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显示 2025年08月06日， 星期三 新的列表
- [1] arXiv:2508.02918 [中文pdf, pdf, 其他]
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标题： 对称性类的中心构型分解标题： Decomposition of Symmetrical Classes of Central ConfigurationsMarcelo P. Santos (1), Leon D. da Silva (1) ((1) Federal Rural University of Pernambuco)主题： 动力系统 (math.DS) ; 符号计算 (cs.SC) ; 表示理论 (math.RT)
我们研究当位置集合具有对称性时的中心配置。 我们使用有限群表示论中的一个定理来探讨中心配置方程的对称性质。 这种方法通过考虑任意数量的物体、对称群和维度，简化了中心配置的方程。 我们讨论如何利用这个定理获得比之前更精细的方程分解。 这里提出的分解方法使用了适应对称性的基方法。 作为应用，我们给出了两个嵌套正四面体、两个嵌套正八面体以及两个嵌套正立方体的中心配置的存在性及可能质量的完整描述。 为此，我们采用了一些有理参数化方法和多元多项式零点的隔离方法。 得到的分解允许使用符号计算来研究这些表达式。 通过这种方式，我们总结了之前的讨论，并通过完成对立方体情况的分析，扩展了这些讨论，包括逆问题和正问题。
We study central configurations when the set of positions is symmetric. We use a theorem from representation theory of finite groups to explore the symmetry properties of equations for central configurations. This approach simplifies equations for central configurations by considering arbitrary numbers of bodies, symmetry groups, and dimensions. We discuss how to use this theorem to obtain a more refined decomposition of the equations than that given before. The decomposition presented here uses the symmetry-adapted basis method. As an application, we give a complete description of the existence and which masses are possible for central configurations of two nested regular tetrahedrons, two nested regular octahedrons, and two nested regular cubes. To do this, we employ some methods of rational parameterizations and isolation of zeros of multivariate polynomials. The decomposition obtained allows symbolic calculations to be used to study the expressions. This way, we summarize previous discussions and extend them by completing the analysis on the cube case, for both the inverse and direct problems.
- [2] arXiv:2508.03206 [中文pdf, pdf, html, 其他]
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标题： 多周期动力学和SIRS流行病模型中高余维分支的三次心理饱和接触率标题： Multicycle dynamics and high-codimension bifurcations in SIRS epidemic models with cubic psychological saturated incidence主题： 动力系统 (math.DS)
本研究探讨了具有三次饱和接触率的SIRS流行病模型中的分岔动力学，扩展了Lu、Huang、Ruan和Yu（《微分方程杂志》，267，2019年）建立的二次饱和框架。 我们严格证明了三阶Bogdanov-Takens分岔和退化Hopf分岔的存在性，首次在流行病建模中展示了三个极限环的共存。 通过创新地应用奇异性理论，我们通过奇点的局部展开和前缘非退化奇点的识别来表征分岔集的拓扑结构。 我们的结果表明，在单调和非单调饱和条件下，三次非线性比二次模型诱导出更丰富的动态结构。 数值模拟验证了单调参数区域中的三个极限环和非单调区域中的两个极限环。 本研究通过引入高阶相互作用和全面的奇异性分析，推进了现有的分岔研究，从而为解码关键的公共卫生策略设计中的复杂传播机制提供了数学基础。
This study investigates bifurcation dynamics in an SIRS epidemic model with cubic saturated incidence, extending the quadratic saturation framework established by Lu, Huang, Ruan, and Yu (Journal of Differential Equations, 267, 2019). We rigorously prove the existence of codimension-three Bogdanov-Takens bifurcations and degenerate Hopf bifurcations, demonstrating, for the first time in epidemiological modeling, the coexistence of three limit cycles. By innovatively applying singularity theory, we characterize the topology of the bifurcation set through the local unfolding of singularities and the identification of nondegenerate singularities for fronts. Our results reveal that cubic nonlinearities induce significantly richer dynamical structures than quadratic models under both monotonic and nonmonotonic saturation. Numerical simulations verify three limit cycles in monotonic parameter regimes and two limit cycles in nonmonotonic regimes. This work advances existing bifurcation research by incorporating higher-order interactions and comprehensive singularity analysis, thereby providing a mathematical foundation for decoding complex transmission mechanisms that are critical to the design of public health strategies.
