Related papers: Dominated chain recurrent class with singularities
It is well known that a $\mathrm{SL}(2,\mathbb{C})$-sequence is uniformly hyperbolic if and only it satisfies a uniform exponential growth condition. Similarly, for $\mathrm{GL}(2,\mathbb{C})$-sequences whose determinants are uniformly…
A vector field X is called a star flow if every periodic orbit, of any vector field C1-close to X, is hyperbolic. It is known that the chain recurrence classes of a generic star flow X on a 3 or 4 manifold are either hyperbolic or singular…
We introduce \emph{residually dominated groups} in pure henselian valued fields of equicharacteristic zero, as an analogue of stably dominated groups introduced by Hrushovski and Rideau-Kikuchi. We show that when $G$ is a residually…
We prove that a general determinantal hypersurface of dimension 3 is nodal. Moreover, in terms of Chern classes associated with bundle morphisms, we derive a formula for the intersection homology Euler characteristic of a general…
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…
We show that in multidimensional gravity vector fields completely determine the structure and properties of singularity. It turns out that in the presence of a vector field the oscillatory regime exists for any number of spatial dimensions…
A 3-tournament is a complete 3-uniform hypergraph where each edge has a special vertex designated as its tail. A vertex set $X$ dominates $T$ if every vertex not in $X$ is contained in an edge whose tail is in $X$. The domination number of…
This article studies germs of holomorphic vector fields at the origin of C3 that are tangent to holomorphic foliations of codimension one. Two situations are considered. First, we assume hypotheses on the reduction of singularities of the…
From a research of several recent papers, in the first part, we are concerned with domination number in cubic graphs and give a sufficient condition of Reed's conjecture. In the second part, from a perspective, we study the structure of a…
Singular hyperbolicity is a weakened form of hyperbolicity that has been introduced for vector fields in order to allow non-isolated singularities inside the non-wandering set. A typical example of a singular hyperbolic set is the Lorenz…
We obtain sufficient conditions for an invariant splitting over a compact invariant subset of a $C^1$ flow $X_t$ to be dominated. In particular, we reduce the requirements to obtain sectional hyperbolicity and hyperbolicity.
In this work we study the existence of singular flows satisfying shadowing-like properties. More precisely, we prove that if C1 -vector field on a closed manifold induces a chain-recurrent flow containing an attached hyperbolic singularity…
We study generic diffeomorphisms with a homoclinc class with non empty interior and in particular those admitting a codimension one dominated splitting. We prove that if in the finest dominated splitting the extreme subbundles are one…
We show an example providing a significance in geometric control theory of the existence of the dependence locus of a system of vector fields in particular, the generic appearance of non-trivial singular trajectories embedded in the…
We study the omega-limit sets $\omega_X(x)$ in an isolating block $U$ of a singular-hyperbolic attractor for three-dimensional vector fields $X$. We prove that for every vector field $Y$ close to $X$ the set $ \{x\in U:\omega_Y(x)$ contains…
A large class of real $3$-dimensional nilpotent polynomial vector fields of arbitrary degree is considered. The aim of this work is to present general properties of the discrete and continuous dynamical systems induced by these vector…
Finitely dominated chain complexes over a Laurent polynomial ring in one indeterminate are characterised by vanishing of their Novikov homology. We present an algebro-geometric approach to this result, based on extension of chain complexes…
This work establishes a strong uniqueness property for a class of planar locally integrable vector fields. A result on pointwise convergence to the boundary value is also proved for bounded solutions.
We prove the results in [1] using Theorem 1 of the recent paper [2] by Crovisier and Yang. References: [1] Arbieto, A., Rojas, C., Santiago, B., Existence of attractors, homoclinic tangencies and singular-hyperbolicity for flows,…
We study the hyperbolicity of singular quotients of bounded symmetric domains. We give effective criteria for such quotients to satisfy Green-Griffiths-Lang's conjectures in both analytic and algebraic settings. As an application, we show…