Related papers: On the Shadowableness of Flows With Hyperbolic Sin…
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}.
It is known that hyperbolic non\-autonomous linear delay differential equations in a finite dimensional space are Hyers--Ulam stable and hence shadowable. The converse result is available only in the special case of autonomous and periodic…
We derive the most general flux-induced superpotential for N=1 M-theory compactifications on seven-dimensional manifolds with SU(3) structure. Imposing the appropriate boundary conditions, this result applies for heterotic M-theory. It is…
Let $M$ be a manifold with a volume form $\omega$ and $f : M \to M$ be a diffeomorphism of class $\mathcal{C}^1$ that preserves $\omega$. In this paper, we do \textit{not} assume $f$ is $\mathcal{C}^1$-generic. We have two main themes in…
We show that for $C^1$ generic diffeomorphisms, an isolated homoclinic class is shadowable if and only if homoclinic class is hyperbolic basic set.
Hamiltonian flows on compact surfaces are characterized, and the topological invariants of such flows with finitely many singular points are constructed from the viewpoints of integrable systems, fluid mechanics, and dynamical systems.…
We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that, when the genus of the surface is two, almost every such locally Hamiltonian flow with…
We consider the Cauchy problem for first order systems. Assuming that the set of the singular points of the characteristic variety is a smooth manifold and the characteristic values are real and semi-simple we introduce a new class which is…
We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two strong different senses. Firstly, the flow is expansive: if two points remain close for all times, possibly with time reparametrization, then their…
In this paper we prove two backward uniqueness theorems for extrinsic geometric flow of possibly non-compact hypersurfaces in general ambient complete Riemannian manifolds. These are applicable to a wide range of extrinsic geometric flow,…
In this article we study geodesic flows on closed Riemannian manifolds without conjugate points and divergence property of geodesic rays. If the fundamental group is Gromov hyperbolic and residually finite we prove, under appropriate…
In this paper, we conduct a comprehensive study on ergodic properties of the geodesic flow on a $C^\infty$ uniform visibility manifold $M$ without conjugate points. If $M$ is a closed surface of genus at least two without conjugate points,…
In this paper, we study the intrinsic mean curvature flow on certain closed spacelike manifolds, and prove the existence of hyperbolic structures on them.
We prove that for any closed manifold of dimension 3 or greater that there is an open set of smooth flows that have a hyperbolic set that is not contained in a locally maximal one. Additionally, we show that the stabilization of the…
In this article we prove that if a flow exhibits a partially hyperbolic attractor and it has two periodic saddles with different indices, and the stable index of one of them coincides with the dimension of strongly stable bundles, then it…
We consider mean curvature flow of an initial surface that is the graph of a function over some domain of definition in $R^n$. If the graph is not complete then we impose a constant Dirichlet boundary condition at the boundary of the…
Stability of inviscid shear shallow water flows with free surface is studied in the framework of the Benney equations. This is done by investigating the generalized hyperbolicity of the integrodifferential Benney system of equations. It is…
The incompressible Navier-Stokes equations in R^3 are shown to admit a unique axisymmetric solution without swirl if the initial vorticity is a circular vortex filament with arbitrarily large circulation Reynolds number. The emphasis is on…
Flows of vector fields are an essential tool in differential geometry, with countless applications in both theory and practice. While they have been extensively studied for ordinary manifolds and supermanifolds, a treatment of flows in…
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…