Postcritical set

With the closed-set convention, the postcritical set of a rational map $f$ iswhere $\mathrm{Crit}(f)$ is the set of critical points of a rational map. Thus one starts at the critical values, takes all their forward orbits, and then takes the closure. Some authors omit the closure; stating the convention matters when a critical orbit accumulates without becoming periodic.

For $f(z)=z^2-1$, the critical points of a rational map are zero and infinity. Their forward images give $P(f)=\{-1,0,\infty \}$, because $0\mapsto -1\mapsto 0$ and infinity is fixed. The postcritical set combines all critical orbits, rather than recording only one chosen starting point.