召集人：范爱华（法国亚眠大学&武汉大学，教授）、王跃飞（中科学数学与系统科学研究院&深圳大学，研究员）、凡石磊（华中师范大学，教授）

时间：2025.03.02—2025.03.08

会议日程

报告题目及摘要

报告人：陈绍示（中科院数学与系统科学研究院）

题目：Arithmetic Dynamics around the Skolem-Mahler-Lech theorem

摘要：The Skolem-Mahler-Lech theorem discrides the zero structure of a linear recurrence over a field of characteristic zero. This classical theorem can be formulated as a special case of the Dynamical Mordell-Lang conjecture in arithmetic dynamics. This talk will first overview some historical developments and extensions of the Skolem-Mahler-Lech theorem and some recent progress in arithmetic dynamics around this theorem.

报告人：崔贵珍（中科院数学与系统科学研究院/深圳大学）

题目：有理函数动力系统

摘要：我们首先简单介绍复动力系统的基本知识，包括Fatou-Julia理论，Sullivan定理，Thurston定理以及McMullen等人的工作。然后介绍我们在这方面的一些结果。最后介绍后续的一些问题。

报告人：丰德军（香港中文大学）

题目：Dimensions of orthogonal projections of typical set-affine sets and measures

摘要：An important problem in fractal geometry and dynamical systems is understanding the geometric properties of orthogonal projections of dynamical-driven sets and measures along specific directions. In this talk, I will present some dimensional results on the orthogonal projections of typical set-affine sets and measures. This work is based on joint research with Yu-Hao Xie.

报告人：高延（深圳大学）

题目：Combinatorial structure of Julia set

摘要：We introduce some problems and recent progresses about the combinatorial structure of Julia set for rational maps.

报告人：何伟鲲（中科院数学与系统科学研究院）

题目：Quantitative equidistribution of random walks on homogeneous spaces

摘要：In this talk, I will present some recent works about random walks on some homogeneous spaces. These can be regarded as quantitative improvements of the work of Benoist-Quint on stationary measures and equidistribution. This talks is based on joint work with Timothée Bénard and Han Zhang.

报告人：黄文（中国科学技术大学）

题目：Mean complexity and Sarnak Conjecture

摘要：In this talk, we will review the mean complexity and the progress related to the Sarnak conjecture.In particular, we will discuss the logarithmic Sarnak conjecture and its equivalent forms, as well as our characterization by the ploynomial mean complexity.

报告人：孔德荣（重庆大学）

题目：Open dynamics with a moving hole

摘要：Motivated by the study of badly approximated numbers in Diophantine approximations, we consider the expanding map T_b: x↦bx (mod 1) on the unit circle with a moving hole. More precisely, given a sequence of open balls {B_n} we study the set K({B_n}) of x such that T_bⁿ(x) avoids the ball B_n for all but finitely many n. We show that K({B_n}) has full Hausdorff dimension if and only if the size of B_n tends to 0 as n goes to infinity. Our study of K({B_n}) is also related to the products of random matrices. Some multifractal results on K({B_n}) are also presented. This is ongoing joint work with Beibei Sun and Zhiqiang Wang.

报告人：赖冠宇（台湾政治大学）

题目：Hausdorff dimension of Beatty multiple shifts

摘要：In this talk, the Beatty multiple shift is introduced, which is a generalization of the multiplicative shift of finite type (multiple SFT) [Kenyon, Peres and Solomyak, Ergodic Theory and Dynamical Systems, 2012] and the affine multiple shift [Ban, Hu, Lai and Liao, arXiv:2402.18822, 2024]. The Hausdorff and Minkowski dimension formulas are obtained, and the coefficients of the formula is closely related to the classical disjoint covering of the positive integers in number theory.

报告人：李兵（华南理工大学）

题目：Exponentially mixing property for  Gibbs measures on self-conformal sets and  applications

摘要：In this talk, we show that  Gibbs measures on self-conformal sets generated by  a $C^{1+\alpha}$ conformal IFS on $\R^d$ satisfying the OSC are exponentially mixing.  We exploit this to obtain essentially sharp asymptotic counting statements for the (self) recurrent  and  the  shrinking target subsets  associated with any such given self-conformal set. In particular, we provide explicit examples of dynamical systems for which the  recurrent sets  exhibit (unexpected) behavior that is not present in the shrinking target setup. This is a joint work with Junjie Huang and Sanju Velani.

