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A $\textit{portrait}$ $\mathcal{P}$ on $\mathbb{P}^N$ is a pair of finitepoint sets $Y\subseteq{X}\subset\mathbb{P}^N$, a map $Y\to X$, and anassignment of weights to the points in $Y$. We construct a parameter space$\operatorname{End}_d^N[\mathcal{P}]$ whose points correspond to degree $d$endomorphisms $f:\mathbb{P}^N\to\mathbb{P}^N$ such that $f:Y\to{X}$ is asspecified by a portrait $\mathcal{P}$, and prove the existence of the GITquotient moduli space$\mathcal{M}_d^N[\mathcal{P}]:=\operatorname{End}_d^N//\operatorname{SL}_{N+1}$under the $\operatorname{SL}_{N+1}$-action$(f,Y,X)^\phi=\bigl(\phi^{-1}\circ{f}\circ\phi,\phi^{-1}(Y),\phi^{-1}(X)\bigr)$relative to an appropriately chosen line bundle. We also investigate thegeometry of $\mathcal{M}_d^N[\mathcal{P}]$ and give two arithmeticapplications.

Moduli Spaces for Dynamical Systems with Portraits