- [3] arXiv:2508.03238 [中文pdf, pdf, html, 其他]
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标题： 模拟大豆豆象种群温湿度耦合动力学及评估PCM-NN算法的预测性能标题： Modeling the Temperature-Humidity Coupling Dynamics of Soybean Pod Borer Population and Assessing the Predictive Performance of the PCM-NN Algorithm主题： 动力系统 (math.DS)
在全球气候变化和农业全球化的背景下，大豆生产日益受到害虫爆发的威胁，其中豆荚螟（学名：Leguminivora glycinivorella，通常称为大豆荚蜂）是一种主要的害虫种类。这种害虫分布广泛，尤其是在中国东北地区，这是中国主要的大豆产区，其爆发对产量和质量都产生了显著影响。尽管统计模型和机制模型已被用于害虫预测，但现有方法往往难以有效整合气候因素与害虫动态，并且表达能力不足。为解决这些局限性，本研究提出了一种基于物理信息神经网络（PINNs）的新颖害虫预测方法。具体而言，我们构建了一个逻辑型常微分方程（ODE），该方程结合了微气候因素、温度、湿度和时间，以描述大豆荚蜂种群的时间动态。该ODE模型被嵌入到PINN框架中，开发出害虫相关性模型神经网络（PCM-NN），用于联合推断由微气候驱动的参数函数alpha(T, H, t）并拟合害虫种群动态。我们使用2020年至2023年7月至9月期间在吉林省长春市收集的大豆荚蜂每日监测数据对PCM-NN进行了评估。实验结果表明，PCM-NN在保持生物可解释性的同时表现出强大的非线性表示能力，为在多因素气候条件下进行害虫建模和预测提供了一条可行的路径。这种方法为农业害虫监测、预防和控制策略提供了有价值的支持。
Against the backdrop of global climate change and agricultural globalization, soybean production is increasingly threatened by pest outbreaks, with Leguminivora glycinivorella (commonly known as the soybean pod borer) being a major pest species. This pest is widely distributed, particularly in northeastern China, the country's primary soybean-producing region, where its outbreaks have significantly affected both yield and quality. Although statistical and mechanistic models have been applied to pest forecasting, existing approaches often fail to effectively integrate climatic factors with pest dynamics and lack sufficient expressive power. To address these limitations, this study proposes a novel pest prediction method based on Physics-Informed Neural Networks (PINNs). Specifically, we formulate a logistic-type ordinary differential equation (ODE) that incorporates microclimate factors, temperature, humidity, and time, to describe the temporal dynamics of the soybean pod borer population. This ODE model is embedded into the PINN framework to develop the Pest Correlation Model Neural Network (PCM-NN), which is used to jointly infer the microclimate-driven parameter function alpha(T, H, t) and fit the pest population dynamics. We evaluate PCM-NN using daily monitoring data of soybean pod borer collected in Changchun, Jilin Province, from July to September during 2020-2023. Experimental results demonstrate that PCM-NN preserves biological interpretability while exhibiting strong nonlinear representational capacity, offering a feasible pathway for pest modeling and forecasting under multi-factor climatic conditions. This approach provides valuable support for agricultural pest monitoring, prevention, and control strategies.
