5. Higher dimensions
Problems related to the notion of PCF for endomorphism of $\mathbb{P}^N$ for $N>1$.
Notions of PCF
$f:\mathbb{P}^N \to \mathbb{P}^N$ morphism. Defn: PCF if the orbit of the ramification locus $C$, $V=\bigcup_{n \geq 0} f^n(C)$ is a proper algebraic subvariety of $\mathbb{P}^N$.
 Defn: “PCF all the way down”: If the restriction of $f$ to every periodic component of the postcritical component $V$ is PCF.
Problem 5.1.
 Does (1) imply (2)?
 Are PCF (1) or (2) Zariski dense in $M_d^N$?
 Is there are more general definition than (it1) such that:
a) “there are more of them”? (i.e. less restrictive)
b) they are dense in any reasonable topology
c) AndreOort property

Thurston Rigidity Exceptions
Problem 5.2.
What are the “obvious” exception to Thurston Rigidity in $M_d^N$? 
Flexible Family of PCF maps
Problem 5.3.
Is there a flexible family of PCF maps that does not come from $\mathbb{P}^1$.
Cite this as: AimPL: Postcritically finite maps in complex and arithmetic dynamics, available at http://aimpl.org/finitedynamics.