Can topological entropy be infinte?

Question

I wonder if the topological entropy as defined by Adler or Bowen can be infinity.

Can you answer that?

Answer 1

This question is actually answered in an example of Adler's original paper. Let $X$ be any compact Hausdorff space with infinitely many points, let $X^\infty$ be the space of two-sided infinite sequences $(x_n)_{n=-\infty}^\infty$ with $x_n \in X$, equipped with the product topology. Then the shift $\sigma:X^\infty \to X^\infty$, $\sigma((x_n)) = (x_{n+1})$ has infinite entropy.