Dynamical correspondences of $L^2$-Betti numbers

Dynamical correspondences for Smale spaces

We initiate the study of correspondences for Smale spaces. Correspondences are shown to provide a notion of a generalized morphism between Smale spaces and are a special case of finite equivalences. Furthermore, for shifts of finite type, a correspondence is related to a matrix which intertwines the adjacency matrices of the shifts. This observation allows us to define an equivalence relation on all Smale spaces which reduces to shift equivalence for shifts of finite type. Several other notions of equivalence are introduced on both correspondences and Smale spaces; a hierarchy between these eq…

ODE振荡爆破解的代数-动力学对应关系

On Algebraic-Dynamical Correspondence of Oscillatory Blow-Up Solutions for ODEs

本文为自治ODE的I型振荡爆破解建立了代数-动力学对应关系，将平衡律周期解与无穷远周期轨道关联，并给出无需显式紧化的存在性判据，同时阐明了两类稳定性信息的差异。

计算与动力学复杂性中的对应关系 II：强制复杂归约

Correspondences in computational and dynamical complexity II: forcing complex reductions

该研究证明不同动力系统间存在自然映射归约时，其行为相似，并指出复杂动力系统中某些问题无法通过简单电路族解决。

计算与动力学复杂性I中的对应关系

Correspondences in computational and dynamical complexity I

本文提出通过计算复杂性理论研究动力系统的方法，利用目标问题的复杂性下界来判断系统拓扑熵的正负。

Dynamical alternating groups, stability, property Gamma, and inner amenability

We prove that the alternating group of a topologically free action of a countably infinite group $Γ$ on the Cantor set has the property that all of its $\ell^2$-Betti numbers vanish and, in the case that $Γ$ is amenable, is stable in the sense of Jones and Schmidt and has property Gamma (and in particular is inner amenable). We show moreover in the realm of amenable $Γ$ that there are many such alternating groups which are simple, finitely generated, and C$^*$-simple. The device for establishing nonisomorphism among these examples is a topological version of Austin's result on the invariance o…

A dynamical proof of the van der Corput inequality

We provide a dynamical proof of the van der Corput inequality for sequences in Hilbert spaces that is based on the Furstenberg correspondence principle. This is done by reducing the inequality to the mean ergodic theorem for contractions on Hilbert spaces. The key difficulty therein is that the Furstenberg correspondence principle is, a priori, limited to scalar-valued sequences. We therefore discuss how interpreting the Furstenberg correspondence principle via the Gelfand--Naimark--Segal construction for C*-algebras allows to study not just scalar but general Hilbert space-valued sequences in…

射影直线上对应的动力系统 II：乘子映射的次数

Dynamical Systems of Correspondences on the Projective Line II: Degrees of Multiplier Maps

本文研究乘子反问题，给出等谱对应数量的显式上界，并以两种计算方法证明 Hutz-Tepper 关于一般三次态射乘子唯一性结果的勘误修正。

动力莫尔斯熵

Dynamical Morse entropy

本文针对可平铺顺从群作用下的拓扑空间连续函数，定义了同调可检测临界点数量的熵，证明其下界为贝蒂数熵之和，推广了微分几何中的标准莫尔斯不等式。

Diophantine exponents of measures: a dynamical approach

We place the theory of metric Diophantine approximation on manifolds into a broader context of studying Diophantine properties of points generic with respect to certain measures on $\Bbb R^n$. The correspondence between multidimensional Diophantine approximation and dynamics of lattices in Euclidean spaces is discussed in an elementary way, and several recent results obtained by means of this correspondence are surveyed.

