Scale function vs Topological entropy

In the realm of topological automorphisms of totally disconnected locally compact groups, the scale function introduced by Willis is compared with the topological entropy. We prove that the logarithm of the scale function is always dominated by the topological entropy and we provide examples showing that this inequality can be strict. Moreover, we give a condition equivalent to the equality between these two invariants. Various properties of the scale function, inspired by those of the topological entropy, are presented.