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Invariant curves from symmetry. (English) Zbl 0783.58065
A very nice result is proved in this short paper. Suppose $$m \geq 2$$ and $$F:\mathbb{R}^ m \to \mathbb{R}^ m$$ is a continuous map. If there are two points in $$\mathbb{R}^ m$$ such that one of them is moved closer to the origin by $$F$$ while the other is moved farther away and if the map $$F$$ is equivariant under the action of a compact subgroup of the orthogonal group that is transitive on the unit sphere, then $$F$$ has an invariant curve such that the action of $$F$$ on this invariant curve is equivalent to a rotation.
##### MSC:
 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems 37B99 Topological dynamics
##### Keywords:
invariant curve; discrete dynamical systems
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