Advances in Mathematics最新文献

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Exponential mixing of all orders and CLT for automorphisms of compact Kähler manifolds 紧态Kähler流形自同构的全阶指数混合及CLT

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-12-01 DOI: 10.1016/j.aim.2025.110689

Fabrizio Bianchi , Tien-Cuong Dinh

We consider the unique measure of maximal entropy of an automorphism of a compact Kähler manifold with simple action on cohomology. We show that it is exponentially mixing of all orders with respect to Hölder observables. It follows that the Central Limit Theorem (CLT) holds for these observables. In particular, our result applies to all automorphisms of compact Kähler surfaces with positive entropy.

考虑上同调上作用简单的紧Kähler流形的自同构最大熵的唯一测度。我们证明了它是关于Hölder可观测值的所有阶的指数混合。由此可见，中心极限定理（CLT）对这些可观测值成立。特别地，我们的结果适用于所有具有正熵的紧致Kähler曲面的自同构。

引用次数: 0

A GL(3) converse theorem via a “beyond endoscopy” approach 基于“超内窥镜”方法的GL(3)逆定理

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-12-01 DOI: 10.1016/j.aim.2025.110716

Valentin Blomer, Wing Hong Leung

We give a new proof of the converse theorem for Maaß forms on

GL (3)

using a technique that is inspired by Langlands' philosophy of “beyond endoscopy”, thereby implementing these ideas for the first time in a higher rank setting.

我们在GL(3)上用一种受Langlands“超越内窥镜”哲学启发的技术给出了maasß型逆定理的新证明，从而首次在更高阶的设置中实现了这些思想。

引用次数: 0

Ladders and squares 梯子和方块

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-11-27 DOI: 10.1016/j.aim.2025.110714

Lorenzo Notaro

In 1984, Ditor asked two questions:

(A)
For each $n \in ω$ and infinite cardinal κ, is there a join-semilattice of breadth $n + 1$ and cardinality $κ^{+ n}$ whose principal ideals have cardinality <κ?
(B)
For each $n \in ω$ , is there a lower finite lattice of cardinality $ℵ_{n}$ whose elements have at most $n + 1$ lower covers?

We show that both questions have positive answers under the axiom of constructibility, and hence consistently with

ZFC

. More specifically, we derive the positive answers from assuming that

□_{κ}

holds for enough κ's.

1984年，Ditor提出了两个问题：(A)对于每个n∈ω和无限基数κ，是否存在一个宽度为n+1，基数为κ+n的连接半格，其主理想的基数为<；κ？(B)对于每个n∈ω，是否存在一个基数为λ n的下有限格，其元素最多有n+1个下覆盖？在构造性公理下，我们证明了这两个问题都有正答案，因此与ZFC一致。更具体地说，我们通过假设□κ具有足够的κ s来得出肯定的答案。

引用次数: 0

Deformation quantization via categorical factorization homology 通过分类分解同调的变形量化

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-11-27 DOI: 10.1016/j.aim.2025.110708

Eilind Karlsson, Corina Keller, Lukas Müller, Ján Pulmann

This paper develops an approach to categorical deformation quantization via factorization homology. We show that a quantization of the local coefficients for factorization homology is equivalent to consistent quantizations of its value on manifolds. To formulate our results we introduce the concepts of shifted almost Poisson and

BD

categories.

Our main example is the character stack of flat principal bundles for a reductive algebraic group G, where we show that applying the general framework to the Drinfeld category reproduces deformations previously introduced by Li-Bland and Ševera. As a direct consequence, we can conclude a precise relation between their quantization and those introduced by Alekseev, Grosse, and Schomerus.

