Sebastian Casalaina-Martin, S. Grushevsky, K. Hulek, R. Laza
We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily–Borel and toroidal compactifications of the ball quotient model, due to Allcock–Carlson–Toledo. Our starting point is Kirwan’s method. We then follow by investigating the behavior of the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli space of cubic surfaces is discussed in an appendix.
{"title":"Cohomology of the Moduli Space of Cubic Threefolds and Its Smooth Models","authors":"Sebastian Casalaina-Martin, S. Grushevsky, K. Hulek, R. Laza","doi":"10.1090/memo/1395","DOIUrl":"https://doi.org/10.1090/memo/1395","url":null,"abstract":"We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily–Borel and toroidal compactifications of the ball quotient model, due to Allcock–Carlson–Toledo. Our starting point is Kirwan’s method. We then follow by investigating the behavior of the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli space of cubic surfaces is discussed in an appendix.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43527483","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}
Let ΩOmega be a smooth bounded domain in Rnmathbb {R}^n (n≥3ngeq 3) such that 0∈∂Ω0in partial Omega. We consider issues of non-existence, existence, and multiplicity of variational solutions in H1,02(Ω)H_{1,0}^2(Omega ) for the borderline Dirichlet problem,
让Ω欧米茄be a smooth bounded域名in R n mathbb {R) ^ n ( n≥3 geq 3)这样的那个 0∈∂Ω0 中部分欧米茄。我们认为non-existence、存在的问题和multiplicity variational的解决方案在 H 1 , 0 2 ( Ω ) H_{1.0) ^ 2(ω)》有点像Dirichlet问题,{ − Δ u − γ u | x | 2 − h ( x ) u a m p ;= m m p;| u | 2 ⋆ ( s ) − 2 u | x| s a m p ;在 Ω , u a m p ;= m m p;零a m p;在 ∂ Ω ∖ { 0 } , 开始{equation *的左派{开始{}{llll阵}-三角洲u u -伽马 frac {} {x | | - h (x) ^ 2的u & = & frac {| | u ^{2 ^ 星(s) - x的u} {| | & ^ s的短信{进来的,我是俄梅戛 hfill u& = &0 & 短信上{}部分 setminus {0},我是俄梅戛end{阵列的 coming right。 end {equation *的地方 0 > s > 2 0 > s > , 2 ⋆ ( s ) ≔ 2 ( n − s ) n − 2 < mml: annotation
{"title":"Multiplicity and Stability of the Pohozaev Obstruction for Hardy-Schrödinger Equations with Boundary Singularity","authors":"N. Ghoussoub, Saikat Mazumdar, F. Robert","doi":"10.1090/memo/1415","DOIUrl":"https://doi.org/10.1090/memo/1415","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega\">\u0000 <mml:semantics>\u0000 <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Omega</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a smooth bounded domain in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript n\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {R}^n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> (<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 3\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">ngeq 3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>) such that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0 element-of partial-differential normal upper Omega\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi>\u0000 <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">0in partial Omega</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We consider issues of non-existence, existence, and multiplicity of variational solutions in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Subscript 1 comma 0 Superscript 2 Baseline left-parenthesis normal upper Omega right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msubsup>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msubsup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">H_{1,0}^2(Omega )</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for the borderline Dirichlet problem, <disp-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartLayout Enlarged left-brace 1st Row 1st Column minus normal upper Delta u minus gamma StartFraction u Over StartAbsoluteValue x EndAbsoluteValue squared EndFraction minus h left-parenthesis x right-parenthesis u 2nd Col","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42325358","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 prove compatibilities of various definitions of relatively unipotent log de Rham fundamental groups for certain proper log smooth integral morphisms of fine log schemes of characteristic zero. Our proofs are purely algebraic. As an application, we give a purely algebraic calculation of the monodromy action on the unipotent log de Rham fundamental group of a stable log curve. As a corollary we give a purely algebraic proof to the transcendental part of Andreatta–Iovita–Kim’s article: obtaining in this way a complete algebraic criterion for good reduction for curves.