- [4] arXiv:2508.03304 [中文pdf, pdf, html, 其他]
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标题： 基于几何奇摄动理论的化学反应网络的坐标无关模型降阶标题： Coordinate-independent model reductions of chemical reaction networks based on geometric singular perturbation theory评论： 37页，6图主题： 动力系统 (math.DS)
准稳态近似（QSSA）是降低化学反应网络（CRN）复杂性的标准技术。任何基于QSSA的模型的有效性仅限于特定的参数范围，这些范围通常会重叠，意味着多个不同的简化模型可以同时有效。这种模糊性使分析变得不必要的复杂，因为选择适当的简化并不总是显而易见的。在这里，我们采用了一种更强大的替代方法：坐标无关的几何奇异摄动理论（ci-GSPT）结合参数化方法。该框架的一个关键优势是其能够推导出不依赖于变量中明显的时间尺度分离的简化模型，从而意味着更少限制的参数范围。我们在两个基准系统上展示了我们的方法。对于Michaelis-Menten（MM）反应，我们为特定参数配置的正常双曲且吸引的临界流形推导出唯一的模型简化。我们系统地探索了三个数量级的参数配置：渐近大、小和“阶一”。因此，我们提供了以下不同的模型简化：(i) 不可逆反应方案的14个相关参数配置和(ii) 可逆反应方案的25个参数配置（其他存在平凡简化或临界流形退化的参数配置简要讨论）。对于更复杂的Kim-Forger模型，我们提供了一种无需坐标变换的新简化，展示了该方法在更大系统中的适用性。
The quasi-steady-state approximation (QSSA) is a standard technique for reducing the complexity of chemical reaction networks (CRNs). The validity of any QSSA-based model is restricted to specific parameter regimes, which often overlap, meaning multiple different reduced models can be simultaneously valid. This ambiguity complicates unnecessarily the analysis, as selecting the appropriate reduction is not always straightforward. Here, we employ a more powerful alternative: coordinate-independent geometric singular perturbation theory (ci-GSPT) accompanied by the parametrization method. A key advantage of this framework is its ability to derive reduced models independent of a clear timescale separation in the variables which, in turn, means less-restricted parameter regimes. We demonstrate our approach on two benchmark systems. For the Michaelis-Menten (MM) reaction, we derive a unique model reduction for normally hyperbolic and attracting critical manifolds of a specific parameter configuration. We systematically explore parameter configurations across three orders of magnitude: asymptotically large, small, and `order one'. Hence, we provide distinct model reductions for: (i) 14 relevant parameter configurations of the irreversible reaction scheme and (ii) 25 for the reversible reaction scheme (other parameter configurations where there are trivial reductions or degeneracies in critical manifolds, are discussed briefly). For the more complex Kim-Forger model, we provide a new reduction without the need of a coordinate transformation showcasing the method's applicability to larger systems.
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- [5] arXiv:2508.02735 (交叉列表自 math.NA) [中文pdf, pdf, html, 其他]
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标题： 周期稳态脉冲在短脉冲光纤激光器中的Floquet稳定性标题： Floquet stability of periodically stationary pulses in a short-pulse fiber laser评论： arXiv管理员注释：与arXiv:2508.01133文本重叠期刊参考： 《应用数学杂志》，第84卷，第3期，第961--987页，2024年主题： 数值分析 (math.NA) ; 动力系统 (math.DS)
现代短脉冲光纤激光器的定量建模和设计无法使用平均模型，因为每个往返过程中脉冲参数有较大的变化。 相反，需要使用通过连接激光器各个组件的模型而得到的集中模型。 由于集中模型中的光脉冲是周期性的，因此使用单值化算子来研究其线性稳定性，该算子是关于脉冲的往返算子的线性化。 开发了一种基于梯度的优化方法来发现周期性脉冲。 目标函数梯度的计算涉及往返算子和单值化算子的伴随算子作用的数值计算。 引入了一种新的傅里叶分裂步法来计算非线性、非局部、刚性方程的线性化方程的解，该方程模拟光纤放大器中的光传播。 该方法是通过对非线性方程的分裂步法中的两个求解算子进行线性化得到的。 单值化算子的谱包括本质谱，对于本质谱有一个解析公式，以及特征值。 在$\lambda=1$处有一个重数为二的特征值，这是由于相位和平移不变性引起的。 其余的特征值由单值化算子的矩阵离散化确定。 仿真结果验证了数值方法的准确性，展示了周期稳态脉冲的例子，它们的谱和本征函数，并讨论了它们的稳定性。
The quantitative modeling and design of modern short-pulse fiber lasers cannot be performed with averaged models because of large variations in the pulse parameters within each round trip. Instead, lumped models obtained by concatenating models for the various components of the laser are required. Since the optical pulses in lumped models are periodic, their linear stability is investigated using the monodromy operator, which is the linearization of the roundtrip operator about the pulse. A gradient-based optimization method is developed to discover periodic pulses. The computation of the gradient of the objective function involves numerical computation of the action of both the round trip operator and the adjoint of the monodromy operator. A novel Fourier split-step method is introduced to compute solutions of the linearization of the nonlinear, nonlocal, stiff equation that models optical propagation in the fiber amplifier. This method is derived by linearizing the two solution operators in a split-step method for the nonlinear equation. The spectrum of the monodromy operator consists of the essential spectrum, for which there is an analytical formula, and the eigenvalues. There is a multiplicity two eigenvalue at $\lambda=1$, which is due to phase and translation invariance. The remaining eigenvalues are determined from a matrix discretization of the monodromy operator. Simulation results verify the accuracy of the numerical methods, show examples of periodically stationary pulses, their spectra and eigenfunctions, and discuss their stability.
