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Does there exist a real quadratic polynomial of Sharkovskys type 2-infinity?

A continuous map from the set of real numbers to itself is said to be Sharkosky type 2-infinity if it has periods of all powers of 2 but no any other period.

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The article states that the logistic map with $\lambda=\lambda^*\approx 3.57$ is such a map. – j.c. Apr 29 '11 at 17:34
@jc: Indeed. If it had points of period $n$ not a power of 2, then for any $m\prec n$, it would have points of period $m$ for some $\lambda < \lambda^*$. As $m$ can be taken to be not a power of 2, this would give a contradiction. – George Lowther Apr 29 '11 at 17:43
@jc could you write your comment as an answer so that i can accept it. – Aliakbar May 2 '11 at 12:03

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