Consider the function $g(x) = -3x^2/2 - x/2 + 1$. When iterated it has a 3 cycle, so therefore, according to Sharkovskii's theorem, it has cycles of all integer lengths (as well as chaotic infinite ones). Is it possible to define a (certainly multivalued) function f(x) that returns a number between -2 and +2 that is one of the destinations along the cycle? Say f(17) would return one of the numbers in the 17 cycle?
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1Yes, you just did define such a function. Were you asking about a finite algorithm to evaluate such a function? – Lutz Lehmann Mar 12 '20 at 00:54
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Note that $f(17)$ is the root of a polynomial of degree $2^{17}-2\sim 100,000$. – Lutz Lehmann Mar 12 '20 at 08:00
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You can only apply Sharkovskii's theorem to maps from a compact interval to itself. But the image of $[−1,1]$ is a larger interval and so you cannot conclude that $g$ has periodic points of all periods. – John B Mar 12 '20 at 23:06