To arrive at our results we compute factorization homology with values in a ribbon category enriched over complete

C [[ħ]]

-modules. More generally, we define enriched skein categories which compute factorization homology for ribbon categories enriched over a general closed symmetric monoidal category

V

本文提出了一种基于因式分解同调的范畴变形量化方法。我们证明了因式同调局部系数的量子化等价于它在流形上的值的一致量子化。为了表述我们的结果，我们引入了移位几乎泊松和BD范畴的概念。我们的主要例子是约化代数群G的平坦主束的特征栈，其中我们展示了将一般框架应用于德林菲尔德范畴再现了Li-Bland和Ševera先前引入的变形。作为直接的结果，我们可以得出它们的量子化与阿列克谢耶夫、格罗斯和舍默鲁斯引入的量子化之间的精确关系。为了得到我们的结果，我们计算了在完全C[[h]]-模上富集的带类中的值的因式同调。更一般地，我们定义了富织带范畴，计算了在一般闭对称单面范畴V上富织带范畴的因式同调。

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引用次数: 0

Exchange matrices of I-boxes 交换i -box矩阵

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-11-27 DOI: 10.1016/j.aim.2025.110711

Masaki Kashiwara , Myungho Kim

Admissible chains of i-boxes are important combinatorial tools in the monoidal categorification of cluster algebras, as they provide seeds of the cluster algebra. In this paper, we explore the properties of maximal commuting families of i-boxes in a more general setting, and define a certain matrix associated with such a family, which we call the exchange matrix. It turns out that, when considering the cluster algebra structure on the Grothendieck rings, this matrix is indeed the exchange matrix of the seed associated with the family, both in certain categories of modules over quantum affine algebras and over quiver Hecke algebras. We prove this by constructing explicit short exact sequences that represent the mutation relations.

可容许i-盒链是聚类代数一元分类中重要的组合工具，因为它提供了聚类代数的种子。本文在更一般的情况下，研究了i-box的极大交换族的性质，并定义了与此族相关的一个矩阵，我们称之为交换矩阵。结果表明，当考虑Grothendieck环上的聚类代数结构时，该矩阵确实是与族相关的种子的交换矩阵，无论是在量子仿射代数上还是在颤振Hecke代数上的模的某些类别中。我们通过构造表示突变关系的显式精确短序列来证明这一点。

引用次数: 0

Open dynamical systems with a moving hole 具有移动孔的开放动力系统

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-11-26 DOI: 10.1016/j.aim.2025.110710

Derong Kong , Beibei Sun , Zhiqiang Wang

<div><div>Given an integer <span><math><mi>b</mi><mo>≥</mo><mn>3</mn></math></span>, let <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>:</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>;</mo><mi>x</mi><mo>↦</mo><mi>b</mi><mi>x</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>1</mn><mo>)</mo></math></span> be the expanding map on the unit circle. For any <span><math><mi>m</mi><mo>∈</mo><mi>N</mi></math></span> and for any <span><math><mi>ω</mi><mo>=</mo><msup><mrow><mi>ω</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>…</mo><mo>∈</mo><msup><mrow><mo>(</mo><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>b</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span> let<span><span><span><math><msup><mrow><mi>K</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>=</mo><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>:</mo><msubsup><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo><mo>∉</mo><msub><mrow><mi>I</mi></mrow><mrow><msup><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mspace></mspace><mo>∀</mo><mi>n</mi><mo>≥</mo><mn>0</mn><mo>}</mo></mrow><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>I</mi></mrow><mrow><msup><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> is the <em>b</em>-adic basic interval generated by <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Then <span><math><msup><mrow><mi>K</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> is called the survivor set of the open dynamical system <span><math><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>,</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>ω</mi></mrow></msub><mo>)</mo></math></span> with respect to the sequence of holes <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>ω</mi></mrow></msub><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>I</mi></mrow><mrow><msup><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>:</mo><mi>n</mi><mo>≥</mo><mn>0</mn><mo>}</mo></mrow></math></span>. We show that the Hausdorff and lower box dimensions of <span><math><msup><mrow><mi>K</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> always coincide, and the packing and upper box dimensions of <span><math><msup><mrow><mi>K</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> also coincide. Moreover, we give sharp lower and upper bounds for the dimensions of <span><math><msup><mrow><mi>K</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>, which can

