## Counterexamples to the Seifert conjecture.(English)Zbl 0924.58086

Summary: 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 exist. This paper surveys presently known types of vector fields without periodic orbits on $$S^3$$ and on other manifolds.

### MSC:

 37C10 Dynamics induced by flows and semiflows 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$
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