Hello I am interested in Sharkovskii's theorem with respect to two aspects:
- Possibilities of generalization for other spaces with some additional structures or dimensions.
- Applications in dynamical systems, other branches of mathematics and real life(!?)
Some references are appreciated.
Just quickly, it is not true in the circle. It has been shown in the context of unique beta expansions. See N. Sidorov et al paper Periodic unique beta-expansions: the Sharkovskiĭ ordering. Perhaps, it might hold on some one-dimensional spaces like finite graphs without circles or hereditarly indecomposable continua such as the pseudoarc. Seems difficult to hold for higher dimensional spaces in my opinion.
As for applications a quick one is related to the structure of some limit spaces of tent maps. Check W. Ingram's work.