给定一个整数b≥3，令Tb:[0,1)→[0,1)；X∑bx（mod1）是单位圆上的展开映射。对于任意m∈N及任意ω=ω0ω1…∈({0,1，…，b−1}m)N0 letKω={x∈[0,1]:Tbn(x)∈Iωn∀N≥0}，其中Iωn为ω N生成的b进基区间。将Kω称为开放动力系统（[0,1），Tb,Iω)相对于空穴Iω={Iωn:n≥0}序列的幸存集。我们证明了Kω的Hausdorff尺寸和下盒尺寸总是重合的，Kω的包装尺寸和上盒尺寸也是重合的。此外，我们给出了Kω维数的明确的下界和上界，这些下界和上界可以显式地计算出来。最后，对于任意允许的α≤β，我们证明存在无穷多个ω（事实上是正维）使得dimH (k) ω=α和dimP (k) ω=β。作为应用，我们首先研究丢芬图近似中的差近似数。对于任意的球序列{Bn}，设K（{Bn}）是x∈[0,1)的集合，使得Tbn(x)∈Bn对于除有限个n≥0以外的所有n≥0。假设极限limn→∞(Bn)存在，我们证明了当且仅当limn→∞(Bn)=0时，dimH (K)=1。同样，我们的结果可以应用于非递归点的集合。对于任意N上的正函数φ，设E（φ）是x∈[0,1]的集合，满足对除有限多个N外的所有N满足|Tbn(x)−x|≥φ (N)。然后证明在极限limn→∞(N)存在的条件下，当且仅当limn→∞(N)=0时，dih (E)=1。我们的结果也可用于研究矩阵的联合谱半径。我们证明了相关邻接矩阵的联合谱半径的有限性是成立的。

引用次数: 0

Partially hyperbolic dynamics with quasi-isometric center 具有准等距中心的部分双曲动力学

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-11-26 DOI: 10.1016/j.aim.2025.110697

Ziqiang Feng

We consider the class of partially hyperbolic diffeomorphisms on a closed 3-manifold with quasi-isometric center. When the non-wandering set is the whole manifold, we prove that the diffeomorphisms are accessible if there is no su-torus. As a consequence, volume-preserving diffeomorphisms in this context are ergodic in the absence of su-tori, thereby confirming the Hertz-Hertz-Ures Ergodicity Conjecture for this class.

For any closed 3-manifold with fundamental group of exponential growth, we show that it supports transitive Anosov flows if and only if it admits non-wandering partially hyperbolic diffeomorphisms with quasi-isometric center. Furthermore, we provide a complete classification of these diffeomorphisms, showing they fall into two categories: skew products and discretized Anosov flows.

研究一类具有拟等距中心的闭3流形上的部分双曲微分同态。当非游走集是整流形时，证明了当不存在苏环面时，差分同态是可达的。因此，在这种情况下，保体积的微分同态在没有苏托里的情况下是遍历的，从而证实了该类的赫兹-赫兹-乌尔斯遍历猜想。对于任何具有指数增长基本群的闭3流形，我们证明当且仅当它允许具有拟等距中心的非漫游部分双曲微分同态时，它支持传递Anosov流。此外，我们提供了这些差分同态的完整分类，表明它们分为两类：偏积和离散的Anosov流。

引用次数: 0

Nonpositively curved 4-manifolds with zero Euler characteristic 具有零欧拉特性的非正弯曲4流形

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-11-26 DOI: 10.1016/j.aim.2025.110680

Chris Connell , Yuping Ruan , Shi Wang

We show that for any closed nonpositively curved Riemannian 4-manifold M with vanishing Euler characteristic, the Ricci curvature must degenerate somewhere. Moreover, for each point

p \in M

, either the Ricci tensor degenerates or else there is a foliation by totally geodesic flat 3-manifolds in a neighborhood of p. As a corollary, we show that if in addition the metric is analytic, then the universal cover of M has a nontrivial Euclidean de Rham factor. Finally we discuss how this result creates an implication among conjectures on simplicial volume in dimension four.

