
2. Moduli Problems

1. Equidistribution Problems

Problem 2.1.

1. $f:\mathbb{P}^1 \to \mathbb{P}^1$. Is the set PCF equidistributed on $M_d$ with respect to the bifurcation measure.
2. Same question for $f:\mathbb{P}^N \to \mathbb{P}^N$ with respect to what measure?
Part (1) is known for polynomials - Thomas Gauthier and Charles Favre
• 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.

1. 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.

1. How big is the PCF locus in $M_d^N$?
2. 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 $\pi: R_d^N \to M_d^{N-1}$. In any given fiber of $\pi$ is there a curve of PCF maps? (Known: Ingram-There 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 $C \to R_d^N \to M_d^{N-1}$ where the map $C \to M_d^{N-1}$ 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 $n$.
Known for abelian varieties and for $d=2$, $n=1,\ldots,5$ $M_d(n)$ is rational and $M_2(6)$ is general type (Blanc-Canci)

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