A. O. Prishlyak
Theorems on the existence of vector fields with given sets of Indexes of isolated Singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a two-dimensional manifold, an index of an isolated Singular point of the gradient field is not greater than one.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Generically, the set of points along which two non-singular vector fields on the three-sphere are positively (resp. negatively) collinear form a link. We prove that the two vector fields are homotopic if and only if the linking number of those links is zero. We use this criterion to give a new proof of a result of Yano: every non-singular vector field on the three-sphere is homotopic to a non-singular Morse-Smale vector field.
We describe all possible topological structures of codimension one gradient vector fields on the shpere with at most ten singular points. To describe structures, we use a graph whose edges are one-dimensional stable manifolds. The saddle-node singularity is specified by selecting a pair of vertices-edge or edge-face, and the saddle connection is specified by a T-vertex.
This paper is concerned with the analysis of a typical singularity of piecewise smooth vector fields on $R^3$ composed by two zones. In our object of study, the cusp-fold singularity, we consider the simultaneous occurrence of a cusp singularity for one vector field and a fold singularity for the other one. We exhibit a normal form that presents one of the most important property searched for in piecewise smooth vector fields: the asymptotical stability.
On physical measures of multi-singular hyperbolic vector fields
本文研究多奇异双曲向量场的物理测度，证明在C^1开且稠密子集中，C^∞向量场有有限多个物理测度，其吸引盆覆盖全勒贝格测度子集，C^1一般情形亦类似。
Our object of study is non smooth vector fields on $\R^2$. We apply the techniques of geometric singular perturbations in non smooth vector fields after regularization and a blow$-$up. In this way we are able to bring out some results that bridge the space between non$-$smooth dynamical systems presenting typical singularities and singularly perturbed smooth systems.
Let $w=(w_1, w_2)$ be a pair of positive real numbers with $w_1+w_2=1$ and $w_1\ge w_2$. We show that the set of $w$-weighted singular vectors in $\mathbb R^2$ has Hausdorff dimension $2- \frac{1}{1+w_1}$. This extends the previous work of Yitwah Cheung on the Hausdorff dimension of the usual (unweighted) singular vectors in $\mathbb R^2$.
We use group representation theory to give algebraic formulae to compute complete transversals of singularities of vector fields, either in the nonsymmetric or in the reversible equivariant contexts. This computation produces normal forms directly, which are used sistematically in the local analysis of symmetric dynamics.
This paper is about the integrability of complex vector fields in dimension three in a neighborhood of a singular point. More precisely, we study the existence of holomorphic first integrals for isolated singularities of holomorphic vector fields in complex dimension three, pursuing the discussion started in \cite{CaSc2009}. Under generic conditions, we prove a topological criteria for the existence of a holomorphic first integral. Our result may be seen as a kind of Reeb stability result for the framework of vector fields singularities in complex dimension three. As a consequence, we prove th…
Singular points in generic two-parameter families of vector fields on 2-manifold
本文刻画紧二维流形上一般双参数向量场族中所有可能的奇点，为研究二维球面双参数向量场族的全局分岔奠定基础。
We prove that any vector field on a three-dimensional compact manifold can be approximated in the C1-topology by one which is singular hyperbolic or by one which exhibits a homoclinic tangency associated to a regular hyperbolic periodic orbit. This answers a conjecture by Palis. During the proof we obtain several other results with independent interest: a compactification of the rescaled sectional Poincaré flow and a generalization of Mañé-Pujals-Sambarino theorem for three-dimensional C2 vector fields with singularities.
The coexistence of singularities and regular orbits in chain transitive sets has been a major obstacle in understanding the hyperbolic/partial hyperbolic nature of robust dynamics. Notably, the vector fields with all periodic orbits robustly hyperbolic (star flows), are hyperbolic in absence of singularities. Morales, Pacifico and Pujals proposed a partial hyperbolicity called "singular hyperbolicity" that characterizes an open and dense subset of three dimensional star flows. In higher dimensions, Bonatti and da Luz characterize an open and dense set of star vector fields by multi-singular hy…
Singular complex analytic vector fields on the Riemann surfaces enjoy several geometric properties (singular means that poles and essential singularities are admissible). We describe relations between singular complex analytic vector fields $\mathbb{X}$ and smooth vector fields $X$. Our approximation route studies three integrability notions for real smooth vector fields $X$ with singularities on the plane or the sphere. The first notion is related to Cauchy-Riemann equations, we say that a vector field $X$ admits an adapted complex structure $J$ if there exists a singular complex analytic vec…
Homoclinic tangencies and singular hyperbolicity are involved in the Palis conjecture for vector fields. Typical three dimensional vector fields are well understood by recent works. We study the dynamics of higher dimensional vector fields that are away from homoclinic tangencies. More precisely, we prove that for \emph{any} dimensional vector field that is away from homoclinic tangencies, all singularities contained in its robustly transitive singular set are all hyperbolic and have the same index. Moreover, the robustly transitive set is {$C^1$-generically }partially hyperbolic if the vector…
We prove inheritance of measure zero property of the set of singular vectors for affine subspaces and submanifolds inside those affine subspaces. We define a notion of $n$-singularity for matrices, which is closely related to the uniform exponent of irrationality. For certain affine subspaces, we show that the set of singular vectors has measure zero if and only if the parametrizing matrix is not $n$-singular. In particular, we show for affine hyperplanes the set of singular vectors has measure zero if and only if the parametrizing matrix is not rational.
