
## 5. Higher dimensions

Problems related to the notion of PCF for endomorphism of $\mathbb{P}^N$ for $N>1$.
1. ### Notions of PCF

$f:\mathbb{P}^N \to \mathbb{P}^N$ morphism.
1. Defn: PCF if the orbit of the ramification locus $C$, $V=\bigcup_{n \geq 0} f^n(C)$ is a proper algebraic subvariety of $\mathbb{P}^N$.
2. Defn: “PCF all the way down”: If the restriction of $f$ to every periodic component of the post-critical component $V$ is PCF.

#### Problem 5.1.

1. Does (1) imply (2)?
2. Are PCF (1) or (2) Zariski dense in $M_d^N$?
3. 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) Andre-Oort 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 $\mathbb{P}^1$.
resolved: yes.

Cite this as: AimPL: Postcritically finite maps in complex and arithmetic dynamics, available at http://aimpl.org/finitedynamics.