**Jung Kyu Canci**^{[1]} and **Laura Paladino**^{[2]}

*
On preperiodic points of rational functions defined over \({\mathbb F}_p(t)\)
*

**Pages:** 193-203

**Received:** 31 December 2015

**Accepted:** 29 February 2016

**Mathematics Subject Classification (2010):** 37P05, 37P35.

**Keywords:** Preperiodic points, function fields.

**Author address:**

[1] : Universität Basel, Mathematisches Institut, Spiegelgasse 1, Basel, CH-4051, Switzerland

[2] : University of Pisa, Department of Mathematics, Largo Bruno Pontecorvo 5, Pisa, 56127, Italy

Laura Paladino's work is supported by Istituto Nazionale di Alta Matematica trough *Assegno di Ricerca Ing. G. Schirillo*

**Abstract:**
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 \({\mathbb F}_p\) 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 \({\mathbb F}_p(t)\cup \{\infty\}\) for periodic and preperiodic points.

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