In the paper, we show that for a generic $C^1$ vector field $X$ on a closed three dimensional manifold $M$, any isolated transitive set of $X$ is singular hyperbolic. It is a partial answer of the conjecture in \cite{MP}.
We classify degenerate singular points of $\C^2$-actions on complex surfaces.
Singularities of 3d vector fields preserving the form of Martinet
研究保持 Martine 形式的 3D 向量场的奇点分类及其横截展开，提供分岔的完整列表。
We consider germs of holomorphic vector fields with an isolated singularity at the origin $0\in\mathbb{C}^2$. We introduce a notion of stability, similar to "Lyapunov stability". For such a germ, called $L$-stable singularity, either the corresponding foliation admits a holomorphic first integral, or it is a real logarithmic foliation singularity. A notion of $L$-stability is also naturally introduced for a leaf of a foliation. In the complex codimension one case, for holomorphic foliations, the holonomy groups of $L$-stable leaves are proved to be abelian, of a suitable type. This implies the…
Quantitative estimates for flows of regular Lagrangian flows for Hörmander singular kernels and LD vector fields
本文针对导数可表示为Hörmander奇异核与可和函数卷积的向量场，给出了正则拉格朗日流的定量估计。
We prove that singular vectors have measure zero with respect to any friendly measure on $\Bbb R^n$ (e.g. the volume measure on a nondegenerate submanifold). This generalizes special cases considered by Davenport-Schmidt, Baker and Bugeaud. The main tool is quantitative nondivergence estimates for quasi-polynomial flows on homogeneous spaces.
In planar analytic vector fields, a monodromic singularity can be distinguished between a focus or a center by means of the Lyapunov coefficients, which are given in terms of the power series coefficients of the first-return map defined around the singularity. In this paper, we are interested in an analogous problem for monodromic tangential singularities of piecewise analytic vector fields $Z=(Z^+ ,Z^-)$. First, we prove that the first-return map, defined in a neighborhood of a monodromic tangential singularity, is analytic, which allows the definition of the Lyapunov coefficients. Then, as a…
Motivated by the wild behavior of isolated essential singularities in complex analysis, we study singular complex analytic vector fields $X$ on arbitrary Riemann surfaces $M$. By vector field singularities we understand zeros, poles, isolated essential singularities and accumulation points of the above kind. In this framework, a singular analytic vector field $X$ has canonically associated; a 1-form, a quadratic differential, a flat metric (with a geodesic foliation), a global distinguished parameter or $\mathbb{C}$-flow box $Ψ_X$, a Newton map $Φ_X$, and a Riemann surface $\mathcal{R}_X$ aris…
It is known that a generic star vector field $X$ on a $3$ or $4$-dimensional manifold is such that its chain recurrence classes are either hyperbolic, or singular hyperbolic ([MPP] and [GSW]). Palis conjectured that every vector field must be approximated either by singular hyperbolic vector fields or by vector fields with homoclinic tangencies or heterodimensional cycles (associated to periodic orbits). We give a counter example in dimension $5$ (and higher). We present here an open set of star vector fields on a $5$-dimensional manifold for which two singular points with different indices be…
Gómez-Mont, Seade and Verjovsky introduced an index, now called GSV-index, generalizing the Poincaré-Hopf index to complex vector fields tangent to singular hypersurfaces. The GSV-index extends to the real case. This is a survey paper on the joint research with Gómez-Mont and Giraldo about calculating the GSV-index $\Ind_{V_\pm,0}(X)$ of a real vector field $X$ tangent to a singular hypersurface $V=f^{-1}(0)$. The index $\Ind_{V_{\pm,0}}(X)$ is calculated as a combination of several terms. Each term is given as a signature of some bilinear form on a local algebra associated to $f$ and $X$. Mai…
An upper bound of the Hausdorff dimension of singular vectors on affine subspaces
本文研究了$\mathbb{R}^n$中仿射子空间上的奇异向量集合，并给出了其豪斯多夫维数的一个上界，该上界由仿射子空间参数的丢番图指数决定。
We generalize Khintchine's method of constructing totally irrational singular vectors and linear forms. The main result of the paper shows existence of totally irrational vectors and linear forms with large uniform Diophantine exponents on certain subsets of $\mathbb{R}^n$, in particular on any analytic submanifold of $\mathbb{R}^n$ of dimension $\ge 2$ which is not contained in a proper rational affine subspace.
