Related papers: On the Transitivity of Invariant Manifolds of Cons…

In this paper, we provide a new criterion for the stable transitivity of volume preserving finite generated group on any compact Riemannian manifold. As one of our applications, we generalised a result of Dolgopyat and Krikorian in…

Helicity is a fundamental conserved quantity in physical systems governed by vector fields whose evolution is described by volume-preserving transformations on a three-manifold. Notable examples include inviscid, incompressible fluid flows,…

We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces, to describe both gradient dynamics and recurrent dynamics. In particular, the abstract weak orbit spaces of flows on…

In this paper it is proved that near a compact, invariant, proper subset of a continuous flow on a compact, connected metric space, at least one, out of twenty eight relevant dynamical phenomena, will necessarily occur. This result shows…

We show that any topologically transitive codimension-one Anosov flow on a closed manifold is topologically equivalent to a smooth Anosov flow that preserves a smooth volume. By a classical theorem due to Verjovsky, any higher dimensional…

We study the flux homomorphism for closed forms of arbitrary degree, with special emphasis on volume forms and on symplectic forms. The volume flux group is an invariant of the underlying manifold, whose non-vanishing implies that the…

Invariant manifolds play an important role in organizing global dynamical behaviors. For example, it is found that in multi-well conservative systems where the potential energy wells are connected by index-1 saddles, the motion between…

First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric…

It is shown that if a non-invertible area preserving local homeomorphism on $\mathbb{T}^2$ is homotopic to a linear expanding or hyperbolic endomorphism, then it must be topologically transitive. This gives a complete characterization, in…

We prove that non-trivial homoclinic classes of $C^r$-generic flows are topologically mixing. This implies that given $\Lambda$ a non-trivial $C^1$-robustly transitive set of a vector field $X$, there is a $C^1$-perturbation $Y$ of $X$ such…

Given a vector field on a manifold M, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the…

This paper meticulously revisit and study the flux geometry of any compact oriented manifold $(M; W)$. We generalize several well-known factorization results, exhibit some orbital conditions for the study of flux geometry, give a proof of…

We prove that every 3-manifold possesses a $C^1$, volume-preserving flow with no fixed points and no closed trajectories. The main construction is a volume-preserving version of the Schweitzer plug. We also prove that every 3-manifold…

In this paper, we prove that for $\mathcal{C}^1$ generic volume-preserving Anosov diffeomorphisms of a compact Riemannian manifold, Liv\v{s}ic measurable rigidity theorem holds. We also prove that for $\mathcal{C}^1$ generic…

We obtain a $C^1$-generic subset of the incompressible flows in a closed three-dimensional manifold where Pesin's entropy formula holds thus establishing the continuous-time version of \cite{T}. Moreover, in any compact manifold of…

Link invariants of long pieces of orbits of a volume-preserving flow can be used to define diffeomorphism invariants of the flow. In this paper, we extend the notions of wrapping number and trunk and define invariants of links with respect…

We study basic properties of flow equivalence on one-dimensional compact metric spaces with a particular emphasis on isotopy in the group of (self-) flow equivalences on such a space. In particular, we show that an orbit-preserving such map…

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.…

In this article we study topological transitivity of Anosov flows on non-compact 3-manifolds. We provide homological conditions under which the lifts of a transitive Anosov flow to certain infinite covers of the manifold remain transitive.…

We study $C^1$-generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices and prove that they exhibit finitely many sinks and sectional-hyperbolic transitive Lyapunov stable sets with…