# Help determining the asymptotic behavior of an integral involving rational functions.

Let $\phi:\mathbb{P}^1\to\mathbb{P}^1$ be a rational function of degree $d\geq2$. How can one prove, using the normalized spherical measure, that $$\int_{\mathbb{P}^1(\mathbb{C})}|(\phi^n)'(z)|\ d\mu (z) \sim d^n$$ as $n\to\infty$?

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Could you give some motivation? –  j.c. Mar 9 '10 at 16:47
I don't know what generalized spherical measure is, but isn't your integrand just $(\phi^n)^* \omega$? Since $\phi$ acts by $d$ on $H^2(\mathbb{P}^1)$ , and any $2$-form on a surface is closed, this should exactly be $d^n \int \omega$, with no asymptotic needed. I'm probably missing something basic, though. –  David Speyer Mar 9 '10 at 20:14
@ David Speyer For a complex operator $A$ $|det A|^2 =det A_{\mathbf R}$, so integrand differs from the preimage under $n$-th iteration of $\phi$. –  Petya Mar 10 '10 at 1:01

If I understand your problem correctly, i.e. that $\mu$ is the usual metric on the Riemann sphere, then you're asking if essentially all orbits are expansive, at least with respect to that metric.