- [6] arXiv:2508.02809 (交叉列表自 math.CV) [中文pdf, pdf, html, 其他]
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标题： 同时线性化和抛物型自映射的中心化子 I：零双曲步标题： Simultaneous linearization and centralizers of parabolic self-maps I: zero hyperbolic step主题： 复变量 (math.CV) ; 动力系统 (math.DS)
设$\varphi:\mathbb D \to \mathbb D$为单位圆盘$\mathbb D$上的抛物型自映射，其具有零双曲步长。我们研究与$\varphi$交换的$\mathbb D$的全纯自映射。 特别地，我们通过证明$\psi\in\mathsf{Hol(\mathbb D,\mathbb D)}$与$\varphi$交换当且仅当这两个自映射具有相同的 Denjoy-Wolff 点，并且$\psi$在 Cowen 的意义上是$\varphi$的伪迭代，从而回答了 Gentili 和 Vlacci (1994) 提出的问题。 此外，我们证明了$\varphi$的中心化子，即半群$\mathcal Z_\forall(\varphi):=\{\psi:\psi\circ\varphi=\varphi\circ\psi\}$是可交换的。 我们还证明，如果$\varphi$是单值的，那么$\mathcal Z_\forall(\varphi)$的所有元素也都是单值的，如果$\varphi$不是单值的，那么恒等映射是$\mathcal Z_\forall(\varphi)$的一个孤立点。 主要工具是同时线性化的机制，我们通过 Cowen 和 Pommerenke 的非椭圆自映射迭代的全纯模型来开发这一机制。
Let $\varphi:\mathbb D \to \mathbb D$ be a parabolic self-map of the unit disc $\mathbb D$ having zero hyperbolic step. We study holomorphic self-maps of $\mathbb D$ commuting with $\varphi$. In particular, we answer a question from Gentili and Vlacci (1994) by proving that $\psi\in\mathsf{Hol(\mathbb D,\mathbb D)}$ commutes with $\varphi$ if and only if the two self-maps have the same Denjoy-Wolff point and $\psi$ is a pseudo-iterate of $\varphi$ in the sense of Cowen. Moreover, we show that the centralizer of $\varphi$, i.e. the semigroup $\mathcal Z_\forall(\varphi):=\{\psi:\psi\circ\varphi=\varphi\circ\psi\}$ is commutative. We also prove that if $\varphi$ is univalent, then all elements of $\mathcal Z_\forall(\varphi)$ are univalent as well, and if $\varphi$ is not univalent, then the identity map is an isolated point of $\mathcal Z_\forall(\varphi)$. The main tool is the machinery of simultaneous linearization, which we develop using holomorphic models for iteration of non-elliptic self-maps originating in works of Cowen and Pommerenke.
- [7] arXiv:2508.02928 (交叉列表自 q-bio.QM) [中文pdf, pdf, html, 其他]
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标题： 用于SEIQR流行病学PDE模型的非标准有限差分格式标题： A nonstandard finite difference scheme for an SEIQR epidemiological PDE mode主题： 定量方法 (q-bio.QM) ; 动力系统 (math.DS) ; 数值分析 (math.NA)
本文介绍了一种非标准有限差分（NSFD）方法，用于反应扩散SEIQR流行病学模型，该模型捕捉传染病传播的时空动态。 作为半线性抛物型偏微分方程（PDEs）系统表述，该模型通过引入空间扩散来考虑人口流动和空间异质性，从而扩展了经典的分 compartment 模型。 所提出的NSFD离散化设计旨在保持连续模型的基本定性特征，如正性、有界性和稳定性，这些特征通常会被标准有限差分方法破坏。 我们严格分析了模型的适定性，为PDE系统构建了一个结构保持的NSFD方案，并研究了其收敛性和局部截断误差。 数值模拟验证了理论结果，并展示了该方案在保持生物上一致动力学方面的有效性。
This paper introduces a nonstandard finite difference (NSFD) approach to a reaction-diffusion SEIQR epidemiological model, which captures the spatiotemporal dynamics of infectious disease transmission. Formulated as a system of semilinear parabolic partial differential equations (PDEs), the model extends classical compartmental models by incorporating spatial diffusion to account for population movement and spatial heterogeneity. The proposed NSFD discretization is designed to preserve the continuous model's essential qualitative features, such as positivity, boundedness, and stability, which are often compromised by standard finite difference methods. We rigorously analyze the model's well-posedness, construct a structure-preserving NSFD scheme for the PDE system, and study its convergence and local truncation error. Numerical simulations validate the theoretical findings and demonstrate the scheme's effectiveness in preserving biologically consistent dynamics.
