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In the book, the proof of Sharkovskiis Theorem is outlined in 11 steps but I’m stuck on step 4. Trying to give all the relevant information here to try explain my problem to someone without the book would be ridiculous, so just if anyone has the book, could you explain why step 4 part(1) must hold? Thanks

Ximenez
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    I don't have the book with available right now, but my advice would be to look in different sources. The proof of Sharkovskii's Theorem is not particularly good in that book. When remember a better place to look, I'll post it here. – Severin Schraven Mar 10 '20 at 23:46
  • Thank you, I’d seriously appreciate that. I’ve seen the other steps and can imagine I’ll have problems with those too. – Ximenez Mar 10 '20 at 23:49
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    I like Devaney's book as an introduction to dynamical systems, but that proof is really terrible. However, it seems to be true for many versions of that proof. People often want to make it shorter then what beginners would need :( – Severin Schraven Mar 10 '20 at 23:54
  • Haha yea, it’s really annoying. They don’t make them easy to follow at all, but hopefully I’ll find a nice proof for beginners somewhere. – Ximenez Mar 11 '20 at 00:03
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    I was running a student seminar on Devaney's book last semester and one of my students found the reference below and found it helpful. Hope it will be helpful for you too. – Severin Schraven Mar 11 '20 at 00:09

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I would suggest the paper "A collection of simple proofs of Sharkovsky’s theorem" by Bau-Sen Du, (https://www.math.sinica.edu.tw/www/file_upload/dubs/dubs7v3.pdf). Even if you don't like the several different proofs of the theorem, then you can still go throught the huge collection of references with other proofs.

Severin Schraven
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