One of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations (\cite{moussu}). In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with "many" periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in di…
On the Hausdorff dimension of Weighted Singular Vectors
该研究利用齐性空间动力学及积分不等式方法证明$\mathbb{R}^m$中加权奇异向量豪斯多夫维数的精确上界，结合已有下界确定其维数，推广二维结果至任意维，并给出分形子集上该维数的首个已知界。
The connection between discrete and continuous dynamical systems through the unit-time map has shown a significant role in bifurcation theory. Recently, it has also been used in fractal analysis of bifurcations. We study fractal properties of the unit-time map near nilpotent nonmonodromic singularities of planar vector fields using normal forms. We are interested in nilpotent singularities because they are nonhyperbolic, and we know that near nonhyperbolic singularities the box dimension is nontrivial. We study discrete orbits generated by the unit-time map, on the separatrices at the bifurcat…
We extend the notion of singular vectors in the context of Diophantine approximation of real numbers with elements of a totally real number field $K$. For $m\geq1$, we establish a version of Dani's correspondence in number fields and prove that under a class of `friendly measures' in $K_S^m$, the set of singular vectors has measure zero. Here $S$ is the set of Archimedean valuations of $K$ and $K_S$ is the product of the completions of $σ(K)$, $σ\in S$. On the other hand, we show the existence of uncountably many non-trivial singular vectors on suitable submanifolds of $K^m_S$ under the action…
Geometry of transcendental singularities of complex analytic functions and vector fields
本文研究了Riemann曲面上多值函数与向量场间的典范对应，将反函数奇点理论从亚纯函数扩展至加性自同构复解析函数，完整刻画了反函数奇点的代数、对数及非零留数零点特征，并应用于复轨迹最大单叶性区域与不完整实轨迹的几何描述。
This article shows how box dimension of the unit-time map can be used in studying the multiplicity of nilpotent singularities of planar vector fields. Using unit-time map on the characteristic curve of nilpotent singularity we define characteristic map and characteristic box dimension of the unit-time map. We study connection between the box dimension of discrete orbits generated by the unit-time map of planar vector fields on the characteristic or invariant curves, and the multiplicity of singularities. Nilpotent singularities which are studied are nilpotent node, focus and cusp. Also, we sho…
We introduce a sl_2-invariant family of nonlinear vector fields with a non-semisimple triple zero singularity. In this paper we are concerned with characterization and normal form classification of these vector fields. We show that the family constitutes a Lie algebra structure and each vector field from this family is solenoidal, completely integrable and rotational. All such vector fields share a common quadratic invariant. We provide a Poisson structure for the Lie algebra from which the second invariant for each vector field can be readily derived. We show that each vector field from this …
We prove that a singular complex surface that admits a complete holomorphic vector field that has no invariant curve through a singular point of the surface is obtained from a Kato surface by contracting some divisor (in particular, it is compact). We also prove that, in a singular Stein surface endowed with a complete holomorphic vector field, a singular point of the surface where the zeroes of the vector field do not accumulate is either a quasihomogeneous or a cyclic quotient singularity. The proofs rely in a combinatorial description of the vector field on a resolution of the singular poin…
We show that the only $ψ$-Dirichlet numbers in a function field over a finite field are rational functions, unlike $ψ$-Dirichlet numbers in $\mathbb{R}$. We also prove that there are uncountably many totally irrational singular vectors with large uniform exponent in quadratic surfaces over a positive characteristic field.
We address the problem of understanding the dynamics around typical singular points of $3D$ piecewise smooth vector fields. A model $Z_0$ in $3D$ presenting a T-singularity is considered and a complete picture of its dynamics is obtained in the following way: \textit{(i)} $Z_0$ has an invariant plane $π_0$ filled up with periodic orbits (this means that the restriction $Z_0 |_{π_0}$ is a center around the singularity), \textit{(ii)} All trajectories of $Z_0$ converge to the surface $π_0$, and such attraction occurs in a very non-usual and amazing way, \textit{(iii)} given an arbitrary integer …
Hausdorff dimension of singular vectors in function fields
本文研究函数域中的奇异向量与 ε-Dirichlet 可改进向量，分别计算并界定其 Hausdorff 维数，给出经典结果的函数域类比。
When a singular point of a vector field passes through resonance, a formal invariant cone appears. In the seventies, Pyartli proved that for $(-1,1)$-resonance the cone is in fact analytic and is the degeneration of a family of invariant cylinders. In his thesis, Stolovitch established a new type of normal form and proved that for a simple resonance and under arithmetic conditions the cone is (the germ of) an analytic variety. In this paper, we prove a versal deformation theorem for analytic vector fields with an isolated singularity over Cantor sets. Our result implies that, under arithmetic …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。