Postcritically finite (PCF) maps have two equivalent finiteness properties. The better-known one, taken as the definition, is that their postcritical set is finite. The other is that a PCF map defined over a number field has only finitely many places at which it has a hyperbolic attracting cycle. This may affect the following questions, for which PCF maps appear special: 1. The spectrum of multipliers of PCF maps. By results of Ghioca-Tucker, if $\phi$ is non-PCF and defined over a number field, then the set of nonzero multipliers of \phi generates an infinitely-generated multiplicative group in $\overline{\mathbb{Q}}^*$; this uses the infinity of primes at which a PCF map has an attracting cycle. The same statement is false for power and Lattes maps. It is not known whether this is true for other PCF maps; very rudimentary computational evidences suggests it should be. 2. PCF maps have very constrained choices of multipliers, and each choice of multiplier further constrains the maps. This is especially easy to verify for fixed points of quadratic polynomials: multipliers are $p$-units for $p > 2$, and lie in the maximal ideal mod 2; for every allowable multiplier with 2-valuation more than 1, there is only one PCF map. For more general maps, there should be similar constraints. The guideline is a conjecture of Baker-DeMarco about the sparseness of PCF maps in $\operatorname{Per}(A)$, the space of rational maps with a periodic cycle of multiplier $A$ ($A \neq 0$). 3. Whenever a map $\phi$ has a $p$-adic hyperbolic attracting cycle, we obtain a Galois cycle permutation in the Galois group of $\phi^n$, where $n$ is the period of the cycle. Past work of Jones has suggested that non-PCF maps' Galois groups, as $n \to \infty$, are as large as they can possibly be (specifically, they're conjectured to be finite-index in the automorphism group of the tree structure on order-up-to-$n$ preimages of 0), while PCF maps' are much smaller. The ability to obtain infinitely many cycles for non-PCF maps may prove parts of this conjecture while explaining what fails for PCF maps.