本文证明了特征为零的精细对数格式的若干适当对数光滑积分态射的相对单幂对数de Rham基群的各种定义的相容性。我们的证明是纯代数的。作为应用，我们给出了稳定对数曲线的幂偶log de Rham基群上的单调作用的纯代数计算。作为推论，我们对Andreatta-Iovita-Kim文章的超越部分给出了一个纯代数证明:由此获得了曲线良好约简的一个完备的代数判据。
{"title":"Comparison of Relatively Unipotent Log de Rham Fundamental Groups","authors":"B. Chiarellotto, V. D. Proietto, Atsushi Shiho","doi":"10.1090/memo/1430","DOIUrl":"https://doi.org/10.1090/memo/1430","url":null,"abstract":"In this paper, we prove compatibilities of various definitions of relatively unipotent log de Rham fundamental groups for certain proper log smooth integral morphisms of fine log schemes of characteristic zero. Our proofs are purely algebraic. As an application, we give a purely algebraic calculation of the monodromy action on the unipotent log de Rham fundamental group of a stable log curve. As a corollary we give a purely algebraic proof to the transcendental part of Andreatta–Iovita–Kim’s article: obtaining in this way a complete algebraic criterion for good reduction for curves.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44522475","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}
The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globally m m -fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long as m ≥ 3. mgeq 3. In this case, all of the angles involved solve a closed ODE system which dictates the global-in-time dynamics of the corners and only depends on the initial locations and sizes of the corners. Along the way we obtain a global well-posedness result for a class of symmetric patches with boundary singular at the origin, which includes logarithmic spirals. On the ill-posedness side, we show that any other type of corner singularity in a vortex patch cannot evolve continuously in time except possibly when all corners involved have precisely the angle π 2 frac {pi }{2} for all time. Even in the case of vortex patches with corners of angle π 2 frac {pi }{2} or with corners which are only locally m m -fold symmetric, we prove that they are generically ill-posed. We expect that in these cases of ill-posedness, the vortex patches actually cusp immediately in a self-similar way and we derive some asymptotic models which may be useful in giving a more precise description of the dynamics. In a companion work from 2020 on singular vortex patches, we discuss the long-time behavior of symmetric vortex patches with corners and use them to construct patches on R 2 mathbb {R}^2 with interesting dynamical behavior such as cusping and spiral formation in infinite time.
{"title":"On Singular Vortex Patches, I: Well-posedness Issues","authors":"T. Elgindi, In-Jee Jeong","doi":"10.1090/memo/1400","DOIUrl":"https://doi.org/10.1090/memo/1400","url":null,"abstract":"The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globally \u0000\u0000 \u0000 m\u0000 m\u0000 \u0000\u0000-fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long as \u0000\u0000 \u0000 \u0000 m\u0000 ≥\u0000 3.\u0000 \u0000 mgeq 3.\u0000 \u0000\u0000 In this case, all of the angles involved solve a closed ODE system which dictates the global-in-time dynamics of the corners and only depends on the initial locations and sizes of the corners. Along the way we obtain a global well-posedness result for a class of symmetric patches with boundary singular at the origin, which includes logarithmic spirals. On the ill-posedness side, we show that any other type of corner singularity in a vortex patch cannot evolve continuously in time except possibly when all corners involved have precisely the angle \u0000\u0000 \u0000 \u0000 π\u0000 2\u0000 \u0000 frac {pi }{2}\u0000 \u0000\u0000 for all time. Even in the case of vortex patches with corners of angle \u0000\u0000 \u0000 \u0000 π\u0000 2\u0000 \u0000 frac {pi }{2}\u0000 \u0000\u0000 or with corners which are only locally \u0000\u0000 \u0000 m\u0000 m\u0000 \u0000\u0000-fold symmetric, we prove that they are generically ill-posed. We expect that in these cases of ill-posedness, the vortex patches actually cusp immediately in a self-similar way and we derive some asymptotic models which may be useful in giving a more precise description of the dynamics. In a companion work from 2020 on singular vortex patches, we discuss the long-time behavior of symmetric vortex patches with corners and use them to construct patches on \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 2\u0000 \u0000 mathbb {R}^2\u0000 \u0000\u0000 with interesting dynamical behavior such as cusping and spiral formation in infinite time.