报告人：李智强（北京大学）

题目：Ergodic optimization for beta-transformations

摘要：Ergodic optimization for beta-transformations Tβ(x) = βx (mod 1) is developed. If β >1 is a beta-number, or such that the orbit-closure of 1 is not minimal, we show that the Typically Periodic Optimization Conjecture holds, establishing that there exists an open dense set of Hölder continuous functions such that for each function in this set, there exists a unique maximizing measure, this measure is supported on a periodic orbit, and the periodic locking property holds. It follows that typical periodic optimization is typical among the class of beta-transformations: it holds for a set of parameters β >1 that is residual, and has full Lebesgue measure. This is a joint work with Zelai Hao, Yinying Huang, and Oliver Jenkinson.

报告人：廖灵敏（武汉大学）

题目：Some recent progress in rational dynamics over the field of p-adic numbers and its finite extensions

摘要：Let K be a finite extension of the field Q_p of p-adic numbers. A rational map φ in K(z) of degree at least 2 is studied as a dynamical system on K or on the field C_p of p-adic complex numbers. We review some recent progress in this area of dynamics, conducted in collaboration with Ai-Hua Fan, Shilei Fan, Hongming Nie, Yuefei Wang, and others. Specifically, we prove that if φ is semi-hyperbolic, then the C_p-Julia set, when restricted to K, coincides with the K-Julia set.  Furthermore, we demonstrate that if φ is sub-hyperbolic, the dynamics on its K-Julia set can be described by a countable state Markov shift, whereas the dynamics on its C_p-Julia set corresponds to a full shift on d symbols.

报告人：吕克宁（四川大学）

题目：Turbulence, Lyapunov exponents, and SRB measures in infinite-dimensional dynamical systems

摘要：In this talk, I will present several results related to Lyapunov exponents, SRB measures, entropy, and horseshoes in the context of infinite-dimensional dynamical systems. I will also discuss recent work on the ergodicity and statistical dynamics of the 2D Navier-Stokes equation, driven by both time-dependent deterministic and stochastic forces. Additionally, I will explore the connection between SRB measures and turbulence.

报告人：乔建永（北京邮电大学）

题目：The BC-conjecture on the areas of Julia sets

摘要：In this topic we consider the area problem on Julia sets of  complex dynamical systems. Firstly, we introduce the concepts of periodic cycle and Bryuno number in the theory of complex dynamics, also formulate Douady’s Plan for researching the area problem. Buff and Chéritat’s theorem is one of the important results in this plan, and the BC-conjecture is an open problem for a long-time posed by Buff and Chéritatand. In the last ,we give an outline of the proof of BC-conjecture by Jianyong Qiao & Hongyu Qu.

报告人：沈维孝（复旦大学）

题目：

摘要：

报告人：史汝西（复旦大学）

题目：Lowering mean topological dimension

摘要：As a new topological invariant, the notion of mean topological dimension was introduced by Gromov (1999). It was developed systematically by Lindenstrauss and Weiss. In this talk, I will discuss how to lower mean topological dimension: we prove that for a topological dynamical system with positive mean topological dimension and marker property, it has factors of arbitrary small mean topological dimension and zero relative mean topological dimension which separate points.

报告人：史逸（四川大学）

题目：Lyapunov spectrum rigidity and simultaneous linearization of random Anosov diffeomorphisms

摘要: Let f be a C^r Anosov diffeomorphism on T^2 and {f_1,...,f_k} be a family of C^r-random perturbations of f with r>2. We show that the family {f_1,...,f_k} has local Lyapunov spectrum rigidity, i.e. if the Lyapunov exponents of the stationary SRB measure of {f_1,...,f_k} is equal to Lyapunov exponents of linearization A in GL(2,Z) of f, then there exists a smooth conjugacy h on T^2, such that h\circ f_i\circ h^{-1}=A+v_i for every i=1,...,k. The same result holds for random perturbations of generic hyperbolic automorphism A in GL(d,Z). Moreover, we show that the random perturbations of a family of Zariski dense positive matrices in SL(2,Z) has the positive Lyapunov exponent rigidity on T^2. This is a joint work with A. Brown.

