Correspondences in complex dynamics

Recurrence in the dynamics of meromorphic correspondences and holomorphic semigroups

This paper studies recurrence phenomena in iterative holomorphic dynamics of certain multi-valued maps. In particular, we prove an analogue of the Poincaré recurrence theorem for meromorphic correspondences with respect to certain dynamically interesting measures associated with them. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result concerning invariance prop…

变量分离仿射对应的临界动力学

Critical dynamics of variable-separated affine correspondences

本文研究了f和g多项式满足deg（g）< deg（f）的形式g（y）=f（x）的仿射对应，其性质是对应的每个临界点至少包含一个有限前向轨道.在g（y）=y的情形下，这就归结为对后临界有限多项式的研究，我们的主要结果推广了作者早期的有限性结果.具体而言，我们表明，这样的对应的一个给定的bidegree的集合相吻合的参数空间的有界Weil高度的一个子集。我们还表明，有没有非平凡的全纯家庭的对应上述性质。

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。

全纯对应动力学中的遍历性

Ergodicity in the dynamics of holomorphic correspondences

本文针对全纯对应动力学中不满足经典不变性的测度，定义新的遍历性概念，证明对应版本的Birkhoff遍历定理，验证紧复流形上此类遍历测度的存在性，并给出一类显式的遍历测度实例。

A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\dot y(t)=g(y(t))$ on $\mathbb{R}^d$ and those of the parabolic equations $\dot u=Δu +f(x,u,\nabla u)$ on a bounded domain $Ω$. We give details on the similarities of these dynamics in the cases $d=1$, $d=2$ and $d\geq 3$ and in the corresponding cases $Ω=(0,1)$, $Ω=\mathbb{T}^1$ and dim($Ω$)$\geq 2$ respectively. In addition to the beauty of such a correspondence, this could serve as a guideline for future research on the dynamics of parabolic equations.

Dynamics of holomorphic correspondences on Riemann Surfaces

We study the dynamics of holomorphic correspondences $f$ on a compact Riemann surface $X$ in the case, so far not well understood, where $f$ and $f^{-1}$ have the same topological degree. Under a mild and necessary condition that we call non weak modularity, $f$ admits two canonical probability measures $μ^+$ and $μ^-$ which are invariant by $f^*$ and $f_*$ respectively. If the critical values of $f$ (resp. $f^{-1}$) are not periodic, the backward (resp. forward) orbit of any point $a \in X$ equidistributes towards $μ^+$ (resp. $μ^-$), uniformly in $a$ and exponentially fast.

A Galois-dynamics correspondence for unicritical polynomials

In an analogy with the Galois homothety property for torsion points of abelian varieties that was used in the proof of the Mordell-Lang conjecture, we describe a correspondence between the action of a Galois group and the dynamical action of a rational map. For nonlinear polynomials with rational coefficients, the irreducibility of the associated dynatomic polynomial serves as a convenient criterion, although we also verify that the correspondence occurs in several cases when the dynatomic polynomial is reducible. The work of Morton, Morton-Patel, and Vivaldi-Hatjispyros in the early 1990s con…

代数对应与Schwarz反射：理性动力学与克莱因群的交汇

Algebraic correspondences and Schwarz reflections: Where rational dynamics meets Kleinian groups

本文探讨了代数对应动力学与克莱因群的交汇，通过理性映射与克莱因群的结合，揭示了参数空间中的统一结构及应用前景。

Holomorphic correspondences related to finitely generated rational semigroups

In this paper, we present a new technique for studying the dynamics of a finitely generated rational semigroup. Such a semigroup can be associated naturally to a certain holomorphic correspondence on $\mathbb{P}^1$. Then, results on the iterative dynamics of such a correspondence can be applied to the study of the rational semigroup. We focus on a certain invariant measure for the aforementioned correspondence---known as the equilibrium measure. This confers some advantages over many of the known techniques for studying the dynamics of rational semigroups. We use the equilibrium measure to ana…

代数与积分的一般对应

A General Correspondence between Averages and Integrals

最近的工作已经广义的Furstenberg对应集的整数和动力系统的版本，其中涉及序列的有限图或序列的$L^\infty$功能。我们给出了一个统一的版本的定理包含所有这些推广。

结构拉姆齐理论和拓扑动力学的综述（英文）

A survey on structural Ramsey theory and topological dynamics with the Kechris-Pestov-Todorcevic correspondence in mind

2005年，Kechris、Pestov和Todorcevic在结构拉姆齐理论和拓扑动力学之间建立了惊人的对应关系。作为一个直接的后果，它引发了新的兴趣，结构拉姆齐理论。本文件的目的是对相应的事态发展作一个独立的概述。

扩展行走液滴动力学中的液滴-波统计对应关系

Extending the Droplet-Wave Statistical Correspondence in Walking Droplet Dynamics

本文扩展了行走液滴动力学中液滴-波统计对应关系，考虑了任意碰撞模型、多液滴相互作用和非共振弹跳，通过数值实验区分相关与不相关液滴对。

共形动力学的镜像：反有理映射、反射群、Schwarz反射与对应之间的相互作用

Mirrors of conformal dynamics: Interplay between anti-rational maps, reflection groups, Schwarz reflections, and correspondences

本文综述了反有理映射迭代、Kleinian反射群作用、正交域中Schwarz反射生成的动力学以及代数对应这四类共形动力学分支之间的密切联系，并通过实例和一般结果展示了它们之间的动力学关系及技术工具的应用。