MINIMALITY OF p-ADIC RATIONAL MAPS WITH GOOD REDUCTION
Résumé
A rational map with good reduction in the field $\mathbb{Q}_p$ of $p$-adic numbers defines a $1$-Lipschitz dynamical system on the projective line $\mathbb{P}^1(\mathbb{Q}_p)$ over $\mathbb{Q}_p$. The dynamical structure of such a system is totally described by a minimal decomposition: $\mathbb{P}^1(\mathbb{Q}_p)$ is decomposed as finitely many periodic orbits; finite or countably many minimal subsystems each consisting of a finite union of balls; and the attracting basins of the periodic orbits and minimal subsystems. For any prime $p$, a minimal criterion for rational maps with good reduction is obtained. In particular, a complete characterization of minimal rational maps with good reduction is given in terms of their coefficients for the case $p = 2$. It is also proved that a rational map of degree $2$ or $3$ can never be minimal on the whole space $\mathbb{P}^1(\mathbb{Q}_2)$.
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