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Register a new userA Criterion for Potentially Good Reduction in Non-archimedean Dynamics
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Authors
Robert L. Benedetto
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Let K be a non-archimedean field, and let f in K(z) be a polynomial or rational function of degree at least 2. We present a necessary and sufficient condition, involving only the fixed points of f and their preimages, that determines whether or not the dynamical system f on P^1 has potentially good reduction.
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Let K be a non-archimedean field, and let f in K(z) be a rational function of degree d>1. If f has potentially good reduction, we give an upper bound, depending only on d, for the minimal degree of an extension L/K such that f is conjugate over L to a map of good reduction. In particular, if d=2 or d is greater than the residue characteristic of K, the bound is d+1. If K is discretely valued, we give examples to show that our bound is sharp.
Let $f_1,...,f_gin {mathbb C}(z)$ be rational functions, let $Phi=(f_1,...,f_g)$ denote their coordinatewise action on $({mathbb P}^1)^g$, let $Vsubset ({mathbb P}^1)^g$ be a proper subvariety, and let $P=(x_1,...,x_g)in ({mathbb P}^1)^g({mathbb C})$ be a nonpreperiodic point for $Phi$. We show that if $V$ does not contain any periodic subvarieties of positive dimension, then the set of $n$ such that $Phi^n(P) in V({mathbb C})$ must be very sparse. In particular, for any $k$ and any sufficiently large $N$, the number of $n leq N$ such that $Phi^n(P) in V({mathbb C})$ is less than $log^k N$, where $log^k$ denotes the $k$-th iterate of the $log$ function. This can be interpreted as an analog of the gap principle of Davenport-Roth and Mumford.
We present a p-adic and non-archimdean version of the Five Islands Theorem for meromorphic functions from Ahlfors theory of covering surfaces. In the non-archimedean setting, the theorem requires only four islands, with explicit constants. We present examples to show that the constants are sharp and that other hypotheses of the theorem cannot be removed. This paper extends an earlier theorem of the author for holomorphic functions.
We formulate a general question regarding the size of the iterated Galois groups associated to an algebraic dynamical system and then we discuss some special cases of our question.
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 $f:mathbb{P}^1_Ktomathbb{P}^1_K$ has simple good reduction outside $S$ if it extends to an $R_S$-morphism $mathbb{P}^1_{R_S}tomathbb{P}^1_{R_S}$. A finite Galois invariant subset $Xsubsetmathbb{P}^1_K(bar{K})$ has good reduction outside $S$ if its closure in $mathbb{P}^1_{R_S}$ is etale over $R_S$. We study triples $(f,Y,X)$ with $X=Ycup f(Y)$. We prove that for a fixed $K$, $S$, and $d$, there are only finitely many $text{PGL}_2(R_S)$-equivalence classes of triples with $text{deg}(f)=d$ and $sum_{Pin Y}e_f(P)ge2d+1$ and $X$ having good reduction outside $S$. We consider refined questions in which the weighted directed graph structure on $f:Yto X$ is specified, and we give an exhaustive analysis for degree $2$ maps on $mathbb{P}^1$ when $Y=X$.
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