Narrow Results

It is well known that the elements of PSL(2, ℂ) are classified as elliptic, parabolic or loxodromic according to the dynamics and their fixed points; these three types are also distinguished by their trace. If we now look at the elements in PU(2,1), then there are the equivalent notions of elliptic, parabolic or loxodromic elements; Goldman classified these transformations by their trace. In this work we extend the classification of elements of PU(2,1) to all of PSL(3, ℂ); we also extend to this setting the theorem that classifies them according to their trace. We use the notion of limit set introduced by Kulkarni, and calculate the limit set of every cyclic subgroup of PSL(3, ℂ) acting on . Given a classical Kleinian group it is possible to "suspend" this group to a subgroup of PSL(3, ℂ); we also calculate the limit set of this suspended group.

Let $\mathrm{\text{SO}}(m,1)$ be the group of orientation preserving isometries of the $m$-dimensional real hyperbolic space ${ℍ}_{ℝ}^m$, and let $\mathrm{\text{SU}}(n,1)$ be the group of holomorphic isometries of the $n$-dimensional complex hyperbolic space ${ℍ}_{ℂ}^n$. The group $\mathrm{\text{SU}}(n,1)$ is naturally a subgroup of $\mathrm{\text{PSL}}(n+1,ℂ)$, so it acts on ${\mathbb{P}}_{ℂ}^n$ preserving a ball that serves as a model for ${ℍ}_{ℂ}^n$. We consider discrete subgroups of $\mathrm{\text{SO}}(m,1)$ with limit set an $(m-1)$-dimensional real sphere and we look at their action on ${\mathbb{P}}_{ℂ}^n$ for $n\ge m$ via the natural embedding of $\mathrm{\text{SO}}(m,1)$ into $\mathrm{\text{SU}}(n,1)$. We describe the Kulkarni limit set of these set of these actions and show that this is a real semi-algebraic set in ${\mathbb{P}}_{ℂ}^n$. We also show that the Kulkarni region of discontinuity can only have either one or three connected components. We use Sylvester’s law of inertia when $n=m$. In the other cases, we use some suitable projections of the $n$-dimensional complex projective space to the $m$-dimensional complex projective space.

This paper is to introduce some analytical tools to characterize the properties of fractal basin boundaries for planar switched systems (with time-dependent switching). The characterizing methods are based on the view point of limit sets and prime ends. By constructing the auxiliary dynamical system, the fractal basin boundaries of planar switched systems can be proved if every diverging path in the basin of associated auxiliary system has the entire basin boundary as its limit set. Fractal property is also verified if every prime end that is defined in the basin of associated auxiliary system is a prime end of type 3 and all other prime ends are of type 1. Bifurcations of fractal basin boundary are investigated by analyzing what types of prime ends in the basin are involved. The fractal basin boundary of switched system is also described by the indecomposable continuum.

This paper investigates a class of dynamic selection processes for n-person normal-form games which includes the Brown-von Neumann-Nash dynamics. For (two-person) zero-sum games and for (n-person) potential games every limit set of these dynamics is a subset of the set of Nash-equilibria. Furthermore, under these dynamics the unique Nash-component of a zero-sum game is minimal asymptotically stable and for a potential game a smoothly connected component which is a local maximizer is minimal asymptotically stable.

Considering the fact that the production and provision of some vaccines are ordered and governed by the government according to the history data of disease, a kind of SIR model with constant vaccination rate and impulsive state feedback control is presented. The dynamical properties of semi-continuous three-dimensional SIR system can be obtained by discussing the properties of the corresponding two-dimensional system in the limit set. The existence and uniqueness of order-1 periodic solution are discussed by using the successive function and the compression mapping theorem. A new theorem for the orbital stability of order-1 periodic solution is proved by geometric method. Finally, numerical simulations are given to verify the mathematical results and some conclusions are given. The results show that the disease can be controlled to a lower level by means of impulsive state feedback control strategy, but cannot be eradicated.

We consider a simple class of Kleinian groups called once punctured torus groups. In this note, we will show how to create a computer program from scratch that can visualize fundamental sets and limit sets of the groups. No knowledge of computer programming is assumed.