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47589347","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":"Fusion of defects","authors":"A. Bartels, Christopher L. Douglas, A. Henriques","doi":"10.1090/memo/1237","DOIUrl":"https://doi.org/10.1090/memo/1237","url":null,"abstract":"","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81497638","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 consider an n n -fold Brylinski–Deligne cover of a reductive group over a p p -adic field. Since the space of Whittaker functionals of an irreducible genuine representation of such a cover is not one-dimensional, one can consider a local coefficients matrix arising from an intertwining operator, which is the natural analogue of the local coefficients in the linear case. In this paper, we concentrate on genuine principal series representations and establish some fundamental properties of such a local coefficients matrix, including the investigation of its arithmetic invariants. As a consequence, we prove a form of the Casselman–Shalika formula which could be viewed as a natural analogue for linear algebraic groups. We also investigate in some depth the behaviour of the local coefficients matrix with respect to the restriction of genuine principal series from covers of G L 2 GL_2 to S L 2 SL_2 . In particular, some further relations are unveiled between local coefficients matrices and gamma factors or metaplectic-gamma factors.
我们考虑p-p-adic域上一个还原群的n-n次Brylinski–Deligne覆盖。由于这种覆盖的不可约真表示的Whittaker泛函的空间不是一维的，因此可以考虑由交织算子产生的局部系数矩阵，这是线性情况下局部系数的自然模拟。本文主要研究真主级数表示，并建立了这种局部系数矩阵的一些基本性质，包括它的算术不变量的研究。因此，我们证明了Casselman–Shalika公式的一种形式，它可以被视为线性代数群的自然类似物。我们还深入地研究了局部系数矩阵关于真主级数从G L 2 GL_ 2到S L 2 SL_。特别地，揭示了局部系数矩阵与伽玛因子或偏辛伽玛因子之间的一些进一步的关系。
{"title":"Local Coefficients and Gamma Factors for Principal Series of Covering Groups","authors":"Fan Gao, F. Shahidi, Dani Szpruch","doi":"10.1090/memo/1399","DOIUrl":"https://doi.org/10.1090/memo/1399","url":null,"abstract":"We consider an \u0000\u0000 \u0000 n\u0000 n\u0000 \u0000\u0000-fold Brylinski–Deligne cover of a reductive group over a \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-adic field. Since the space of Whittaker functionals of an irreducible genuine representation of such a cover is not one-dimensional, one can consider a local coefficients matrix arising from an intertwining operator, which is the natural analogue of the local coefficients in the linear case. In this paper, we concentrate on genuine principal series representations and establish some fundamental properties of such a local coefficients matrix, including the investigation of its arithmetic invariants. As a consequence, we prove a form of the Casselman–Shalika formula which could be viewed as a natural analogue for linear algebraic groups. We also investigate in some depth the behaviour of the local coefficients matrix with respect to the restriction of genuine principal series from covers of \u0000\u0000 \u0000 \u0000 G\u0000 \u0000 L\u0000 2\u0000 \u0000 \u0000 GL_2\u0000 \u0000\u0000 to \u0000\u0000 \u0000 \u0000 S\u0000 \u0000 L\u0000 2\u0000 \u0000 \u0000 SL_2\u0000 \u0000\u0000. In particular, some further relations are unveiled between local coefficients matrices and gamma factors or metaplectic-gamma factors.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46486772","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 develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation which shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems that our results apply to include those arising from Bernoulli convolutions, the {0,1,3}{0,1,3} problem, and affine contractions with varying translation parameter. As a by-product of our analysis we obtain new proofs of some well known results due to Solomyak on the absolute continuity of Bernoulli convolutions, and when the attractor in the {0,1,3}{0,1,3} problem has positive Lebesgue measure.
