Possibilites for Sharkovskii's theorem.

Question

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.

Could you please be a bit more specific as to what generalizations you are looking for? As to the applications, Sharkovskii's theorem is a result on periodicity of dynamical systems. — David_Shmij, Jul 21 '18 at 01:49
I would recommend you read the paper "Period Three Implies Chaos" by Tien-Yien Li & James A. Yorke — David_Shmij, Jul 21 '18 at 02:02
Almost the same question was posted in MO References and background on Sharkovskii's theorem. I think that simultaneous cross-posting, without mentioning it, is considered a bad practice — user539887, Jul 21 '18 at 09:49

score 2 · Accepted Answer · answered Jul 21 '18 at 20:43

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.

Possibilites for Sharkovskii's theorem.

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