Is the topological entropy time-reversal invariant?

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It is, when the dynamics is a homeomorphism. This is a simple exercise since a family of sets is a cover if and only if its image or preimage (under a homeomorphism!) is a cover.

However, in case you are talking about the topological entropy on a noncompact set, in general it is not true even if the dynamics is invertible. At some point it was introduced the notion of two-sided topological entropy for which what you ask holds for any noncompact set.