Pub Date : 2018-09-24DOI: 10.21711/217504322019/em331
Aaron W. Brown, S. Alvarez, Dominique Malicet, Davi Obata, M. Rold'an, B. Santiago, Michele Triestino
This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled Workshop for young researchers: Groups acting on manifolds held in Teresopolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory-particularly Lya-punov exponents and metric entropy-as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [114] and recent the work of the author with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang on actions of lattices in higher-rank semisimple Lie groups on manifolds [31, 36]. We give complete proofs of these results and present sufficient background in smooth ergodic theory needed for the proofs. A unifying theme in this text is the use of metric entropy and its relation to the geometry of conditional measures along foliations as a mechanism to verify invariance of measures. i
{"title":"Entropy, Lyapunov exponents, and rigidity of group actions","authors":"Aaron W. Brown, S. Alvarez, Dominique Malicet, Davi Obata, M. Rold'an, B. Santiago, Michele Triestino","doi":"10.21711/217504322019/em331","DOIUrl":"https://doi.org/10.21711/217504322019/em331","url":null,"abstract":"This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled Workshop for young researchers: Groups acting on manifolds held in Teresopolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory-particularly Lya-punov exponents and metric entropy-as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [114] and recent the work of the author with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang on actions of lattices in higher-rank semisimple Lie groups on manifolds [31, 36]. We give complete proofs of these results and present sufficient background in smooth ergodic theory needed for the proofs. A unifying theme in this text is the use of metric entropy and its relation to the geometry of conditional measures along foliations as a mechanism to verify invariance of measures. i","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"1993 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131134494","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 : 2018-02-15DOI: 10.21711/217504322018/em321
J. Ripoll, F. Tomi
In these notes we study the Dirichlet problem for critical points of a convex functional of the form% [ F(u)=int_{Omega}phileft( leftvert nabla urightvert right) , ] where $Omega$ is a bounded domain of a complete Riemannian manifold $mathcal{M}.$ We also study the asymptotic Dirichlet problem when $Omega=mathcal{M}$ is a Cartan-Hadamard manifold. Our aim is to present a unified approach to this problem which comprises the classical examples of the $p-$Laplacian ($phi(s)=s^{p}$, $p>1)$ and the minimal surface equation ($phi(s)=sqrt{1+s^{2}}$). Our approach does not use the direct method of the Calculus of Variations which seems to be common in the case of the $p-$Laplacian. Instead, we use the classical method of a-priori $C^{1}$ estimates of smooth solutions of the Euler-Lagrange equation. These estimates are obtained by a coordinate free calculus. Degenerate elliptic equations like the $p-$Laplacian are dealt with by an approximation argument. These notes address mainly researchers and graduate students interested in elliptic partial differential equations on Riemannian manifolds and may serve as a material for corresponding courses and seminars.
在这些笔记中，我们研究了一类凸泛函的临界点的狄利克雷问题% [ F(u)=int_{Omega}phileft( leftvert nabla urightvert right) , ] where $Omega$ is a bounded domain of a complete Riemannian manifold $mathcal{M}.$ We also study the asymptotic Dirichlet problem when $Omega=mathcal{M}$ is a Cartan-Hadamard manifold. Our aim is to present a unified approach to this problem which comprises the classical examples of the $p-$Laplacian ($phi(s)=s^{p}$, $p>1)$ and the minimal surface equation ($phi(s)=sqrt{1+s^{2}}$). Our approach does not use the direct method of the Calculus of Variations which seems to be common in the case of the $p-$Laplacian. Instead, we use the classical method of a-priori $C^{1}$ estimates of smooth solutions of the Euler-Lagrange equation. These estimates are obtained by a coordinate free calculus. Degenerate elliptic equations like the $p-$Laplacian are dealt with by an approximation argument. These notes address mainly researchers and graduate students interested in elliptic partial differential equations on Riemannian manifolds and may serve as a material for corresponding courses and seminars.
