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法群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移动到哪里，那么上述两个条件是非常自然的。(i)说t=0是流动的起始时间，（ii）说如果一个粒子在t个时间单位内从p移动到q，在s个时间单位内从q移动到r，那么它将在int+s个时间单位内从p移动到r。每一个可微动力系统通过给每一个点分配一个质点根据系统的作用通过该点的速度向量，导致M上的一个向量场。 相反，对一个C^1向量场进行积分会得到一个动力系统。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下的像称为轨线、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或简单闭曲线上，从而证明了H. Seifert在1950年提出的第一个定理是，在三球面上的每个动力系统都必须有一个紧轨道。在1974年至1995年期间，发现了几个具有逐渐更好光滑性的猜想的反例。本系列讲座将介绍必要的背景，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.