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Here are four little results about fixed points, that I can't prove.

We say that $x$ is a point of period $n$ under a function $f$, if $f^n$ has $x$ as a fixed point, but $f^k$ for $k \lt n$ don't.

  1. If $f$ has a point with period $n$, where $n$ is odd, then $f$ has a point with period $k>n$, $k$ odd.

  2. If $f$ has a point with period $n$, $n$ odd, then $f$ has a point with period $k$, $k$ even.

  3. If $f$ has a point with period $n$, $n$ even, then $f$ has a point with period $2$.

  4. If $f$ has a point of period $3$, then given any $n$, $f$ has a point of period $n$.

EDITED: for more details $ f^n $ denotes the composition of f , n times . And f is a function $$ R \to R $$

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by $f^n$ do you mean the composition of $f$ $n$ times? –  uforoboa Oct 18 '11 at 11:53
    
Could you make your hypotheses more explicit? You must be in a special situation for these results to hold (e.g. interval maps or something like that). –  t.b. Oct 18 '11 at 12:25
3  
It looks as if you were trying to prove Sarkovski's theorem. –  Julián Aguirre Oct 18 '11 at 12:31
    
According to the Wikipedia article you need $f$ to be continuous. Certainly without that I can define an $f$ that violates each of your four statements. –  Ross Millikan Oct 18 '11 at 13:56
    
Although I do not see a question here, the paper "Period three implies chaos" (jstor.org/stable/2318254) will provide the answer. –  Dirk Oct 18 '11 at 14:03

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