Degree Growth, Linear Independence and Periods of a Class of Rational Dynamical Systems

TL;DR

This paper investigates algebraic dynamical systems from rational functions, analyzing degree growth, linear independence, and trajectory lengths over finite fields, extending known polynomial case results and introducing new findings.

Contribution

It introduces a comprehensive study of rational function-based dynamical systems, generalizing polynomial case results and providing new insights into their algebraic properties.

Findings

Degree growth patterns characterized

Linear independence of iterates established

Trajectory length bounds over finite fields derived

Abstract

We introduce and study algebraic dynamical systems generated by triangular systems of rational functions. We obtain several results about the degree growth and linear independence of iterates as well as about possible lengths of trajectories generated by such dynamical systems over finite fields. Some of these results are generalisations of those known in the polynomial case, some are new even in this case.