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In 'Dynamics in One Complex Variable' is states that a polynomial $f$ of degree $n$ maps the equipotential $G^{-1}(c) = \{z; G(z)=c\}$ to $G^{-1}(nc)$. I have been thinking about this and I can not immediately see why. Could someone explain this?

Thanks

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The potential is $$ G(z)=\lim_{d\to\infty}\frac{\log(|f^d(z)|)}{n^d} $$ so that $$ G(f(z))=\lim_{d\to\infty}\frac{\log(|f^d(f(z))|)}{n^d}=n\lim_{d\to\infty}\frac{\log(|f^{d+1}(f(z))|)}{n^{d+1}}=nG(z). $$ Now use this identity in the description of the level sets, $$ f(G^{-1}(c))=\{f(z):G(z)=c\}=\{f(z):G(f(z))=nc\}\subset G^{-1}(nc). $$ And as $f(z)=w$ always has solutions $z\in\Bbb C$, you get also equalitly in the last relation.

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