- [8] arXiv:2508.03200 (交叉列表自 nlin.SI) [中文pdf, pdf, html, 其他]
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标题： 赫斯-斯雷滕斯基问题中刚体运动的Liouvillan解的存在性标题： Existence of Liouvillan solutions in the Hess-Sretensky case of the problem of motion of a gyrostat with a fixed point评论： arXiv管理员备注：与arXiv:2011.14183存在大量文本重叠主题： 精确可解与可积系统 (nlin.SI) ; 动力系统 (math.DS)
1890年，W. Hess发现了刚体绕固定点运动的欧拉-泊松方程可积性的新特殊情况。1963年，L.N. 斯列滕斯基证明了类似于Hess情况的可积特殊情况也存在于一个带有旋转均匀转子的重陀螺（即具有固定点的重刚体）的运动问题中。随后提出了许多经典的Hess情况的推广，这些推广在刚体和陀螺在各种力场中的运动过程中成立。几乎在发现这一情况后立即发表了提供重刚体在可积Hess情况下运动的定性描述的首次研究。1892年，P.A.涅克拉索夫证明，在Hess情况下，具有固定点的重刚体运动问题的解可以归结为对具有变系数的二阶线性齐次微分方程的积分。斯列滕斯基则给出了关于Hess-斯列滕斯基情况下重陀螺运动问题的类似结果。在本文中，我们推导了相应的二阶线性微分方程，并将该方程的系数简化为有理函数的形式。然后，使用Kovacic算法研究了相应二阶线性微分方程的Liouvillian解的存在性问题。我们得到了问题参数的条件，使得相应线性微分方程的Liouvillian解存在。在这些条件下，Hess-斯列滕斯基情况下具有固定点的重陀螺的运动方程可以通过积分求解。
In 1890 W. Hess found the new special case of integrability of the Euler - Poisson equations of motion of a heavy rigid body with a fixed point. In 1963 L.N. Sretensky proved that the special case of integrability, similar to the Hess case, also exists in the problem of the motion of a heavy gyrostat - a heavy rigid body with a fixed point, which contains a rotating homogeneous rotor. Further numerous generalizations of the classical Hess case were proposed, which take place during the motion of a heavy rigid body and a gyrostat with a fixed point in various force fields. The first studies that provided a qualitative description of the motion of a heavy rigid body in the integrable Hess case were published almost immediately after this case was found. In 1892 P.A. Nekrasov proved, that the solution of the problem of motion of a heavy rigid body with a fixed point in the Hess case is reduced to the integration the second order linear homogeneous differential equation with variable coefficients. A similar result regarding the problem of the motion of a heavy gyrostat in the Hess - Sretensky case was presented by Sretensky. In this paper we present the derivation of the corresponding second order linear differential equation and reduce the coefficients of this equation to the form of rational functions. Then, using the Kovacic algorithm we study the problem of the existence of liouvillian solutions of the corresponding second order linear differential equation. We obtain the conditions for the parameters of the problem, under which the liouvillian solutions of the corresponding linear differential equation exist. Under these conditions equations of motion of a heavy gyrostat with a fixed point in the Hess - Sretensky case can be integrated in quadratures.
