## Instability of equilibria in dimension three.(English)Zbl 0930.37025

The author studies analytic vector fields defined on a neighborhood of $$0$$ in $$n$$ dimensional Euclidean space, which have isolated singular point at $$0$$. Such a vector field is called stable, if there exists a fundamental system of neighborhoods of $$0$$ which are invariant by $$v$$. There are no known examples for stable vector fields in odd dimension, whereas many examples exist in even dimension. The author proves that if $$M$$ is an 3-dimensional irreducible analytic subvariety of some open neighborhood $$0$$, whose link has nonvanishing Euler characteristic, then $$0$$ is an unstable point.

### MSC:

 37F75 Dynamical aspects of holomorphic foliations and vector fields 37F15 Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems
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### References:

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