  We construct suitable metrics for two classes of topological dynamical
systems (linear maps on the torus and non-invertible expansive maps on compact
spaces) in order to get a lower bound for topological entropy in terms of the
resulting Hausdorff dimensions and Lipschitz constants. This reverses an old
inequality of Dai, Zhou, and Gheng and leads to a short proof of a well-known
theorem on expansive mappings. It also suggests a new invariant of topological
conjugacy for dynamical systems on compact metric spaces.
