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

I agree with the answer of jc in the case of real polynomial, for this question Does there exist a real quadratic polynomial of Sharkovskys type 2-infinity?

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It is customary over here to invest a certain amount of care in the questions. –  Franz Lemmermeyer Apr 30 '11 at 7:18
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This would seem to be a follow-up to mathoverflow.net/questions/63441/… –  Gerry Myerson Apr 30 '11 at 8:17
    
Here it is quadratic polynomial with rational coefficient. –  Aliakbar Apr 30 '11 at 9:16
    
Unlikely. All such examples usually involve the Thue-Morse sequence, which usually leads to transcendental numbers. (Its binary sum is transcendental, say.) –  Nikita Sidorov Apr 30 '11 at 13:07
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