Title: Counterexamples to the Seifert Conjecture

Since H.~Seifert proved in 1950 the existence of a periodic orbit for a vector field on the 3-dimensional sphere $S^3$ which forms small angles with the fibers of the Hopf fibration, several examples of aperiodic vector fields on $S^3$ have been produced as well as results showing that in some situations a compact orbit must exists. This paper surveys presently known types of vector fields without periodic orbits on $S^3$ and on other manifolds.

1991 Mathematics Subject Classification: Primary 58F25; Secondary 57R25, 35B10, 58F18

Keywords and Phrases: dynamical system, plug, periodic orbit, minimal set, PL foliation

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