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Let $\phi\in \mathbb{C}(z)$ be a degree $d\geq 2$ rational map, which we can write as $\phi = \frac{f}{g}$ for $f,g\in \mathbb{C}[z]$. Let $\omega_{FS}$ denote the Fubini-Study form on $\mathbb{P}^1(\mathbb{C})$. I recently came across the following integral expression:

$\int_{\mathbb{P}^1(\mathbb{C})} ||\phi(z) - z||^2 \omega_{FS}$,

where $||\cdot - \cdot||$ denotes the chordal metric on $\mathbb{P}^1(\mathbb{C})$. There is an analogous expression over $\mathbb{R}$:

$\int_{\mathbb{P}^1(\mathbb{R})} ||\phi(x) - x||^2 \omega$,

where here $\omega = \frac{1}{\pi(1+x^2)} dx$.

My question is whether anyone has seen this sort of expression before, and if so whether there is a closed form for this integral expression given in terms of, say, the coefficients of $f$ and $g$? Or in terms of the fixed points of $\phi$?

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Only a partial and rather distant answer: There is ``this sort of expression" in dynamical Nevanlinna theory. The mean proximity of $f$ with respect to $a \in \hat{\mathbb{C}}$ is defined as $m(a,f)=\int_{\hat{\mathbb{C}}}\log\frac{1}{[a,f(w)]}d\sigma(w)$, where $\sigma$ is the (normalized) spherical area measure on $\hat{\mathbb{C}}$ and $[\cdot,\cdot]$ the chordal distance normalized so that $[0,\infty]=1$. It can be generalized to a proximity of two holomorphic functions. It appears in the works of M. Sodin, @AlexandreEremenko (who is on MO and might provide a more insightful answer) and, particularly in applications to non-archimedean dynamics, of Y. Okuyama.

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  • $\begingroup$ The connection to Nevanlinna theory is interesting here; the integral expression above is derived as a Laplacian of an `averaged mean proximity function'. $\endgroup$
    – Ken Jacobs
    Commented May 24, 2016 at 16:53

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