- [9] arXiv:2508.03308 (交叉列表自 math.NT) [中文pdf, pdf, html, 其他]
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标题： 形如$\boldsymbol{x^d+c}$的后临界有限多项式的迭代标题： Iterates of post-critically finite polynomials of the form $\boldsymbol{x^d+c}$评论： 17页主题： 数论 (math.NT) ; 动力系统 (math.DS)
固定一个素数$d$。 形式为$f_{d,c} = x^d+c\in \mathbb{C}[x]$的后临界有限多项式在多项式动力系统中起着基础性的作用。 虽然在复动力系统环境下已知许多结果，但对这些多项式的算术性质了解甚少。 在本文中，我们描述了后临界有限多项式$f_{d,c}$在其定义域上的迭代的分解情况。 作为推论，我们证明了 Andrews 和 Petsche 关于阿贝尔树状伽罗瓦表示的一个猜想的新情况。
Fix a prime number $d$. The post-critically finite polynomials of the form $f_{d,c} = x^d+c\in \mathbb{C}[x]$ play a fundamental role in polynomial dynamics. While many results are known in the complex dynamical setting, much less is understood about the arithmetic properties of these polynomials. In this paper, we describe the factorization of the iterates of post-critically finite polynomials $f_{d,c}$ over their fields of definition. As a consequence, we prove new cases of a conjecture of Andrews and Petsche on abelian arboreal Galois representations.
- [10] arXiv:2508.03359 (交叉列表自 math.NT) [中文pdf, pdf, html, 其他]
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标题： 从球到开集的维数质量传递原理及其在动力学丢番图逼近中的应用标题： A dimensional mass transference principle from balls to open sets and applications to dynamical Diophantine approximation主题： 数论 (math.NT) ; 动力系统 (math.DS)
伯斯涅维奇和维拉尼的质量传递原理是确定在丢番图逼近中自然出现的$\limsup$集合的豪斯多夫维数/测度的强大机制。 然而，在动力学丢番图逼近的背景下，这一原理通常无法有效应用，因为定义动力学$\limsup$集合的球的半径通常依赖于点$x$的轨道本身。 在本文中，我们发展了一个维数质量传递原理，使我们能够恢复并扩展关于收缩目标问题的经典结果，特别是针对$\beta$变换和高斯映射。 此外，我们的结果表明，相应的$\limsup$集合具有大交集性质。 我们方法的一个潜在有趣的特点是，在许多情况下，收缩目标问题与寻找适当的吉布斯测度密切相关，这可能揭示热力学形式主义与动力学丢番图逼近之间联系的新方面。
The mass transference principle of Beresnevich and Velani is a powerful mechanism for determining the Hausdorff dimension/measure of $\limsup$ sets that arise naturally in Diophantine approximation. However, in the setting of dynamical Diophantine approximation, this principle often fails to apply effectively, as the radii of the balls defining the dynamical $\limsup$ sets generally depend on the orbit of the point $x$ itself. In this paper, we develop a dimensional mass transference principle that enables us to recover and extend classical results on shrinking target problems, particularly for the $\beta$-transformation and the Gauss map. Moreover, our result shows that the corresponding $\limsup$ sets have large intersection properties. A potentially interesting feature of our method is that, in many cases, shrinking target problems are closely related to finding an appropriate Gibbs measure, which may reveal new aspects of the link between thermodynamic formalism and dynamical Diophantine approximation.