Relative dynamical degrees of correspondences over a field of arbitrary characteristic

Let $K$ be an algebraically closed field of arbitrary characteristic, $X$ an irreducible variety and $Y$ an irreducible projective variety over $K$, both are not necessarily smooth. Let $f:X\rightarrow X$ and $g:Y\rightarrow Y$ be dominant correspondences, and $π:X\rightarrow Y$ a dominant rational map such that $π\circ f=g\circ π$. We define relative dynamical degrees $λ_p(f|π)$ ($p=0,\ldots ,\dim (X)-\dim (Y)$). These degrees measure the relative growth of positive algebraic cycles, satisfy a product formula when $Y$ is smooth and $g$ is a multiple of a rational map, and are birational invar…

满射局部同胚与一族分离图之间的对应关系

A correspondence between surjective local homeomorphisms and a family of separated graphs

本文通过双色Bratteli分离图建立满射局部同胚动力系统与分离图子类（l-图）的双射对应，推广有限型移位，并将相关Steinberg与C*代数表示为广义有限移位代数的余极限。

A new relation between geometric and dynamical properties of objects such as tilings and Delone sets

Let P be an object such as tiling, Delone set and weighted Dirac comb. There corresponds a dynamical system to P, called the corresponding dynamical system. Such dynamical systems are geometric analogues of symbolic dynamics. It is well-known that there are correspondences between geometric properties of P and properties of the corresponding dynamical system. In this article we give a new correspondence. In other words, we characterize the property that the group of topological eigenvalues for the corresponding dynamical system is not discrete, in terms of a geometric property of P.

On Bertelson-Gromov Dynamical Morse Entropy

In this mainly expository paper we present a detailed proof of several results contained in a paper by M. Bertelson and M. Gromov on Dynamical Morse Entropy. This is an introduction to the ideas presented in that work. Suppose $M$ is compact oriented connected $C^\infty$ manifold of finite dimension. Assume that $f_0 :M \to [0,1]$ is a surjective Morse function. For a given natural number $n$, consider the set $M^n$ and for $x=(x_0,x_1,...,x_{n-1}) \in M^n$, denote $ f_n (x) = \frac{1}{n} \, \sum_{j=0}^{n-1} f_0 (x_j).$ The Dynamical Morse Entropy describes for a fixed interval $I\subset [0,1]…

Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems

Practically and intrinsically, inclusions of operator algebras are of fundamental interest. The subject of this paper is intermediate operator algebras of inclusions. There are two previously known theorems which naturally and completely describe all intermediate operator algebras: the Galois Correspondence Theorem and the Tensor Splitting Theorem. Here we establish the third, new complete description theorem which gives a canonical bijective correspondence between intermediate operator algebras and intermediate extensions of dynamical systems. One can also regard this theorem as a crossed pro…

二维自治动力系统的对称性与其相平面实现之间的对应关系

On the correspondence between symmetries of two-dimensional autonomous dynamical systems and their phase plane realisations

本文建立二维自治动力系统微分方程表述与相平面单方程表述之间对称生成元的对应关系，提出并求解线性偏微分形式的提升条件，且以线性质量守恒模型和非线性振子模型验证。

代数曲线中台球的动力学次数

The dynamical degree of billiards in an algebraic curve

本研究拓展复台球理论提出代数闭域上平面曲线的代数台球表述，证明其动力学次数的显式上界，推导实代数曲线经典台球映射的拓扑熵上界，验证奇点约束与2-形式保持性质，并通过双有理模型完成证明。

Dynamical Pose Estimation

We study the problem of aligning two sets of 3D geometric primitives given known correspondences. Our first contribution is to show that this primitive alignment framework unifies five perception problems including point cloud registration, primitive (mesh) registration, category-level 3D registration, absolution pose estimation (APE), and category-level APE. Our second contribution is to propose DynAMical Pose estimation (DAMP), the first general and practical algorithm to solve primitive alignment problem by simulating rigid body dynamics arising from virtual springs and damping, where the s…

全纯对应图的等分布

Equidistribution of graphs of holomorphic correspondences

研究紧黎曼曲面上的全纯自对应，在动力学度不相等或非弱模条件下，证明迭代图的指数快速等分布。

Dynamical characterization of C-sets and its application

In this paper, we set up a general correspondence between the algebra properties of $\bN$ and the sets defined by dynamical properties. In particular, we obtain a dynamical characterization of C-sets, where C-sets are the sets satisfying the strong Central Sets Theorem. As an application, we show that Rado systems are solvable in C-sets.