necessity of uniform continuity for topological entropy

Question

I am referring to Walters' book "Introduction to Ergodic Theory."

When he defines the concept of topological entropy he always assumes that

the transformation $T: X \rightarrow X$ is uniformly continuous

where $X$ is a not-necessarily compact metric space.

However, it is unclear from the text why we need this for the definition of the entropy.

Answer 1

A study of the non-uniformly continuous case was recently undertaken by Hasselblatt, Nitecki, and Propp in Topological entropy for non-uniformly continuous maps (preprint version; journal version). In the preprint version they write, referring to Bowen's definition (Definition 7.10 in the book, which they call Bowen compacta entropy, because of the $\sup_K h(T,K)$ step):

It is not clear that the assumption of uniform continuity plays any role in this definition, although the examples motivating Bowen were all uniformly continuous. [...] We shall avoid the assumption of uniform continuity for our purposes. However, Remark 10, as well as examples outlined by Walters [Wal82, p. 176], show that when the assumption of uniform continuity is dropped, a number of useful properties of entropy that hold for uniformly continuous maps can fail.

This looked promising. But:

Remark 10 of the preprint is the observation that the Bowen compacta entropy of a homeomorphism need not be the same as of its inverse. This is indeed true, but equality fails already for uniformly continuous maps on noncompact spaces. (Take $Tx=2x$ on $\mathbb R$: then $h(T)=\log 2$ but $h(T^{-1})=0$.) So this does not explain why the lack of uniform continuity is a problem.

[Wal82, p. 176] (the book An introduction to Ergodic theory by Walters) mentions that there are examples of homeomorphisms $T_1,T_2$ on noncompact metric spaces $X_1,X_2$ such that $h(T_1\times T_2)<h(T_1)\times h(T_2)$. The example is an incomplete sketch, but it seems (to me) that it also is based on the lack of compactness and not of uniform continuity.

So, I'm left underwhelmed by the "number of useful properties" failing. Interestingly, in the journal version the first sentence of the quotation above was changed (at referee's suggestion?):

Bowen made the assumption of uniform continuity in this definition since his motivating examples were all uniformly continuous.

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Okay, I'm done with quotations. An explanation from my own point of view: the idea of topological entropy is to have a topological invariant. Well, now we involved a metric, so it's not topological anymore. But, if the definition relies only on the uniformity of the space (the class of uniformly equivalent metrics), that's not so far from being topological. Once one begins to consider metrics up to uniform equivalence, it is natural to focus on the class UC of uniformly continuous maps.

Besides, the natural non-UC maps, like $T(x)=x^2$ and $T(x)=x^3$, have infinite entropy under Bowen's compacta definition, so we can't tell them apart using this tool.