报告人：佟志成（吉林大学）

题目：Exponential convergence of the weighted Birkhoff average

摘要：It is well-known that in the classical Birkhoff Ergodic Theorem, the rate at which the time average of length N converges to the spatial average is at most O(1/N), and in some cases, it can be arbitrarily slow. Building on the work of J. Laskar, J. Yorke, and others, we prove that for quasi-periodic and almost-periodic systems with certain analytic properties, the weighted Birkhoff average can converge at an exponential rate under specific weighting functions, and this is universal in the sense of full measure. We also obtain some quantitative results, which are optimal in a universal sense.

报告人：王晓光（浙江大学）

题目：中心双曲分支与边界延拓问题

摘要：介绍d次多项式空间中心双曲分支的边界上对应的动力系统性质，Milnor映射的边界延拓性质。

报告人：吴军（华中科技大学）

题目：The Cartesian product of limsup sets in Diophantine approximation

摘要：We give a general principle for Hausdorff dimension of the Cartesian product of limsup sets arising in Diophantine approximation. As an application, it yields that

dim_H W(ψ)×…×W(ψ)=d-1+dim_H W(ψ)

where W(ψ) is the set of ψ-well approximable points in R and ψ: N→R₊ is a positive non-increasing function. This is a joint work with Bao-Wei Wang.

报告人：谢俊逸（北京大学）

题目：动力系统的算术复杂性：算术次数

摘要：算术次数刻画了一条轨道的算术复杂性。报告中我们将介绍其定义，基本性质和最新进展。同时我们会通过例子介绍如何将其应用于算术动力系统的其他问题。

报告人：杨文元（北京大学）

题目：Counting problems in groups with contracting elements

摘要：In this talk, we shall study a class of groups with contracting elements and survey several counting results in this class of groups. We will explain a basic tool called extension lemma in obtaining those counting results. This tool could be thought of as an orbit closing lemma in group theory. 

报告人：张国华（复旦大学）

题目：Tiling Structure of Amenable Groups, with Applications to the Comparison Property

摘要：Since the introduction of the concept of amenable groups by von Neumann in 1929 while studying the well-known Banach-Tarski paradox, the structure of amenable groups has remained a bit mysterious. In their seminal work published in 1987, Donald Ornstein and Benjamin Weiss developed the machinery of quasitiling for amenable groups. To further understand the structure of amenable groups, my colleagues and I proved a finitileability theorem, and then solved a question about the tileability of countable amenable groups using finitely many tiles with good invariance properties. This question had remained open for a long time. In this talk, I will present our finitileability theorem, which enhances the Ornstein-Weiss quasitiling machinery for amenable groups. Additionally, if time permits, I will discuss the application of our finitileability theorem to the study of the comparison property of amenable groups. This talk is based on joint works with Tomasz Downarowicz and Dawid Huczek (both from Poland).

报告人：张宏坤（大湾区大学）

题目：Deep Learning for Chaos: Prediction, Structure, and Conservation

摘要：Chaotic systems pose significant challenges due to their sensitivity to initial conditions. This talk explores recent advances in deep learning for modeling chaos, including Discrete-Temporal Sobolev Networks for trajectory prediction, Euler Difference Neural Networks for capturing system dynamics, and neural deflation techniques for discovering conservation laws. By leveraging these methods, we improve predictive accuracy, reveal hidden structures, and enhance our understanding of complex dynamical behavior.

报告人：张志鸿（高雄大学）

题目：Bohr chaoticity of number-conserving shifts

摘要：Let X be a.compact metric space and T: X → X be a continuous transformation. A dynamical system (X, T) is called Bohr chaotic if for each weight sequence (w_n)∈ℓ^∞(N,R) there are f ∈C(X) and x∈X such that (w_n) is orthogonal to {f∘Tⁿ(x)}. For 2 ≤ m∈N and a₀,a₁,...,a_n, c∈Z_m, a number-conserving shift (X, σ) is defined as

X = {x∈Z_m: a₀x_i + a₁x_i+1 +...+ a_nx_i+n = c (mod m), i∈Z},

where σ is the shift map. In this talk, we consider the Bohr chaoticity and topological properties of X. It is seen that X is Bohr chaos if and only if X has a horseshoe. Furthermore, a closed form of the topological entropy of X is determined.