We continue the discussion of topological entropy started yesterday.

-1. Basic properties-

So, recall that we attached an invariant {h_{top}(T) \in [0, \infty]} to a transformation {T: X \rightarrow X} of a compact metric space {X}. We showed that it was purely topological, i.e. invariant under semiconjugacies. However, we have yet to establish its basic properties and compute some examples.

In general, we can’t expect topological entropy to be additive, i.e. {h_{top}(T \circ S) = h_{top}(T) + h_{top}(S)}, even if {T} and {S} commute. The reason is that the identity—or any isometry—has zero entropy, while there are homeomorphisms with nonzero entropy.

However, we do have:

Theorem 1 If {m \in \mathbb{Z}}, then {h_{top}(T^m) = |m| h_{top}(T)}.

(Here if {m \geq 0}, this makes sense even for {T} noninvertible.)

We handle the two cases {m>0} and {m=-1} (which together imply the result). In each, we will use the second definition of entropy that we gave in terms of coverings (which historically actually came first). Namely, the definition was

\displaystyle h_{top}(T) = \sup_{\mathfrak{A}} \lim_{n} \frac{1}{n} \log \mathcal{N}( \mathfrak{A} \vee T^{-1}\mathfrak{A} \vee \dots \vee T^{-n+1}\mathfrak{A}). (more…)