Annals of Mathematics最新文献_第8页

The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables 中心对称台球桌的Birkhoff—Poritsky猜想

IF 4.9 1区数学

Annals of Mathematics Pub Date : 2020-08-08 DOI: 10.4007/annals.2022.196.1.2

M. Bialy, A. Mironov

{"title":"The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables","authors":"M. Bialy, A. Mironov","doi":"10.4007/annals.2022.196.1.2","DOIUrl":"https://doi.org/10.4007/annals.2022.196.1.2","url":null,"abstract":"In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric $C^2$-smooth convex planar billiards. We assume that the domain $mathcal A$ between the invariant curve of $4$-periodic orbits and the boundary of the phase cylinder is foliated by $C^0$-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. Other versions of Birkhoff-Poritsky conjecture follow from this result. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a $C^1$-smooth foliation by convex caustics of rotation numbers in the interval (0; 1/4] then the boundary curve is an ellipse. In the language of first integrals one can assert that {if the billiard inside a centrally-symmetric $C^2$-smooth convex curve $gamma$ admits a $C^1$-smooth first integral which is not singular on $mathcal A$, then the curve $gamma$ is an ellipse. } \u0000The main ingredients of the proof are : (1) the non-standard generating function for convex billiards discovered in [8], [10]; (2) the remarkable structure of the invariant curve consisting of $4$-periodic orbits; and (3) the integral-geometry approach initiated in [6], [7] for rigidity results of circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiard in ellipse.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2020-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44514204","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}

引用次数: 23

Pointwise ergodic theorems for non-conventional bilinear polynomial averages 非常规双线性多项式平均的点态遍历定理

IF 4.9 1区数学

Annals of Mathematics Pub Date : 2020-08-03 DOI: 10.4007/annals.2022.195.3.4

Ben Krause, Mariusz Mirek, T. Tao

{"title":"Pointwise ergodic theorems for non-conventional bilinear polynomial averages","authors":"Ben Krause, Mariusz Mirek, T. Tao","doi":"10.4007/annals.2022.195.3.4","DOIUrl":"https://doi.org/10.4007/annals.2022.195.3.4","url":null,"abstract":"We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages [ A_N(f,g)(x) := frac{1}{N} sum_{n =1}^N f(T^nx) g(T^{P(n)}x)] as $N to infty$, where $T colon X to X$ is a measure-preserving transformation of a $sigma$-finite measure space $(X,mu)$, $P(mathrm{n}) in mathbb Z[mathrm{n}]$ is a polynomial of degree $d geq 2$, and $f in L^{p_1}(X), g in L^{p_2}(X)$ for some $p_1,p_2 > 1$ with $frac{1}{p_1} + frac{1}{p_2} leq 1$. We also establish an $r$-variational inequality for these averages (at lacunary scales) in the optimal range $r > 2$. We are also able to ``break duality'' by handling some ranges of exponents $p_1,p_2$ with $frac{1}{p_1}+frac{1}{p_2} > 1$, at the cost of increasing $r$ slightly. \u0000This gives an affirmative answer to Problem 11 from Frantzikinakis' open problems survey for the Furstenberg--Weiss averages (with $P(mathrm{n})=mathrm{n}^2$), which is a bilinear variant of Question 9 considered by Bergelson in his survey on Ergodic Ramsey Theory from 1996. Our methods combine techniques from harmonic analysis with the recent inverse theorems of Peluse and Prendiville in additive combinatorics. At large scales, the harmonic analysis of the adelic integers $mathbb A_{mathbb Z}$ also plays a role.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2020-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48469027","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}

引用次数: 22

Chow groups and $L$-derivatives of automorphic motives for unitary groups Chow群和酉群自同构动机的$L$-导数

IF 4.9 1区数学

Annals of Mathematics Pub Date : 2020-06-11 DOI: 10.4007/annals.2021.194.3.6

Chao Li, Yifeng Liu

引用次数: 21

Smooth mixing Anosov flows in dimension three are exponentially mixing 平滑混合三维的阿诺索夫流是指数混合

IF 4.9 1区数学

Annals of Mathematics Pub Date : 2020-06-08 DOI: 10.4007/annals.2023.197.1.2

M. Tsujii, Zhiyuan Zhang

引用次数: 19

The Hasse principle for random Fano hypersurfaces 随机Fano超曲面的Hasse原理

IF 4.9 1区数学

Annals of Mathematics Pub Date : 2020-06-03 DOI: 10.4007/annals.2023.197.3.3

T. Browning, Pierre Le Boudec, W. Sawin

引用次数: 9

Rigid local systems and the multiplicative eigenvalue problem 刚性局部系统与乘性特征值问题

IF 4.9 1区数学

Annals of Mathematics Pub Date : 2020-05-26 DOI: 10.4007/annals.2022.195.3.3

P. Belkale

{"title":"Rigid local systems and the multiplicative eigenvalue problem","authors":"P. Belkale","doi":"10.4007/annals.2022.195.3.3","DOIUrl":"https://doi.org/10.4007/annals.2022.195.3.3","url":null,"abstract":"We give a construction which produces irreducible complex rigid local systems on $Bbb{P}_{Bbb{C}}^1-{p_1,dots,p_s}$ via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups $operatorname{SU}(n)$ (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals of the simplest vertices of these polytopes give (all) possible unitary irreducible rigid local systems. As a consequence we obtain that the ranks of unitary irreducible rigid local systems (including those with finite global monodromy) on $Bbb{P}^1-S$ are bounded above if we fix the cardinality of the set $S={p_1,dots,p_s}$ and require that the local monodromies have orders which divide $n$, for a fixed $n$. Answering a question of N. Katz, we show that there are no irreducible rigid local systems of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing $n$, when $n$ is a prime number. \u0000We also show that all unitary irreducible rigid local systems on $Bbb{P}^1_{Bbb{C}} -S$ with finite local monodromies arise as solutions to the Knizhnik-Zamalodchikov equations on conformal blocks for the special linear group. Along the way, generalising previous works of the author and J. Kiers, we give an inductive mechanism for determining all vertices in the multiplicative eigenvalue problem for $operatorname{SU}(n)$.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":null,"pages":null},"PeriodicalIF":4.9,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42682066","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}

引用次数: 6

The rectangular peg problem 矩形钉问题

IF 4.9 1区数学

Annals of Mathematics Pub Date : 2020-05-19 DOI: 10.4007/annals.2021.194.2.4

J. Greene, A. Lobb

引用次数: 16

Special subvarieties of non-arithmetic ball quotients and Hodge theory 非算术球商的特殊子变量与Hodge理论

IF 4.9 1区数学

Annals of Mathematics Pub Date : 2020-05-07 DOI: 10.4007/annals.2023.197.1.3

G. Baldi, E. Ullmo

引用次数: 6

There is no Enriques surface over the integers 整数上没有恩里克曲面