交叉提交 (展示 6 之 6 条目 )
- [11] arXiv:2507.21605 (替换) [中文pdf, pdf, html, 其他]
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标题： 关于随机伯努利卷积的傅里叶变换标题： On the Fourier transform of random Bernoulli convolutions主题： 动力系统 (math.DS) ; 概率 (math.PR)
我们研究随机伯努利卷积，即由无限卷积给出的概率测度\[ \mu_\omega = \mathop{\circledast}_{k=1}^{\infty} \left( \frac{\delta_0 + \delta_{\lambda_1 \lambda_2 \ldots \lambda_{k-1} \lambda_k}}{2} \right), \]，其中$\omega=(\lambda_k)$是一系列独立同分布的随机变量，每个变量都服从某个固定区间的均匀分布。我们研究这些测度的正则性，并证明当$\exp\mathbb{E}\left( \log \lambda_1\right)>\frac{2}{\pi}, $时，傅里叶变换$\widehat{\mu}_\omega$几乎必然是一个$L^{1}$函数。这进一步意味着支撑$\mu_{\omega}$的相应随机自相似集几乎必然具有非空内部。这改进了 Peres、Simon 和 Solomyak 之前的界限。 此外，在不对$\exp \mathbb{E}(\log \lambda_1), $的值做出任何假设的情况下，我们证明$\widehat \mu_\omega$将几乎必然以多项式速率衰减到零。
We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution \[ \mu_\omega = \mathop{\circledast}_{k=1}^{\infty} \left( \frac{\delta_0 + \delta_{\lambda_1 \lambda_2 \ldots \lambda_{k-1} \lambda_k}}{2} \right), \] where $\omega=(\lambda_k)$ is a sequence of i.i.d. random variables each following the uniform distribution on some fixed interval. We study the regularity of these measures and prove that when $\exp\mathbb{E}\left( \log \lambda_1\right)>\frac{2}{\pi}, $ the Fourier transform $\widehat{\mu}_\omega$ is an $L^{1}$ function almost surely. This in turn implies that the corresponding random self-similar set supporting $\mu_{\omega}$ has non-empty interior almost surely. This improves upon a previous bound due to Peres, Simon and Solomyak. Furthermore, under no assumptions on the value of $\exp \mathbb{E}(\log \lambda_1), $ we prove that $\widehat \mu_\omega$ will decay to zero at a polynomial rate almost surely.
- [12] arXiv:2508.00631 (替换) [中文pdf, pdf, html, 其他]
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标题： 关于哈雷方法的动力学标题： On the dynamics of Halley's method评论： 25页，14图主题： 动力系统 (math.DS)
在本文中，我们研究了将Halley方法应用于复多项式时的全局动力学。 具体而言，我们分析了该方法的Julia集的结构和连通性。 对于各种类型的多项式，包括具有非平凡对称群的单临界多项式、三次多项式和四次多项式，研究了相应的Fatou集和Julia集的收敛行为、对称性质和拓扑特征。 特别是，我们证明了当$p$属于上述类别之一时，Halley方法$H_p$是收敛的，其Julia集是连通的，瞬时吸引域是无界的，并且其对称群与多项式的对称群一致。 我们将结果进一步扩展到更广泛的多项式类别。 结果显示，对应于$p$的根的Halley方法$H_p$的瞬时吸引域可能是有界的。 我们还对一般三次多项式应用Halley方法的动力学做出了一些备注。
In this article, we study the global dynamics of Halley's method applied to complex polynomials. Specifically, we analyze the structure and connectivity of the Julia set of this method. The convergence behavior, symmetry properties, and topological features of the corresponding Fatou and Julia sets are studied for various classes of polynomials, including unicritical, cubic, and quartic polynomials with non-trivial symmetry groups. In particular, we prove that the Halley's method $H_p$ is convergent, its Julia set is connected, the immediate basins are unbounded and the symmetry group of it coincides with that of the polynomial whenever $p$ belongs to one of the above classes. We further extend our results to a broader class of polynomials. It is shown that the immediate basin of the Halley's method $H_p$ corresponding to a root of $p$ can be bounded. We also make some remarks on the dynamics of the Halley's method applied to a cubic polynomial in general.
- [13] arXiv:2305.14545 (替换) [中文pdf, pdf, html, 其他]
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标题： 群的Liouville性质和共形维数标题： Liouville property for groups and conformal dimension评论： v4：42页，5张图，最终版本主题： 群论 (math.GR) ; 动力系统 (math.DS) ; 概率 (math.PR)
共形维数是度量空间的基本不变量，特别适合于自相似空间的研究，例如具有扩张自覆盖的空间（例如复有理函数的Julia集）。这些系统的动力学由相关的迭代单值群编码，这些是收缩自相似群的例子。它们的可约性是一个众所周知的开放问题。我们证明，如果$G$是一个迭代单值群，并且底层空间的（Alfhors-正则）共形维数严格小于2，那么在$G$上的每个具有有限二阶矩的对称随机游走都具有Liouville性质。作为推论，每个这样的群都是可约的。此准则适用于所有之前已知的可约收缩群例子以及许多新的例子。特别是，它意味着对于每一个子双曲复有理函数$f$，其Julia集不是整个球面，那么$f$的迭代单值群是可约的。
Conformal dimension is a fundamental invariant of metric spaces, particularly suited to the study of self-similar spaces, such as spaces with an expanding self-covering (e.g. Julia sets of complex rational functions). The dynamics of these systems are encoded by the associated iterated monodromy groups, which are examples of contracting self-similar groups. Their amenability is a well-known open question. We show that if $G$ is an iterated monodromy group, and if the (Alfhors-regular) conformal dimension of the underlying space is strictly less than 2, then every symmetric random walk with finite second moment on $G$ has the Liouville property. As a corollary, every such group is amenable. This criterion applies to all examples of contracting groups previously known to be amenable, and to many new ones. In particular, it implies that for every sub-hyperbolic complex rational function $f$ whose Julia set is not the whole sphere, the iterated monodromy group of $f$ is amenable.