For each t∈[0,1]tin [0,1] we let ΦtPhi _t be the iterated f
本文提出了一种研究重叠迭代函数系统的新方法。这种方法的灵感来自于Khintchine关于Diophantine近似的一个著名结果，该结果表明，对于一类limsup集合，它们的勒贝格测度是由自然发生的体积和的收敛或发散决定的。对于许多重叠迭代函数系统的参数化族，我们证明了一个典型的成员将表现出类似钦钦机的行为。我们的结果适用的迭代函数系统族包括伯努利卷积、{0,1,3｛0,1,3}｝问题和具有变化平移参数的仿射收缩引起的迭代函数系统族。作为我们分析的副产品，我们得到了关于Bernoulli卷积绝对连续性的一些著名结果的新的证明，当{0,1,3｛0,1,3}｝问题中的吸引子具有正的Lebesgue测度时。对于每一个t∈[0,1]t in[0,1]，设Φ t Phi ＿t为Φ t的迭代函数系统:{φ 2 (x) = x + 1 2， φ 3 (x) = x + t 2， φ 4 (x) = x + 1 + t 2}。begin{equation*} Phi _{t}≔Big {phi _1(x)=frac {x}{2},phi _2(x)=frac {x+1}{2},phi _3(x)=frac {x+t}{2},phi _{4}(x)=frac {x+1+t}{2}Big }. end{equation*}我们证明要么Φ t Phi ＿t包含一个精确的重叠，要么我们观察到类似钦钦的行为。我们的分析表明，通过研究limsup集的度量性质，我们可以区分迭代函数系统的重叠行为，这是我们通过简单地研究自相似度量的性质所无法做到的。最后，我们引入迭代函数系统的一个性质，我们称之为相对于度量的一致分离。我们证明了这一性质意味着测度的推进是绝对连续的。我们包含了几个一致分离的迭代函数系统的显式示例。
{"title":"Overlapping Iterated Function Systems from the Perspective of Metric Number Theory","authors":"S. Baker","doi":"10.1090/memo/1428","DOIUrl":"https://doi.org/10.1090/memo/1428","url":null,"abstract":"<p>In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation which shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems that our results apply to include those arising from Bernoulli convolutions, the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartSet 0 comma 1 comma 3 EndSet\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">{0,1,3}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> problem, and affine contractions with varying translation parameter. As a by-product of our analysis we obtain new proofs of some well known results due to Solomyak on the absolute continuity of Bernoulli convolutions, and when the attractor in the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartSet 0 comma 1 comma 3 EndSet\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">{0,1,3}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> problem has positive Lebesgue measure.</p>\u0000\u0000<p>For each <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t element-of left-bracket 0 comma 1 right-bracket\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">tin [0,1]</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> we let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Phi Subscript t\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi mathvariant=\"normal\">Φ<!-- Φ --></mml:mi>\u0000 <mml:mi>t</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">Phi _t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be the iterated f","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60559782","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 generalizations of McShane’s identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen, which generalize the notion of horocycle lengths. In particular, we obtain McShane-type identities for finite-area cusped convex real projective surfaces by generalizing the Birman–Series geodesic scarcity theorem. More generally, we establish McShane-type identities for positive surface group representations with loxodromic boundary monodromy, as well as McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are systematically expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple and cross ratios. We apply our identities to derive the simple spectral discreteness of unipotent-bordered positive representations, collar lemmas, and generalizations of the Thurston metric.
