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

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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.