Counting distinct functional graphs from linear finite dynamical systems

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