We further continue the discussion of topological entropy. Here, we discuss various results that bound above and below the entropy of a given map.

**1. The topological entropy of Lipschitz maps **

Many of the dynamical systems of interest are actually given by compact manifolds and smooth maps . These are always Lipschitz with respect to appropriate metrics. Indeed, choose a Riemannian metric on and let denote the induced norm on the tangent spaces. Then is a Lipschitz constant for with respect to the metric on induced by the Riemannian metric. In this case, the entropy is always finite. We shall prove this in a more general context.

Let be a compact metric space, and for , let denote the the number of -balls necessary to cover (which is always finite). Then we call

the **ball dimension** of . For instance, an -cube has ball dimension . It follows more generally that a Riemannian -manifold has ball dimension .

The reason we shall use this concept below is that gives an upper bound for a minimal -spanning set of the space . (Recall that an -spanning set means that every point is -close to it.) In fact, if the -balls cover , then the centers of these form an -spanning set.

Theorem 1Let be a compact metric space with finite ball dimension . Suppose is a Lipschitz continuous mapping with constant , i.e. for all . Then the entropy of is finite and (more…)