Is there a bijection $\phi: \mathbb Q \to \mathbb Q$ such that
- $\phi$ is nonlinear: different from $ax+b$,
- $\phi$ is smooth: the extension $\hat{\phi}$ of $\phi$ over $\mathbb R$ is $\mathcal C^2$ ?
What if we require $\mathcal C^\infty$?
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6http://mathoverflow.net/questions/48910/smooth-functions-for-which-fx-is-rational-if-and-only-if-x-is-rational?rq=1 – znt Mar 22 '16 at 22:11
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$x \mapsto {x^3 \over 1+x^2}$ – Martin Hairer Mar 22 '16 at 22:16
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1@MartinHairer That doesn't look surjective to me (no rational value of $x$ maps to $1$). – Todd Trimble Mar 22 '16 at 23:16
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As written, this is a special-case of the question znt links to. – Ryan Budney Mar 22 '16 at 23:37
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Sort of, except this question imposes bijectivity. Fortunately, this condition was treated by Pietro Majer in his solution over there. – Todd Trimble Mar 23 '16 at 00:36
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@ToddTrimble That's what happens when one types too fast and too late at night... – Martin Hairer Mar 23 '16 at 13:00