{"title":"McShane Identities for Higher Teichmüller Theory and the Goncharov–Shen Potential","authors":"Yi Huang, Zhe Sun","doi":"10.1090/memo/1422","DOIUrl":"https://doi.org/10.1090/memo/1422","url":null,"abstract":"We derive generalizations of McShane’s identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen, which generalize the notion of horocycle lengths. In particular, we obtain McShane-type identities for finite-area cusped convex real projective surfaces by generalizing the Birman–Series geodesic scarcity theorem. More generally, we establish McShane-type identities for positive surface group representations with loxodromic boundary monodromy, as well as McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are systematically expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple and cross ratios. We apply our identities to derive the simple spectral discreteness of unipotent-bordered positive representations, collar lemmas, and generalizations of the Thurston metric.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43474587","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 topological dynamics, the Symbolic Extension Entropy Theorem (SEET) (Boyle and Downarowicz, 2004) describes the possibility of a lossless digitalization of a dynamical system by extending it to a subshift on finitely many symbols. The theorem gives a precise estimate on the entropy of such a symbolic extension (and hence on the necessary number of symbols). Unlike in the measure-theoretic case, where Kolmogorov–Sinai entropy serves as an estimate in an analogous problem, in the topological setup the task reaches beyond the classical theory of measure-theoretic and topological entropy. Necessary are tools from an extended theory of entropy, the theory of entropy structures developed in Downarowicz (2005). The main goal of this paper is to prove the analog of the SEET for actions of (discrete infinite) countable amenable groups:
Let a countable amenable group GG act by homeomorphisms on a compact metric space XX and let MG(X)mathcal {M}_{G}(X) denote the simplex of all GG-invariant Borel probability measures on XX. A function
在拓扑动力学中，符号扩展熵定理(SEET) (Boyle和Downarowicz, 2004)描述了通过将动力系统扩展到有限多个符号上的子移来实现无损数字化的可能性。该定理给出了这种符号扩展的熵的精确估计(因此也给出了必要的符号数量)。不像在测量理论的情况下，Kolmogorov-Sinai熵作为一个类似问题的估计，在拓扑设置的任务超越了测量理论和拓扑熵的经典理论。必要的工具来自于熵的扩展理论，即Downarowicz(2005)发展的熵结构理论。本文的主要目的是证明(离散无限)可数可调群作用的SEET的类似性:设一个可数可调群G G在紧度量空间X X上通过同胚作用，设M G(X) mathcal M＿G{(}X)表示X X上所有G G不变Borel概率测度的单纯形。函数E A {E＿}{}{mathsf A{在}}M G(X) mathcal M＿G{(X)}上等于符号扩展π的扩展熵函数h π h^ {}pi:(Y,G)→(X,G) pi:(Y,G) to (X,G)，其中h π (μ)= sup h ν {(Y,G): ν∈π−1 (μ) h^}pi (mu)= sup ｛h＿ nu (Y,G):nuinpi ^-1{(}mu)} (μ∈M G(X) muinmathcal M＿G{(}X{)})，当且仅当E A E＿ {}{mathsf A是(X,G) (X,G)的熵结构的有限{仿射}}超包络。当然，在陈述之前，先介绍了熵结构及其超包络的概念，这些概念改编自Z mathbb Z -actions的情况{。}总的来说，我们能够证明一个稍弱的SEET版本，其中符号扩展被替换
{"title":"Symbolic Extensions of Amenable Group Actions and the Comparison Property","authors":"T. Downarowicz, Guohua Zhang","doi":"10.1090/memo/1390","DOIUrl":"https://doi.org/10.1090/memo/1390","url":null,"abstract":"<p>In topological dynamics, the <italic>Symbolic Extension Entropy Theorem</italic> (SEET) (Boyle and Downarowicz, 2004) describes the possibility of a lossless digitalization of a dynamical system by extending it to a subshift on finitely many symbols. The theorem gives a precise estimate on the entropy of such a symbolic extension (and hence on the necessary number of symbols). Unlike in the measure-theoretic case, where Kolmogorov–Sinai entropy serves as an estimate in an analogous problem, in the topological setup the task reaches beyond the classical theory of measure-theoretic and topological entropy. Necessary are tools from an extended theory of entropy, the <italic>theory of entropy structures</italic> developed in Downarowicz (2005). The main goal of this paper is to prove the analog of the SEET for actions of (discrete infinite) countable amenable groups:</p>\u0000\u0000<p><disp-quote>\u0000<p>\u0000 <italic>Let a countable amenable group <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> act by homeomorphisms on a compact metric space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper M Subscript upper G Baseline left-parenthesis upper X right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">M</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>G</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {M}_{G}(X)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> denote the simplex of all <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-invariant Borel probability measures on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. A function <inline-formula content-type=\"math/mathml\">\u0000<mml:math","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43127393","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}