{"title":"Notes on the Dirichlet problem of a class of second order elliptic partial differential equations on a Riemannian manifold","authors":"J. Ripoll, F. Tomi","doi":"10.21711/217504322018/em321","DOIUrl":"https://doi.org/10.21711/217504322018/em321","url":null,"abstract":"In these notes we study the Dirichlet problem for critical points of a convex functional of the form% [ F(u)=int_{Omega}phileft( leftvert nabla urightvert right) , ] where $Omega$ is a bounded domain of a complete Riemannian manifold $mathcal{M}.$ We also study the asymptotic Dirichlet problem when $Omega=mathcal{M}$ is a Cartan-Hadamard manifold. Our aim is to present a unified approach to this problem which comprises the classical examples of the $p-$Laplacian ($phi(s)=s^{p}$, $p>1)$ and the minimal surface equation ($phi(s)=sqrt{1+s^{2}}$). Our approach does not use the direct method of the Calculus of Variations which seems to be common in the case of the $p-$Laplacian. Instead, we use the classical method of a-priori $C^{1}$ estimates of smooth solutions of the Euler-Lagrange equation. These estimates are obtained by a coordinate free calculus. Degenerate elliptic equations like the $p-$Laplacian are dealt with by an approximation argument. \u0000These notes address mainly researchers and graduate students interested in elliptic partial differential equations on Riemannian manifolds and may serve as a material for corresponding courses and seminars.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131315688","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 : 2017-11-28DOI: 10.21711/217504322017/em311
V. Sokolov
The survey is devoted to algebraic structures related to integrable ODEs and evolution PDEs. A description of Lax representations is given in terms of vector space decomposition of loop algebras into a direct sum of Taylor series and a complementary subalgebra. Examples of complementary subalgebras and corresponding integrable models are presented. In the framework of the bi-Hamiltonian approach compatible associative algebras related affine Dynkin diagrams are considered. A bi-Hamiltonian origin of the classical elliptic Calogero-Moser models is revealed.
{"title":"Algebraic structures related to integrable differential equations","authors":"V. Sokolov","doi":"10.21711/217504322017/em311","DOIUrl":"https://doi.org/10.21711/217504322017/em311","url":null,"abstract":"The survey is devoted to algebraic structures related to integrable ODEs and evolution PDEs. A description of Lax representations is given in terms of vector space decomposition of loop algebras into a direct sum of Taylor series and a complementary subalgebra. Examples of complementary subalgebras and corresponding integrable models are presented. In the framework of the bi-Hamiltonian approach compatible associative algebras related affine Dynkin diagrams are considered. A bi-Hamiltonian origin of the classical elliptic Calogero-Moser models is revealed.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133772419","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 : 2015-12-31DOI: 10.21711/217504322016/em302
É. Ghys, A. Ranicki
We survey the 19th century development of the signature of a quadratic form, and the applications in the 20th and 21st century to the topology of manifolds and dynamical systems. Version 2 is an expanded and corrected version of Version 1, including an Appendix by the second named author "Algebraic L-theory of rings with involution and the localization exact sequence".
{"title":"Signatures in algebra, topology and dynamics","authors":"É. Ghys, A. Ranicki","doi":"10.21711/217504322016/em302","DOIUrl":"https://doi.org/10.21711/217504322016/em302","url":null,"abstract":"We survey the 19th century development of the signature of a quadratic form, and the applications in the 20th and 21st century to the topology of manifolds and dynamical systems. Version 2 is an expanded and corrected version of Version 1, including an Appendix by the second named author \"Algebraic L-theory of rings with involution and the localization exact sequence\".","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"171 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125662003","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 : 2015-01-31DOI: 10.21711/217504322015/em291
Massimiliano Gubinelli
These are the notes for a course at the 18th Brazilian School of Probability held from August 3rd to 9th, 2014 in Mambucaba. The aim of the course is to introduce the basic problems of non‐linear PDEs with stochastic and irregular terms. We explain how it is possible to handle them using two main techniques: the notion of energy solutions [GJ10, GJ13] and that of paracontrolled distributions, recently introduced in [GIP13]. In order to maintain a link with physical intuitions, we motivate such singular SPDEs via a homogenization result for a di usion in a random potential.
