# Proof that Julia set is contained in closure of repelling periodic points

Kenneth Falconer's Fractal Geometry gives a proof that the Julia set of a polynomial $$f$$ is the closure of the repelling periodic points of $$f$$. To show that $$J(f)$$ is contained in the closure of the repelling periodics, he gives the following argument, which I understand except for the claim that $$w$$ must be in the closure of repelling periodics.

Let $$E=\{w\in J(f) : \exists v\neq w \text{ with } f(v)=w \text{ and } f'(v)\neq 0\}$$. Suppose that $$w\in E.$$ Then there is an open neighborhood $$V$$ of $$w$$ on which we may find a local analytic inverse $$f^{-1}: V\rightarrow \mathbb{C}\setminus V$$ so that $$f^{-1}(w)=v\neq w$$ (just choose values of $$f^{-1}(z)$$ in a continuous manner). Define a family of analytic functions $$\{h_k\}$$ on $$V$$ by $$\begin{equation*} h_k(z) = \frac{f^k(z)-z}{f^{-1}(z)-z}. \end{equation*}$$ Let $$U$$ be any open neighbourhood of $$w$$ with $$U\subset V.$$ Since $$w\in J(f)$$ the family $$\{f_k\}$$ and thus, from the definition, the family $$\{h_k\}$$ is not normal on $$U.$$ By Montel's theorem 14.5, $$h_k(z)$$ must take the value 0 or 1 for some $$k$$ and $$z\in U.$$ In the first case $$f^k(z)=z$$ for some $$z\in U$$; in the second case, $$f^k(z)=f^{-1}(z)$$ so $$f^{k+1}(z)=z$$ for some $$z\in U.$$ Thus $$U$$ contains a periodic point of periodic point of $$f$$, so $$w$$ is in the closure of the repelling periodic points for all $$w\in E.$$ Since $$f$$ is a polynomial, $$E$$ contains all of $$J(f)$$ except for a finite number of points. Since $$J(f)$$ contains no isolated points, by Proposition 14.9, $$J(f)\subset \overline{E}$$ is a subset of the closure of the repelling periodic points.

Why must $$w$$ be in the closure of the repelling periodics? Didn't we only show that there is a periodic point (not necessarily repelling) in every neighborhood of $$w$$? Is the periodic point we found in $$U$$ repelling, or is there another point that we must find? I feel like the point $$v$$ is an important piece here that I don't understand.