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