{"title":"Lectures on singular stochastic PDEs","authors":"Massimiliano Gubinelli","doi":"10.21711/217504322015/em291","DOIUrl":"https://doi.org/10.21711/217504322015/em291","url":null,"abstract":"These are the notes for a course at the 18th Brazilian School of Probability held from August 3rd to 9th, 2014 in Mambucaba. The aim of the course is to introduce the basic problems of non‐linear PDEs with stochastic and irregular terms. We explain how it is possible to handle them using two main techniques: the notion of energy solutions [GJ10, GJ13] and that of paracontrolled distributions, recently introduced in [GIP13]. In order to maintain a link with physical intuitions, we motivate such singular SPDEs via a homogenization result for a di usion in a random potential.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123361483","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 : 2012-07-11DOI: 10.21711/217504322012/em221
Pierre-Antoine Guihéneuf
This memoir is concerned with the generic dynamical properties of conservative homeomorphisms of compact manifolds. Several important techniques allowing to prove genericity results are presented: we emphasize the important role played by periodic approximations of homeomorphisms, and by the embedding of the space of homeomorphisms in the space of bi-measurable automorphisms.
{"title":"Propriétés dynamiques génériques des homéomorphismes conservatifs","authors":"Pierre-Antoine Guihéneuf","doi":"10.21711/217504322012/em221","DOIUrl":"https://doi.org/10.21711/217504322012/em221","url":null,"abstract":"This memoir is concerned with the generic dynamical properties of conservative homeomorphisms of compact manifolds. Several important techniques allowing to prove genericity results are presented: we emphasize the important role played by periodic approximations of homeomorphisms, and by the embedding of the space of homeomorphisms in the space of bi-measurable automorphisms.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123127368","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 : 2012-06-24DOI: 10.21711/217504322015/em281
Graham Smith
Lecture notes for a minicourse to given in the XVII Brazilian School of Geometry, UFAM (Amazonas), Brazil, July 2012.
2012年7月，在UFAM(亚马逊)，巴西第十七届巴西几何学院提供的迷你课程讲义。
{"title":"Global singularity theory for the Gauss curvature equation","authors":"Graham Smith","doi":"10.21711/217504322015/em281","DOIUrl":"https://doi.org/10.21711/217504322015/em281","url":null,"abstract":"Lecture notes for a minicourse to given in the XVII Brazilian School of Geometry, UFAM (Amazonas), Brazil, July 2012.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124725563","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 : 2011-08-19DOI: 10.21711/217504322012/em211
Julie D'eserti
We recall some properties, unfortunately not all, of the Cre- mona group. We first begin by presenting a nice proof of the amalgamated product structure of the well-known subgroup of the Cremona group made up of the polynomial automorphisms of C 2 . Then we deal with the classification of birational maps and some applications (Tits alternative, non-simplicity...) Since any birational map can be written as a composition of quadratic birational maps up to an automorphism of the complex projective plane, we spend time on these special maps. Some questions of group theory are evoked: the classification of the finite subgroups of the Cremona group and related problems, the description of the automorphisms of the Cremona group and the representations of some lattices in the Cremona group. The description of the centralizers of discrete dynamical systems is an important problem in real and complex dynamic, we describe the state of the art for this problem in the Cremona group. Let S be a compact complex surface which carries an automorphism f of positive topological entropy. Either the Kodaira dimension of S is zero and f is conjugate to an automorphism on the unique minimal model of S which is either a torus, or a K3 surface, or an Enriques surface, or S is a non-minimal rational surface and f is conjugate to a birational map of the complex projective plane. We deal with results obtained in this last case: construction of such automorphisms, dynamical properties (rotation domains...).
{"title":"Some properties of the Cremona group","authors":"Julie D'eserti","doi":"10.21711/217504322012/em211","DOIUrl":"https://doi.org/10.21711/217504322012/em211","url":null,"abstract":"We recall some properties, unfortunately not all, of the Cre- mona group. We first begin by presenting a nice proof of the amalgamated product structure of the well-known subgroup of the Cremona group made up of the polynomial automorphisms of C 2 . Then we deal with the classification of birational maps and some applications (Tits alternative, non-simplicity...) Since any birational map can be written as a composition of quadratic birational maps up to an automorphism of the complex projective plane, we spend time on these special maps. Some questions of group theory are evoked: the classification of the finite subgroups of the Cremona group and related problems, the description of the automorphisms of the Cremona group and the representations of some lattices in the Cremona group. The description of the centralizers of discrete dynamical systems is an important problem in real and complex dynamic, we describe the state of the art for this problem in the Cremona group. Let S be a compact complex surface which carries an automorphism f of positive topological entropy. Either the Kodaira dimension of S is zero and f is conjugate to an automorphism on the unique minimal model of S which is either a torus, or a K3 surface, or an Enriques surface, or S is a non-minimal rational surface and f is conjugate to a birational map of the complex projective plane. We deal with results obtained in this last case: construction of such automorphisms, dynamical properties (rotation domains...).","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131365129","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 : 2010-04-13DOI: 10.21711/217504322010/em181
T. Seppäläinen
This review article discusses limit distributions and variance bounds for particle current in several dynamical stochastic systems of par- ticles on the one-dimensional integer lattice: independent particles, inde- pendent particles in a random environment, the random average process, the asymmetric simple exclusion process, and a class of totally asymmet- ric zero range processes. The first three models possess linear macroscopic flux functions and lie in the Edwards-Wilkinson universality class with scaling exponent 1/4 for current fluctuations. For these we prove Gaus- sian limits for the current process. The latter two systems belong to the Kardar-Parisi-Zhang class. For these we prove the scaling exponent 1/3 in the form of upper and lower variance bounds.
