Cycles for rational maps over global function fields with one prime of bad reduction

TL;DR

This paper establishes optimal bounds on cycle lengths and finite orbit sizes for rational maps over global function fields with limited bad reduction primes, advancing understanding of dynamical systems in positive characteristic.

Contribution

It provides the first optimal bounds for cycle lengths depending only on the characteristic and degree, using a novel analysis of p-adic distances and polynomial families.

Findings

Bound on cycle lengths depending only on p and D

Bound on the size of finite orbits

Structural insights into periodic points for polynomials

Abstract

Let $K$ be a global function field of characteristic $p$ and degree $D$ over $\mathbb F_{p}(t)$. We consider dynamical systems over the projective line $\mathbb P^1(K)$ defined by rational maps with at most one prime of bad reduction. The main result is an optimal bound for cycle lengths that only depends on $p$ and $D$. A bound for the cardinality of finite orbits is given as well. Our method is based on a careful analysis (for every prime of good reduction) of the $\mathfrak p$-adic distances between points belonging to the same finite orbit, in part motivated by previous work by Canci and Paladino. Valuable insight is provided by a certain family of polynomials. In this case we also gain a good deal of information about the structure and size of the set of periodic points for polynomials of given degree.