Periodic orbits of Linear and invariant flows on connected Lie groups

Periodic first integrals for Hamiltonian systems of Lie type

We prove the existence of a Lie algebra of first integrals for time dependent Hamiltonian systems of Lie type. Moreover, applying the Floquet theory for periodic Euler systems on Lie algebras, we show the existence of an abelian Lie algebra of periodic first integrals for periodic Hamiltonian systems. An application to the dynamics of a nonlinear oscillator is given.

Pointwise-recurrent maps on uniquely arcwise connected locally arcwise connected spaces

We prove that self-mappings of uniquely arcwise connected locally arcwise connected spaces are pointwise-recurrent if and only if all their cutpoints are periodic while all endpoints are either periodic or belong to what we call "topological weak adding machines". We also introduce the notion of a \emph{ray complete} uniquely arcwise connected locally arcwise connected space and show that for them the above "topological weak adding machines" coincide with classical adding machines (e.g., this holds if the entire space is compact).

Parameter rigid actions of simply connected nilpotent Lie groups

We show that for a locally free action of a simply connected nilpotent Lie group on a compact manifold, if every real valued cocycle is cohomologous to a constant cocycle, then the action is parameter rigid. The converse is true if the action has a dense orbit. Using this, we construct parameter rigid actions of simply connected nilpotent Lie groups whose Lie algebras admit rational structures with graduations. This generalizes the results of dos Santos [Parameter rigid actions of the Heisenberg groups. Ergod. Th. & Dynam. Sys. 27 (2007), 1719-1735] concerning the Heisenberg groups.

Periodic trajectory tracking for control-affine driftless systems on compact Lie groups

We treat the periodic trajectory tracking problem: given a periodic trajectory of a control-affine, left-invariant driftless system in a compact and connected Lie group $G$ and an initial condition in $G$, find another trajectory of the system satisfying the initial condition given and that asymptotically tracks the periodic trajectory. We solve this problem locally (for initial conditions in a neighborhood of some point of the periodic trajectory) when $G$ is semisimple and the system is Lie-determined (i.e. controllable), and only for a class of periodic trajectories (which we call regular).…

Topologically Conjugate Classifications of the Translation Actions on Compact Connected Lie Groups ${\rm SU}(2) \times T^n$

In this article, we focus on the left (translation) actions on noncommutative compact connected Lie groups ${\rm SU}(2) \times T^n$. We define the rotation vectors of the left actions induced by the elements in the maximal tori of ${\rm SU}(2) \times T^n$, and utilize rotation vectors to give the complete topologically conjugate classifications of left actions. Algebraic conjugacy and smooth conjugacy are also considered.

When a smooth self-map of a semi-simple Lie group can realize the least number of periodic points

There are two algebraic lower bounds of the number of n-periodic points of a self-map f:M\to M of a compact smooth manifold of dimension at least 3 : NF_n(f)=min {#Fix(g^n) ;g\sim f; g continuous} and NJD_n(f)=min {#Fix}(g^n) ;g\sim f; g smooth}}. In general NJD_n(f) may be much greater than NF_n(f). We show that for a self-map of a semi-simple Lie group, inducing the identity fundamental group homomorphism, the equality NF_n(f)=NJD_n(f) holds for all n iff all eigenvalues of a quotient cohomology homomorphism induced by f have moduli \le 1.

多项式增长Lie群的自由作用与可分类C*-代数

Free actions of polynomial growth Lie groups and classifiable C*-algebras

本文证明了多项式增长的连通Lie群在有限维局部紧空间上的自由作用具有有限管维数，进而得出相关交叉积C*-代数具有有限核维数，并证明了与某些非周期点集相关的C*-代数是可分类的。

半李群上的周期测地线

On Periodic Geodesics of Half-Lie Groups

研究希尔伯特半李群上强右不变黎曼度量下周期测地线的存在性，证明非平凡自由同伦类包含周期测地线，并建立Lyusternik-Fet型定理。