Counterexample of Julia Set definition

Question

So I'm following a book by Falconer on Fractals, and it defines the Julia set of a complex function as $$J(f)=\overline{\{z\in\mathbb{C}:z \text{ is a repulsive periodic point of }f\}}$$ Where the overline denotes closure. It then defines: $$J_0(f) = \{z\in\mathbb{C}:\{f^k\}_k\text{ is not normal at }z\}$$ And goes on to prove that if $f$ is a polynomial, then $J(f)=J_0(f)$.
I've seen that the same is true for rational functions and I was wondering if there are any simple counterexamples (where the 2 sets are different)?

Answer 1

By theorem 2.15 on page 14 of the paper Complex Exponential Dynamics by Bob Devaney this equivalence holds for all complex analytic functions on $\mathbb C$. By theorem 4 on page 160 of Iteration of Meromorphic Functions by Walter Bergweiler, this extends to meromorphic functions.

If you're interested in complex dynamics beyond rational functions, these are great papers and pretty much required reading. The first is easier going than the second.