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- W2974292007 abstract "Let $K$ be a number field, let $S$ be a finite set of places of $K$, and let $R_S$ be the ring of $S$-integers of $K$. A $K$-morphism $f:mathbb{P}^N_Ktomathbb{P}^N_K$ has simple good reduction outside $S$ if it extends to an $R_S$-morphism $mathbb{P}^N_{R_S}tomathbb{P}^N_{R_S}$. A finite Galois invariant subset $Xsubsetmathbb{P}^N_K(bar{K})$ has good reduction outside $S$ if its closure in $mathbb{P}^N_{R_S}$ is 'etale over $R_S$. The set $X$ is a preperiodic structure for $f$ if it satisfies $f(X)subseteq X$. We conjecture that there is an $n=n(N,d)$ such that there are only finitely many $text{PGL}_{N+1}(R_S)$-equivalence classes of pairs $(f,X)$, where $text{deg}(f)=d$, $#Xge n$, $X$ is a preperiodic structure for $f$, and both $f$ and $X$ have good reduction outside $S$. We prove our conjecture for $N=1$ with $n(1,d)=2d+1$. We consider refined questions in which the (weighted) directed graph structure on $f:Xto X$ is specified, we construct associated moduli spaces, and we give an exhaustive analysis for degree $2$ maps on $mathbb{P}^1$." @default.
- W2974292007 created "2019-09-26" @default.
- W2974292007 creator A5058156788 @default.
- W2974292007 date "2017-03-02" @default.
- W2974292007 modified "2023-09-27" @default.
- W2974292007 title "Good reduction of dynamical systems with preperiodic level structure and a Shafarevich-type conjecture" @default.
- W2974292007 hasPublicationYear "2017" @default.
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