- [14] arXiv:2407.17701 (替换) [中文pdf, pdf, html, 其他]
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标题： 几何、算术和混沌标题： On Geometry, Arithmetics and Chaos主题： 代数几何 (math.AG) ; 微分几何 (math.DG) ; 动力系统 (math.DS)
我们的主要结果是，维度$n+1$中的混沌是一个一维的几何对象，嵌入在一个维度为$n$的几何对象中，这对应于一个维度为$n$的对象，该对象可能是奇异的或非奇异的。 我们的主要结果是，在第一种情况下，这种混沌发生在孤立奇点或非孤立奇点上。 在第一种情况下，这种混沌要么是边界混沌，要么是球面混沌，这在非奇异情况下也发生。 在孤立奇点几何的情况下，会出现混沌，这种混沌可以是边界混沌、球面混沌或管状混沌。 我们进一步证明，质数表现出量子行为。
Our main result is that chaos in dimension $n+1$ is a one-dimensional geometrical object embedded in a geometrical object of dimension $n$ which corresponds to a $n$ dimensional object which is either singular or non-singular. Our main result is then that this chaos occurs in the first case as either on an isolated or non-isolated singularity. In the first case this chaos is either boundary chaos or spherical chaos which is what happens also in the non-singular case. In the case of an isolated singular geometry one has chaos which can either be boundary, spherical or tubular chaos. We furthermore prove that the prime numbers display quantum behaviour.
- [15] arXiv:2504.05941 (替换) [中文pdf, pdf, html, 其他]
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标题： 离散时间继电器反馈系统中的自持振荡标题： Self-sustained oscillations in discrete-time relay feedback systems评论： 根据审稿人的意见更新一些图表主题： 优化与控制 (math.OC) ; 系统与控制 (eess.SY) ; 动力系统 (math.DS)
我们研究在离散时间线性时不变继电器反馈系统中确定自持振荡的问题。 具体来说，我们感兴趣的是预测这样的系统何时具有单峰振荡，即输出具有单峰值周期。 在假设线性系统是稳定且其脉冲响应在其无限支持上严格单调递减的前提下，我们采用总正性框架来解决我们的主要问题，这是一种新的方法。 结果表明，只有当一个周期内的正元素和负元素的数量相同时，才可能存在单峰自持振荡。 基于这一结果，我们推导了此类振荡存在的条件，确定了它们周期的界限，并解决了唯一性问题。
We study the problem of determining self-sustained oscillations in discrete-time linear time-invariant relay feedback systems. Concretely, we are interested in predicting when such a system admits unimodal oscillations, i.e., when the output has a single-peaked period. Under the assumption that the linear system is stable and has an impulse response that is strictly monotonically decreasing on its infinite support, we take a novel approach in using the framework of total positivity to address our main question. It is shown that unimodal self-oscillations can only exist if the number of positive and negative elements in a period coincides. Based on this result, we derive conditions for the existence of such oscillations, determine bounds on their periods, and address the question of uniqueness.
- [16] arXiv:2508.01347 (替换) [中文pdf, pdf, 其他]
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标题： 廉价嵌入原理：同调增长的动力学上界标题： The cheap embedding principle: Dynamical upper bounds for homology growth评论： 104页，标题已更正主题： 代数拓扑 (math.AT) ; 动力系统 (math.DS) ; 群论 (math.GR) ; 算子代数 (math.OA)
我们提供了群的对数扭同调增长和贝蒂数增长的上界，用测度群理论的语言来表述。
We provide upper bounds for logarithmic torsion homology growth and Betti number growth of groups, phrased in the language of measured group theory.