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From: wheierman on 22 Dec 2007 00:35
I have been reading the paper "A Graph Theoretic Proof of Sharkovsky's Theorem on the Periodic Points of Continuous functions" by Chun-wu Ho and Charles Morris Pac. J. Math, 96, No. 2, (1981), pp. 361-370, and have a question about one of the proofs. Perhaps one of you is familiar with this result, and can cue me in.
The question regards a step in the proof of Proposition 3.3 at the bottom of the first paragraph on page 366. I do not follow the argument in the case s=t, i.e., when h and k have the same power of 2 as their even parts. To illustrate the concern, I use the example: x is a fixed point of F^4 of order 5, for which the authors seem to assert it would be "straightforward to show" that x would be a fixed point of order 20. Clearly, the order of x for F would have to be a divisor of 20, but why not 5 or 10?
Is this a gap in the paper, is it implicitly covered by the cited papers (which I do not have), or am I having a senior moment?
Thank you all, and Happy Holidays!
From: wheierman on 22 Dec 2007 01:49
It appears the paper may have a gap, but I just realized it can be patched up by considering the "first" member m of the Sarkovskii ordering for which F has a fixed point of order m in the computations. This seems to eliminate the divisors of k which have lesser even parts (lower powers of two in their factorizations), and which seemed to cause the problem.
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