We begin this lecture by showing that for a piecewise monotone dynamical system on the interval, one can compute the topological entropy by looking at the exponential growth rate of the number of intervals of monotonicity of the iterated function.

This gives an easy proof that the topological entropy of the tent map $ \tau \colon [0,1] \to [0,1]$ is $ \log 2$.

We then introduce the notion of the ball dimension of a metric space. The ball dimension measures the exponential growth rate (in $ \varepsilon$) of the cardinality of coverings by balls of radius $ \varepsilon$. The ball dimension is not necessarily an integer (and can be infinite). However it agrees with the usual concept of dimension for all our standard spaces (cubes, spheres, tori, and more generally any topological manifold). For more exotic spaces, however, the ball dimension is harder to guess. (Example: the ball dimension of the Cantor Set is $\frac{\log 2}{\log 3}$, as you will prove on Problem Sheet F.)

We conclude by proving that Lipschitz continuous dynamical systems on compact metric spaces with finite ball dimension always have finite entropy. In particular, this shows that differentiable dynamical systems on compact manifolds have finite entropy. This will be useful next semester.

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