@Person

First of all, I really should have conjugated so $\infty$ mapped to zero and zero mapped to some other finite point, like $1$. Here's a conjugacy that does that: $$\phi(z)=\frac{1}{z+1}$$ We want to show three things: that $\infty$ is a fixed point of $N$, that $\infty$ is a repulsive, and that zero maps to $\infty$.

Equivalently, we must show that zero is a fixed point of $\phi\circ N\circ\phi^{-1}$, that $zero$ is a repulsive, and that $1$ maps to zero.

Recall that the Julia set of $N$ is the closed set of repelling periodic points under iteration of $N$. So we want to show that $\infty$ has repulsive behavior of some kind. Showing that it is a repulsive fixed point of $N$ suffices.

If we then show that $N(0)=\infty$, we also show that zero is in the Julia set of $N$ since the Julia set of $N$ is closed under iteration of $N$. This related to why composing $N$ with itself any finite number of times gives the same Julia set.

I haven't tried sowing that zero is in the Julia set of $N$ using a direct method (like showing it is a periodic point), but I assume you'll get something messy like a $1/0$ or indeterminant form.