Invariant curves from symmetry. (English) Zbl 0783.58065

Fečkan, Michal

Math. Bohem. 118, No. 2, 171-174 (1993).

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.

Reviewer: C.Chicone (Columbia)

MSC:

37G15	Bifurcations of limit cycles and periodic orbits in dynamical systems
37B99	Topological dynamics

Keywords:

invariant curve; discrete dynamical systems