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added comment on Noam's answer.
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Igor Rivin
Igor Rivin
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This is a generalization of Integrality of iterates of rational functions.

The question is: given an infinite sequence of rational numbers $a_0, a_1, \dotsc, a_n, \dotsc,$ is it always an orbit of $a_0$ under an iteration of a rational function? If the answer is in the affirmative, it will answer the original question. I am guessing it is undecidable, however.

EDIT Noam points out that the answer is not in general affirmative, but how does one decide (maybe if the growth is not too fast the answer is always yes?)

This is a generalization of Integrality of iterates of rational functions.

The question is: given an infinite sequence of rational numbers $a_0, a_1, \dotsc, a_n, \dotsc,$ is it always an orbit of $a_0$ under an iteration of a rational function? If the answer is in the affirmative, it will answer the original question. I am guessing it is undecidable, however.

This is a generalization of Integrality of iterates of rational functions.

The question is: given an infinite sequence of rational numbers $a_0, a_1, \dotsc, a_n, \dotsc,$ is it always an orbit of $a_0$ under an iteration of a rational function? If the answer is in the affirmative, it will answer the original question. I am guessing it is undecidable, however.

EDIT Noam points out that the answer is not in general affirmative, but how does one decide (maybe if the growth is not too fast the answer is always yes?)

Source Link
Igor Rivin
Igor Rivin
  • 96.4k
  • 11
  • 153
  • 366

Orbits of rational functions

This is a generalization of Integrality of iterates of rational functions.

The question is: given an infinite sequence of rational numbers $a_0, a_1, \dotsc, a_n, \dotsc,$ is it always an orbit of $a_0$ under an iteration of a rational function? If the answer is in the affirmative, it will answer the original question. I am guessing it is undecidable, however.