Counting distinct functional graphs from linear finite dynamical systems

Abstract

Let Fq be the finite field with q elements and, for each positive integer n, let Aq(n) be the number of non isomorphic functional graphs arising from Fq-linear maps T:Fqn→Fqn. In 2013, Bach and Bridy proved that, for every prime power q, we have that 12≤liminfn→+∞log⁡log⁡Aq(n)log⁡n≤limsupn→+∞log⁡log⁡Aq(n)log⁡n≤1. By combining some ideas from linear algebra, combinatorics and number theory, in this paper we provide sharper estimates on the function Aq(n) and, in particular, we prove that limn→+∞⁡log⁡log⁡Aq(n)log⁡n=1 for every prime power q.