具有多数值不变量的向量场族

平面多项式向量场族的达布型中心条件

Darboux-type center conditions for families of planar polynomial vector fields

研究平面多项式向量场的中心焦点问题，基于李雅普诺夫函数法，给出两类微分系统中心焦点条件新结果，找到由李雅普诺夫常数生成的鲍廷理想的一个包含关系，结果适用于任意次数\(n\geq2\)。

Berger domains and Kolmogorov typicality of infinitely many invariant circles

Using the novel notion of parablender, P. Berger proved that the existence of finitely many attractors is not Kolmogorov typical in parametric families of diffeomorphisms. Here, motivated by the concept of Newhouse domains we define Berger domains for families of diffeomorphisms. As an application, we show that the coexistence of infinitely many attracting invariant smooth circles is Kolmogorov typical in certain non-sectionally dissipative Berger domains of parametric families in dimension three or greater.

Compact families and typical entropy invariants of measure-preserving actions

For a compact set of actions, an invariant of Kushnirenko's entropy type is chosen in such a way that on this set it is equal to zero, but will be infinity for typical actions. As a consequence, we show that typical measure-preserving transformations are not isomorphic to geometric shape exchange transformations. This problem arose in connection with the result of Chaika and Davis about the atypical nature of IETs.

Common zeroes of families of smooth vector fields on surfaces

Let Y and X denote C^k vector fields on a possibly noncompact surface with empty boundary, k >0. Say that Y tracks X if the dynamical system it generates locally permutes integral curves of X. Let K be a locally maximal compact set of zeroes of X. THEOREM Assume the Poincar'e-Hopf index of X at K is nonzero, and the k-jet of X at each point of K is nontrivial. If g is a supersolvable Lie algebra of C^k vector fields that track X, then the elements of g have a common zero in K. Applications are made to attractors and transformation groups.

关于扭转映射台球解析族周期不变曲线的存在性

On the existence of periodic invariant curves for analytic families of twist maps and billiards

本文证明环面扭转映射解析族中给定旋转数的周期不变曲线要么对所有参数值存在，要么至多存在于离散子集中，并将此结论推广并应用于台球的有理焦散线模型。

关于二心问题中lemniscate运动族不变量的研究

On invariants of families of lemniscate motions in the two-center problem

本文研究了二心问题中lemniscate运动的拓扑不变量，基于Arnold的$J^+$-不变量确定了四个新的不变量。

通过辛不变量研究Hill问题的对称周期轨道族网络

Studying network of symmetric periodic orbit families of the Hill problem via symplectic invariants

本文在空间圆形Hill三体问题框架下，应用辛不变量分析对称周期轨道族网络结构，通过计算Conley-Zehnder指标构建分岔图，揭示周期解族间的相互关联及对称性的关键作用。

Global bifurcations in the two-sphere: a new perspective

We construct an open set of structurally unstable three parameter families whose weak and so called moderate topological classification defined below has a numerical invariant that may take an arbitrary positive value. Here and below "families" are "families of vector fields in the two-sphere". This result disproves an Arnold's conjecture of 1985. Then we construct an open set of six parameter families whose moderate topological classification has a functional invariant. This invariant is an arbitrary germ of a smooth map $(\mathbb R_+, a)\to(\mathbb R_+, b)$. More generally, for any positive …

Parameter Families of Iterated Function Systems and Continuity

In this paper, we investigate parameter families of iterated function systems and continuity. Specifically, if we have a set of iterated function systems that depend continuously on a parameter, which properties of the invariant sets will vary continuously? We show here that the invariant sets will vary continuously as a function of the parameter. Furthermore, we give some sufficient conditions under which the Hausdorff dimension will depend continuously on the parameter. Lastly, we give a counter-example where the dimension is not a continuous function of the parameter.

参数化映射族的周期点不变量比较

Comparing Periodic Point Invariants for Parameterized Families of Maps

本文比较紧流形参数化映射族的周期点不变量，指出纤维Fuller迹比迭代的纤维Reidemeister迹集合更敏感，解决了Malkiewich和Ponto的相关猜想。

New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant

We show that an orientable pseudo-Anosov homeomorphism has vanishing Sah-Arnoux-Fathi invariant if and only if the minimal polynomial of its dilatation is not reciprocal. We relate this to works of Margalit-Spallone and Birman, Brinkmann and Kawamuro. Mainly, we use Veech's construction of pseudo-Anosov maps to give explicit pseudo-Anosov maps of vanishing Sah-Arnoux-Fathi invariant. In particular, we give new infinite families of such maps in genus 3.

识别拟平稳近不变集族的起始与衰减及其在大气阻塞事件中的应用

Identifying the onset and decay of quasi-stationary families of almost-invariant sets with an application to atmospheric blocking events

针对时变动力系统，提出一种扩展转移算子方法（含Ulam格式），识别空间拟平稳的近不变集族及其起始与衰减，应用于2003年欧洲热浪大气阻塞事件，为相空间动力学组织提供高层信息。

Common hypercyclic vectors for high dimensional families of operators

Let $(T\_λ)\_{λ\inΛ}$ be a family of operators acting on a $F$-space $X$, where the parameter space $Λ$ is a subset of $\mathbb R^d$. We give sufficient conditions on the family to yield the existence of a vector $x\in X$ such that, for any $λ\inΛ$, the set $\big\{T\_λ^n x;\ n\geq 1\big\}$ is dense in $X$. We obtain results valid for any value of $d\geq 1$ whereas the previously known results where restricted to $d=1$. Our methods also shed new light on the one-dimensional case.