CBMS Regional Conference in the Mathematical Sciences--The existence and non-existence of periodic orbits in smooth dynamical systems--July 10-14, 2000

闭流形M上的动力系统或流是实数的加群R在M上的作用。换句话说，M上的动力系统是从R和M的笛卡尔积到M上的映射T，使得(I)T(0，p)=p和(Ii)T(t S，p)=T(S，T(t，p))。如果我们将参数t解释为时间，并将值T(t，p)视为告诉我们从位置p开始的粒子在时间t移动到了哪里，那么上述两种情况是非常自然的。(I)说t=0是流的开始时间，(Ii)说如果一个粒子以t个时间单位从p运动到q，以S时间单位从q运动到r，那么它将以S的时间单位从p运动到r。每个可微动力系统根据系统的作用给每个点分配通过该点的质点的速度向量，从而得到M上的一个矢量场。反之，积分一个C^1矢量场就会得到一个动力学系统。对于流形上的每个点p，R在由T_p(T)=T(t，p)定义的函数T_p：R-M下的像称为轨迹、轨道或解。紧凑轨道也称为周期轨道。当p的轨道为周期时，T_p将R映射到单点p或简单的闭曲线上。1950年H.Seifert的结果提出了Seifert猜想，即三球上的每个动力系统必有紧轨道。从1974到1995年间，发现了几个对猜想具有越来越好的光滑性的反例。这一系列讲座将介绍必要的背景知识，然后介绍一些反例以及光滑动力系统周期解研究的进一步发展。

A dynamical system, or a flow, on a closed manifold M is an action of the additive group R of the reals on M. Stated another way, a dynamical system on M is a map T from the Cartesian product of R and M onto M such that (i) T(0,p)=p and (ii) T(t+s,p)=T(s,T(t,p)). If we interpret the parameter t as time and consider the value T(t,p) as telling us where a particle starting at location p has moved to at time t, then the two conditions above are quite natural. (i) says that t=0 is the starting time for the flow, and (ii) says that if a particle moves from p to q in t units of time and from q to r in s units of time then it will move from p to r int+s units of time. Each differentiable dynamical system leads to a vector field on M by assigning to each point the velocity vector of a particle moving through that point according to the action of the system. Conversely, integrating a C^1 vector field results in a dynamical system. For each point p of the manifold, the image of R under the function T_p:R-M defined by T_p(t)=T(t,p) is called a trajectory, an orbit, or a solution. A compact orbit is also called periodic. When the orbit of pis periodic, T_p maps R either to the single point p or onto a simple closed curve.The Seifert conjecture, which was suggested by results of H. Seifert in 1950, is the assertion that every dynamical system on the three-sphere must possess a compact orbit. In the period from 1974 to 1995, several counterexamples to the conjecture with progressively better smoothness properties were discovered.This lecture series will develop the necessary background and then present some of the counterexamples as well as further developments in the study of periodic solutions of smooth dynamical systems.