2. Moduli Problems

Equidistribution Problems
Problem 2.1.
 $f:\mathbb{P}^1 \to \mathbb{P}^1$. Is the set PCF equidistributed on $M_d$ with respect to the bifurcation measure.
 Same question for $f:\mathbb{P}^N \to \mathbb{P}^N$ with respect to what measure?

Critical orbit relations
Conjecture 2.2.
[Lauran DeMarco/Patrick Ingram] An algebraic subvariety in $M_d$ has a Zariski dense subset of PCF maps if and only if that subvariety is defined by critical orbit relations.
 Let $X \subset \mathbb{A}^2$ be a curve. Is it true that you have infinitely many points $(c_1,c_2) \in X$ such that $z^2+c_1$ and $z^2+c_2$ are both PCF if and only if it is vertical line, a horizontal line, or the diagonal?

PCF locus
Problem 2.3.
 How big is the PCF locus in $M_d^N$?
 Given a “critical portrait” when is the set of PCF maps with that portrait zero dimensional? (what are the obvious counter examples?)

$p$adic PCF locus
Problem 2.4.
Is $M_d(\Q_p) \cap PCF$ (or polynomials $P_d(\Q_p) \cap PCF$) infinite for all $p$? (maybe no for $p > d$, but what about $p < d$?) What are the accumulation points? 
regular polynomial endomorphisms
Problem 2.5.
[Patrick Ingram] Fix $d,N$. $f:\mathbb{P}^N \to \mathbb{P}^N$ is a regular polynomial endomorphism if it leaves a hyperplane invariant, call the space of them $R_d^N$. If we restrict each $f$ to its hyperplane it gives a projection $\pi: R_d^N \to M_d^{N1}$. In any given fiber of $\pi$ is there a curve of PCF maps? (Known: IngramThere are horizontal families, i.e. intersect every fiber in finitely many places. There are no PCF maps above the power map).
Alternate formulation: If you have a map from a curve $C \to R_d^N \to M_d^{N1}$ where the map $C \to M_d^{N1}$ is constant and $C$ lands in the PCF locus of $R_d^N$. Is the map $C \to R_d^N$ already constant? 
field of definition vs field of moduli
Problem 2.6.
Are all PCF maps defined over their field of moduli? 
Classification of moduli space
Problem 2.7.
[Joseph Silverman] Fix $d$. If $n >>_d 1$, then is $M_d(n)$ of general type, where $M_d(n)$ marks periodic points of formal period $n$.
Cite this as: AimPL: Postcritically finite maps in complex and arithmetic dynamics, available at http://aimpl.org/finitedynamics.