{"title":"Current fluctuations for stochastic particle systems with drift in one spatial dimension","authors":"T. Seppäläinen","doi":"10.21711/217504322010/em181","DOIUrl":"https://doi.org/10.21711/217504322010/em181","url":null,"abstract":"This review article discusses limit distributions and variance bounds for particle current in several dynamical stochastic systems of par- ticles on the one-dimensional integer lattice: independent particles, inde- pendent particles in a random environment, the random average process, the asymmetric simple exclusion process, and a class of totally asymmet- ric zero range processes. The first three models possess linear macroscopic flux functions and lie in the Edwards-Wilkinson universality class with scaling exponent 1/4 for current fluctuations. For these we prove Gaus- sian limits for the current process. The latter two systems belong to the Kardar-Parisi-Zhang class. For these we prove the scaling exponent 1/3 in the form of upper and lower variance bounds.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121964359","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 : 2007-12-07DOI: 10.21711/217504322008/em151
A. Ny
In this monograph, we survey some key issues of the theory of Gibbsian and non-Gibbsian measures in flnite-spin lattice systems. While non-Gibbsian measures are truly only the object of the last chapter, the material of the flrst chapters is selected with generalized Gibbs measures in mind. The topics of Gibbsian theory are then chosen for their foun- dational or contrasting role with respect to the measures analyzed in the flnal chapter, including in particular more detailed parts as e.g. the proof of the Choquet decomposition of Gibbs measures in Chapter 2, a proof of the Kozlov theorem in Chapter 3, under a slightly novel presentation that serves to introduce a telescoping procedure needed for generalized Gibbs measures in Chapter 5, and a careful discussion of the variational princi- ple in Chapter 4. This monograph covers also the contents of mini-courses given in 2007 at the universities UFMG (Belo Horizonte) and UFRGS (Porto Alegre), whose aim was to convey, in a relatively short number of lectures, the heart of the theory needed to understand Gibbsianness and non-Gibbsianness.
{"title":"Introduction to (generalized) Gibbs measures","authors":"A. Ny","doi":"10.21711/217504322008/em151","DOIUrl":"https://doi.org/10.21711/217504322008/em151","url":null,"abstract":"In this monograph, we survey some key issues of the theory of Gibbsian and non-Gibbsian measures in flnite-spin lattice systems. While non-Gibbsian measures are truly only the object of the last chapter, the material of the flrst chapters is selected with generalized Gibbs measures in mind. The topics of Gibbsian theory are then chosen for their foun- dational or contrasting role with respect to the measures analyzed in the flnal chapter, including in particular more detailed parts as e.g. the proof of the Choquet decomposition of Gibbs measures in Chapter 2, a proof of the Kozlov theorem in Chapter 3, under a slightly novel presentation that serves to introduce a telescoping procedure needed for generalized Gibbs measures in Chapter 5, and a careful discussion of the variational princi- ple in Chapter 4. This monograph covers also the contents of mini-courses given in 2007 at the universities UFMG (Belo Horizonte) and UFRGS (Porto Alegre), whose aim was to convey, in a relatively short number of lectures, the heart of the theory needed to understand Gibbsianness and non-Gibbsianness.","PeriodicalId":359243,"journal":{"name":"Ensaios Matemáticos","volume":"195 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132530242","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}
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