schoen generalises appartenant aux groupes de Chow de produits de trois varietes de Kuga-Sato, et aux systemes d’Euler qui leur sont associes. La serie au complet repose sur une variante p-adique de la formule de Gross-Zagier qui relie l’image des cycles de Gross-Kudla-Schoen par l’application d’Abel-Jacobi p-adique aux valeurs speciales de certaines fonctions L p-adiques attachees a la convolution de Garrett-Rankin de trois familles de Hida de formes modulaires cuspidales. L’objectif principal de cet article est de decrire et de demontrer cette variante. MSC: 11F12, 11G05, 11G35, 11G40.
本文是一系列致力于研究三个Kuga-Sato变积中的广义Gross-KudlaSchoen对角环和相关Selmer类的欧拉系统性质的文章中的第一篇，特别强调它们在Birch-Swinnerton-Dyer猜想和Stark-Heegner点理论中的应用。整个研究的基础是一个p进的Gross-Zagier型公式，该公式将p进Abel-Jacobi映射下这些对角环的像与附加在三个模形式的Hida族的Garrett-Rankin三重卷积上的某些p进l函数的特殊值联系起来。本文的主要目的是描述和证明这个公式。这篇文章测试了le premier d 'une serie consacree aux cycles de Gross-Kudla-Schoen概括了明显的aux groups de Chow de producties de trois vartes de Kuga-Sato，以及aux systemes d 'Euler quur sontassocies。在Gross-Zagier的公式中，我们得到了一个完整的函数，我们得到了一个完整的函数，我们得到了一个完整的函数，我们得到了一个完整的函数，我们得到了一个完整的函数，我们得到了一个完整的函数，我们得到了一个完整的函数。客观原则决定了文章的性质，决定了论证的性质，决定了变量。Msc: 11f12, 11g05, 11g35, 11g40。
{"title":"Diagonal cycles and Euler systems I: A $p$-adic Gross-Zagier formula","authors":"H. Darmon, V. Rotger","doi":"10.24033/ASENS.2227","DOIUrl":"https://doi.org/10.24033/ASENS.2227","url":null,"abstract":"This article is the first in a series devoted to studying generalised Gross-KudlaSchoen diagonal cycles in the product of three Kuga-Sato varieties and the Euler system properties of the associated Selmer classes, with special emphasis on their application to the Birch–Swinnerton-Dyer conjecture and the theory of Stark-Heegner points. The basis for the entire study is a p-adic formula of Gross-Zagier type which relates the images of these diagonal cycles under the p-adic Abel-Jacobi map to special values of certain p-adic Lfunctions attached to the Garrett-Rankin triple convolution of three Hida families of modular forms. The main goal of this article is to describe and prove this formula. Cet article est le premier d’une serie consacree aux cycles de Gross-Kudla-Schoen generalises appartenant aux groupes de Chow de produits de trois varietes de Kuga-Sato, et aux systemes d’Euler qui leur sont associes. La serie au complet repose sur une variante p-adique de la formule de Gross-Zagier qui relie l’image des cycles de Gross-Kudla-Schoen par l’application d’Abel-Jacobi p-adique aux valeurs speciales de certaines fonctions L p-adiques attachees a la convolution de Garrett-Rankin de trois familles de Hida de formes modulaires cuspidales. L’objectif principal de cet article est de decrire et de demontrer cette variante. MSC: 11F12, 11G05, 11G35, 11G40.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"134 1","pages":"779-832"},"PeriodicalIF":1.9,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76721502","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 M̆ be an unramified Rapoport-Zink space of EL type or unitary/symplectic PEL type. Let (M̆K)K be the tower of Berkovich’s analytic spaces classifying the level structures over the generic fiber of M̆. In [Che13], we have defined a determinant morphism detK from the tower (M̆K)K to a tower of Berkovich’s analytic spaces of dimension 0 associated to the cocenter of the reductive group related to the space M̆. Suppose that the Newton polygon and Hodge polygon related to M̆ don’t touch each other except their end point. And suppose that a conjecture on the set of connected components of the reduced special fiber of M̆ holds. Then we prove that the geometric fibers of the determinant morphism detK are the geometrically connected components of M̆K . The conjecture in the hypothesis will be confirmed in a paper in preparation by Kisin, Viehmann and the author.
{"title":"Composantes connexes géométriques de la tour des espaces de modules de groupes $p$-divisibles","authors":"Miaofen Chen","doi":"10.24033/ASENS.2225","DOIUrl":"https://doi.org/10.24033/ASENS.2225","url":null,"abstract":"Let M̆ be an unramified Rapoport-Zink space of EL type or unitary/symplectic PEL type. Let (M̆K)K be the tower of Berkovich’s analytic spaces classifying the level structures over the generic fiber of M̆. In [Che13], we have defined a determinant morphism detK from the tower (M̆K)K to a tower of Berkovich’s analytic spaces of dimension 0 associated to the cocenter of the reductive group related to the space M̆. Suppose that the Newton polygon and Hodge polygon related to M̆ don’t touch each other except their end point. And suppose that a conjecture on the set of connected components of the reduced special fiber of M̆ holds. Then we prove that the geometric fibers of the determinant morphism detK are the geometrically connected components of M̆K . The conjecture in the hypothesis will be confirmed in a paper in preparation by Kisin, Viehmann and the author.","PeriodicalId":50971,"journal":{"name":"Annales Scientifiques De L Ecole Normale Superieure","volume":"24 1","pages":"723-764"},"PeriodicalIF":1.9,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78600408","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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