On Sussmann theorem for orbits of sets of vector fields on Banach manifolds

Banach 空间上超循环算子轨道的三分法

Trichotomy for the orbits of a hypercyclic operator on a Banach space

本文在可分 Banach 空间上建立了超循环算子轨道的三分法，区分均值渐近于零、绝对均值不规则和均值发散到无穷三种情形，并用加权后移算子说明三种情况均可实现。

Monotone semiflows with respect to high-rank cones on a Banach space

We consider semiflows in general Banach spaces motivated by monotone cyclic feedback systems or differential equations with integer-valued Lyapunov functionals. These semiflows enjoy strong monotonicity properties with respect to cones of high ranks, which imply order-related structures on the $ω$-limit sets of precompact semi-orbits. We show that for a pseudo-ordered precompact semi-orbit the $ω$-limit set $Ω$ is either ordered, or is contained in the set of equilibria, or possesses a certain ordered homoclinic property. In particular, we show that if $Ω$ contains no equilibrium, then $Ω$ its…

Closedness of the tangent spaces to the orbits of proper actions

In this note we show that for any proper action of a Banach--Lie group $G$ on a Banach manifold $M$, the corresponding tangent maps $\g \to T_x(M)$ have closed range for each $x \in M$, i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient $M/G$ carries a natural manifold structure.

Dynamics of the semigroup of contractive automorphisms of Banach spaces

Motivated by some recent twaddles on Mazur rotations problem, we study the "dynamics" of the semigroup of contractive automorphisms of Banach spaces, mostly in finite-dimensional spaces. We focus on the metric aspects of the "action" of such semigroups, the size of the orbits and semitransitivity properties, and their impact on the geometry of the unit ball of the underlying space.

The Poincaré-Bendixson theory for certain compact semi-flows in Banach spaces

We study semiflows satisfying a certain squeezing condition with respect to a quadratic functional in some Banach space. Under certain compactness assumptions from our previous results it follows that there exists an invariant manifold, which is under more restrictive conditions is an inertial manifold. In the case of a two-dimensional manifold we obtain an analog of the Poincaré-Bendixson theorem on the trichotomy of $ω$-limit sets. Moreover, we obtain conditions for the existence of an orbitally stable periodic orbit. Our approach unifies a series of papers by R.~A.~Smith, establishes their …

Snakes and articulated arms in an Hilbert space

The purpose of this paper is to give an illustration of results on integrability of distributions and orbits of vector fields on Banach manifolds obtained in [Pe] and [LaPe]. Using arguments and results of these papers, in the context of a separable Hilbert space, we give a generalization of a Theorem of accessibility contained in [Ha], [Ro] and proved for a finite dimensional Hilbert space