Sharkovskii's theorem does not immediately apply to dynamical systems on other topological spaces. It is easy to find a circle map with periodic points of period 3 only: take a rotation by 120 degrees, for example. But some generalizations are possible, typically involving the mapping class group of the space minus a periodic orbit. For example, Peter Kloeden showed that Sharkovskii's theorem holds for triangular mappings, i.e., mappings for which the component f i depends only on the first i components x 1,..., x i.