Introduction. This series of lectures involves the interplay between local group theory and the theory of fusion systems, with the focus of interest the possibility of using fusion systems to simplify part of the proof of the theorem classifying the finite simple groups. For our purposes, the classification of the finite simple groups begins with the GorensteinWalter Dichotomy Theorem (cf. [ALSS]) which says that each finite group G of 2-rank at least 3 is either of component type or of characteristic 2-type. This supplies a partition of the finite groups into groups of odd and even characteristic, from the point of view of their 2-local structure. We will be concerned almost exclusively with the groups of odd characteristic: the groups of component type. However Ulrich Meierfrankenfeld’s lectures can be thought of as being concerned with the groups of even characteristic. In the case of a saturated fusion system F , the situation vis-a-vis the GorensteinWalter dichotomy is nicer: F is either of characteristic p-type or component type, irrespective of rank. Further the Dichotomy Theorem for saturated fusion systems is much easier to prove than the theorem for groups; indeed once the notion of the generalized Fitting subsystem F ∗(F) of a saturated fusion system F is put in place, and suitable properties of F ∗(F) are established, including E-balance, the proof of the Dichotomy Theorem for fusion systems is easy. But of more importance, it seems easier to work with 2-fusion systems of component type than with groups of component type. This is because in a group G of component type, a 2-local subgroup H of G may have a nontrivial core, where the core of H is the largest normal subgroup O(H) of H of odd order. The existence of these cores introduces big problems into the analysis of groups of component type. These problems can be minimized if one can prove the B-Conjecture, which says that, in a simple group,
{"title":"On Fusion Systems of Component Type","authors":"M. Aschbacher","doi":"10.1090/memo/1236","DOIUrl":"https://doi.org/10.1090/memo/1236","url":null,"abstract":"Introduction. This series of lectures involves the interplay between local group theory and the theory of fusion systems, with the focus of interest the possibility of using fusion systems to simplify part of the proof of the theorem classifying the finite simple groups. For our purposes, the classification of the finite simple groups begins with the GorensteinWalter Dichotomy Theorem (cf. [ALSS]) which says that each finite group G of 2-rank at least 3 is either of component type or of characteristic 2-type. This supplies a partition of the finite groups into groups of odd and even characteristic, from the point of view of their 2-local structure. We will be concerned almost exclusively with the groups of odd characteristic: the groups of component type. However Ulrich Meierfrankenfeld’s lectures can be thought of as being concerned with the groups of even characteristic. In the case of a saturated fusion system F , the situation vis-a-vis the GorensteinWalter dichotomy is nicer: F is either of characteristic p-type or component type, irrespective of rank. Further the Dichotomy Theorem for saturated fusion systems is much easier to prove than the theorem for groups; indeed once the notion of the generalized Fitting subsystem F ∗(F) of a saturated fusion system F is put in place, and suitable properties of F ∗(F) are established, including E-balance, the proof of the Dichotomy Theorem for fusion systems is easy. But of more importance, it seems easier to work with 2-fusion systems of component type than with groups of component type. This is because in a group G of component type, a 2-local subgroup H of G may have a nontrivial core, where the core of H is the largest normal subgroup O(H) of H of odd order. The existence of these cores introduces big problems into the analysis of groups of component type. These problems can be minimized if one can prove the B-Conjecture, which says that, in a simple group,","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2018-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83058871","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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