# On preperiodic points for rational functions defined over $mathbb{F}_p(t)$

Canci, Jung Kyu and Paladino, Laura. (2016) On preperiodic points for rational functions defined over $mathbb{F}_p(t)$. Rivista di Matematica della Università di Parma, 7 (1). p. 12.

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Let $Pin mathbb(P)_1(mathbb{Q})$ be a periodic point for a monic polynomial with coefficients in $mathbb{Z}$. With elementary techniques one sees that the minimal periodicity of $P$ is at most 2. Recently we proved a generalization of this fact to the set of all rational functions defined over $mathbb{Q}$ with good reduction everywhere (i.e. at any finite place of $mathbb{Q}$). The set of monic polynomials with coefficients in $mathbb{Z}$ can be characterized, up to conjugation by elements in PGL$_2(mathbb{Z}), as the set of all rational functions defined over$mathbb{Q}$with a totally ramified fixed point in$mathbb{Q}$and with good reduction everywhere. Let$p$be a prime number and let$mathbb{F}_p$be the field with$p$elements. In the present paper we consider rational functions defined over the rational global function field$mathbb{F}_p(t)$with good reduction at every finite place. We prove some bounds for the cardinality of orbits in$mathbb{F}_pcup{infty}\$ for periodic and preperiodic points.