On preperiodic points of rational functions defined over F_p(t)

Let $P\in\mathbb{P}_1(\mathbb{Q})$ 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 $\mathbb{F}_p$ bethe field with p elements. In the present paper we consider rational functions defined over the rational global function field $\mathbb{F}_p$ with good reduction at every finite place. We provesome bounds for the cardinality of orbits in $\mathbb{F}_p\cup\{\infty\}$ for periodic and preperiodic points.