Let $f$ be a rational map acting on the Riemann sphere $\widehat{\mathbb{C}}$. The Julia set $\mathcal{J}_f$ is the complement of the Fatou set $\mathcal{F}_f$, defined to be the union of all open neighborhoods such that the iterates of $f$ form a normal family.
Wikipedia states that the Julia set is the smallest closed set with at least three points that is completely invariant under $f$. Would it hold true were we to replace the phrase completely invariant with either "forward-invariant" or "backward-invariant"?
No. The set $$ \{e^{-\frac{4 i \pi }{5}},e^{-\frac{2 i \pi }{5}},e^{\frac{2 i \pi }{5}},e^{\frac{4 i \pi }{5}}\} $$ is forward invariant under the action of $z\to z^2$, as you can easily check. Of course, these are exactly the primitive $5^{\text{th}}$ roots of unity so one might suppose this could be extended to a more general observation.