我们证明了对于任何具有消失欧拉特征的闭非正弯曲黎曼4流形M，里奇曲率必须在某处简并。此外，对于每个点p∈M，要么里奇张量退化，要么在p的邻域内存在完全测地平面3流形的叶化。作为推论，我们证明了如果度量是解析的，则M的全称覆盖具有非平凡的欧几里得德拉姆因子。最后，我们讨论了这个结果如何在四维简单体积的猜想中产生一个暗示。

引用次数: 0

Rauzy fractals of random substitutions 随机替换的杂乱分形

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-11-26 DOI: 10.1016/j.aim.2025.110713

P. Gohlke , A. Mitchell , D. Rust , T. Samuel

We develop a theory of Rauzy fractals for random substitutions, which are a generalisation of deterministic substitutions where the substituted image of a letter is determined by a Markov process. We show that a Rauzy fractal can be associated with a given random substitution in a canonical manner, under natural assumptions on the random substitution. Further, we show the existence of a natural measure supported on the Rauzy fractal, which we call the Rauzy measure, that captures geometric and dynamical information. We provide several different constructions for the Rauzy fractal and Rauzy measure, which we show coincide, and ascertain various analytic, dynamical and geometric properties. While the Rauzy fractal is independent of the choice of (non-degenerate) probabilities assigned to a given random substitution, the Rauzy measure captures the explicit choice of probabilities. Moreover, Rauzy measures vary continuously with the choice of probabilities, thus provide a natural means of interpolating between Rauzy fractals of deterministic substitutions. Additionally, we highlight connections between Rauzy fractals and Rauzy measures of random substitutions and related S-adic systems.

我们发展了随机替换的Rauzy分形理论，这是确定性替换的推广，其中字母的替换图像由马尔可夫过程确定。我们证明了在随机替换的自然假设下，Rauzy分形可以以规范的方式与给定的随机替换相关联。进一步，我们证明了Rauzy分形支持的自然测度的存在性，我们称之为Rauzy测度，它捕获几何和动态信息。我们对Rauzy分形和Rauzy测度给出了几种不同的构造，并证明了它们是一致的，并确定了各种解析性质、动力学性质和几何性质。虽然Rauzy分形与分配给给定随机替换的（非退化）概率的选择无关，但Rauzy度量捕获了概率的显式选择。此外，Rauzy测度随概率的选择而连续变化，从而提供了一种在确定性替换的Rauzy分形之间进行插值的自然方法。此外，我们强调了Rauzy分形和随机替换的Rauzy测度以及相关的S-adic系统之间的联系。

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引用次数: 0

Assouad spectrum of Gatzouras–Lalley carpets 各种各样的Gatzouras-Lalley地毯

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-11-26 DOI: 10.1016/j.aim.2025.110707

Amlan Banaji , Jonathan M. Fraser , István Kolossváry , Alex Rutar

We study the fine local scaling properties of a class of self-affine fractal sets called Gatzouras–Lalley carpets. More precisely, we establish a formula for the Assouad spectrum of all Gatzouras–Lalley carpets as the concave conjugate of an explicit piecewise-analytic function combined with a simple parameter change. Our formula implies a number of novel properties for the Assouad spectrum not previously observed for dynamically invariant sets; in particular, the Assouad spectrum can be a non-trivial differentiable function on the entire domain

(0, 1)

and can be strictly concave on open intervals. Our proof introduces a general framework for covering arguments using techniques developed in the context of multifractal analysis, including the method of types from large deviations theory and Lagrange duality from optimisation theory.

研究了一类自仿射分形集Gatzouras-Lalley地毯的精细局部标度性质。更准确地说，我们建立了所有Gatzouras-Lalley地毯的assad谱的公式，作为显式分段解析函数与简单参数变化相结合的凹共轭。我们的公式暗示了一些以前未观察到的动态不变集的assad谱的新性质；特别地，assad谱在整个（0,1）域上可以是一个非平凡的可微函数，并且在开区间上可以是严格凹的。我们的证明引入了一个通用框架，用于涵盖在多重分形分析背景下开发的技术，包括来自大偏差理论的类型方法和来自优化理论的拉格朗日对偶。

引用次数: 0

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