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Let $f_1,g_1,f_2,g_2$ be non-constant rational functions on the Riemann sphere (i.e. elements of $\Bbb{C}(z)-\Bbb{C}$) satisfying $f_1\circ g_1=f_2\circ g_2$. Suppose there is a prime number $p$ such that at least one of $f_1$ and $g_1$ belongs to $\Bbb{C}(z^p)$. Can the same be said about $f_2$ and $g_2$?

Update: The example from the comments is a very special one in which $g_1=f_2$ is linear, hence a conjugacy between dynamical systems $f_1=z^2$ and $g_2=z^2+2z$. I am interested in examples where the maps are all non-linear. There is a very special family of maps in complex dynamics that includes power maps, Chebyshev polynomials and Lattès maps. I am in particular interested to see if there is any example where $f_1$ or $g_1$ is in $\Bbb{C}(z^p)$, but on the other side $f_2$ and $g_2$ are not conjugate to any map from the family mentioned above.

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    $\begingroup$ No. For example $p=2$, $f_1=x+1$, $g_1=x^2$, $f_2=x^2+x$, $g_2=x+1$. $\endgroup$
    – LeechLattice
    Commented Dec 10, 2021 at 6:32
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    $\begingroup$ @LeechLattice: I think you mean $f_1=x^2$, $g_1=f_2=x+1$, $g_2=x^2+2x$. $\endgroup$
    – abx
    Commented Dec 10, 2021 at 7:27
  • $\begingroup$ Thank you both of you. $\endgroup$
    – KhashF
    Commented Dec 10, 2021 at 17:18

1 Answer 1

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This question is discussed in the paper by S. Lysenko On the functional equation f(p(z))=g(q(z)), where f and g are meromorphic functions, and p and q are polynomials. (Russian. English, Russian, Ukrainian summary) Mat. Fiz. Anal. Geom. 2 (1995), no. 1, 68–86.

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  • $\begingroup$ Thank you professor Eremenko. $\endgroup$
    – KhashF
    Commented Dec 10, 2021 at 17:50
  • $\begingroup$ arxiv.org/abs/math/9606217 $\endgroup$
    – Adam
    Commented Apr 7, 2023 at 8:15

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