
3. PCF Maps in Higher Dimensions

1. Problem 3.1.

In the definition of PCF maps in higher dimensions, is the “all the way down” condition needed?
1. Remark. A map $f:\Bbb{P}^N\rightarrow\Bbb{P}^N$ is called post-critically finite (PCF) if, denoting the ramification locus by $C$, its post-critical set $V:=\bigcup_{n\geq 0}f^n(C)$ is a proper algebraic subvariety of $\Bbb{P}^N$. We call $f$ “PCF all the way down” if the restriction of $f$ to every periodic component of $V$ is PCF too. (http://aimpl.org/finitedynamics/5/)
• Problem 3.2.

Is the set of PCF maps in $\mathcal{M}_d^N$ Zariski dense for $N\geq 2$?
• Problem 3.1.

[P. Ingram] For which $(N+1)\times (N+1)$ matrices $A$ the map $\mathbf{X}\in\Bbb{P}^N\mapsto A\mathbf{X}^d\in\Bbb{P}^N$ is PCF for some $d\geq 2$?

Cite this as: AimPL: Moduli spaces for algebraic dynamical systems, available at http://aimpl.org/modalgdyn.