Related papers: On the singular hyperbolicity of star flows

We prove that the singular set of a multiplicity $2$ integral hypercurrent that is stationary in the sense of varifolds has a singular set of measure zero.

Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the disjoint union of a nonempty discrete set,…

For an integer $e$ and hyperbolic curve $X$ over $\overline{\mathbb Q}$, Mochizuki showed that there are only finitely many isomorphism classes of hyperbolic curves $Y$ of Euler characteristic $e$ with the same universal cover as $X$. We…

We prove a unique ergodicity theorem for singular holomorphic foliations of $\mathbb{P}^3(\mathbb{C})$ with hyperbolic singularities and with an invariant plane with no foliation cycle, in analogy with a result of Dinh-Sibony concerning…

A real matrix is Hurwitz if its eigenvalues have negative real parts. The following generalisation of the Bidimensional Global Asymptotic Stability Problem (BGAS) is provided: Let $X:R^2-->R^2$ be a C^1 vector field whose derivative DX(p)…

We consider a chain of oscillators with hyperbolic chaos coupled via diffusion. When the coupling is strong the chain is synchronized and demonstrates hyperbolic chaos so that there is one positive Lyapunov exponent. With the decay of the…

We elaborate on a previous attempt to prove the irreversibility of the renormalization group flow above two dimensions. This involves the construction of a monotonically decreasing $c$-function using a spectral representation. The missing…

Using the framework of Colombeau algebras of generalized functions, we prove the existence and uniqueness results for global generalized solvability of semilinear hyperbolic systems with nonlinear nonlocal boundary conditions. We admit…

In this paper we find conditions which guarantee that a given flow $\Phi$ on a compact metric space $X$ admits a Lyapunov one-form $\omega$ lying in a prescribed \v{C}ech cohomology class $\xi\in \check H^1(X;\R)$. These conditions are…

A proof for a non-perturbative C-theorem in four dimensions, capturing the irreversibility of the renormalization group flow in the space of unitary quantum field theories, has not been accomplished, yet. We test the conjectured C-theorems…

For any algebraic scheme $X$ and every $(n,\mathscr{L})\in \mathbb{Z}\times \text{Pic}(X)$ we define an associated involution of its Chow group $A_*X$, and show that certain characteristic classes of (possibly singular) hypersurfaces in a…

In this paper we study structurally stable homoclinic classes. In a natural way, the structural stability for an individual homoclinic class is defined through the continuation of periodic points. Since the homoclinic classes is not…

Let $f:M\to M$ be a $C^1$ map of a compact manifold $M$, with dimension at least $2$, admitting some point whose future trajectory has only negative Lyapunov exponents. Then this trajectory converges to a periodic sink. We need only assume…

In this paper we provide the stability of generic polycycles of hybrid planar vector fields, extending previous known results in the literature. The polycycles considered here may have hyperbolic saddles, tangential singularities and jump…

Hyperbolicity plays an important role in the study of dynamical systems, and is a key concept in the iteration of rational functions of one complex variable. Hyperbolic systems have also been considered in the study of transcendental entire…

For a continuous flow on a compact metric space, the aim of this paper is to prove a Conley-type decomposition of the strong chain recurrent set. We first discuss in details the main properties of strong chain recurrent sets. We then…

This article is dedicated to the characterisation of the relative hyperbolicity of Haglund and Wise's special groups. More precise, we introduce a new combinatorial formalism to study (virtually) special groups, and we prove that, given a…

We show, for some classes of diffusion coefficients that, generically in f, all equilibria of the reaction-diffusion equation u_t = (a(x)u_x)_x + f(u) with homogeneous Neumann boundary conditions are hyperbolic.

We use the framework of Colombeau algebras of generalized functions to study existence and uniqueness of global generalized solutions to mixed non-local problems for a semilinear hyperbolic system. Coefficients of the system as well as…

Let \Fc be a holomorphic foliation by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: (1) the singular points of \Fc are all hyperbolic; (2) \Fc is Brody hyperbolic. Then we…