# Equipotential curves of a Julia set

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

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.