For a compact metric space $(X,d)$ and a continuous map $f\colon X\to X$, the book Introduction to Dynamical Systems written by M. Brin and G. Stuck uses the expression $$h_{\varepsilon}(f):=\limsup_{n\to\infty}\log(\text{cov}(n,\varepsilon,f))$$ in their definition of topological entropy. Later they prove that you can replace limsup by lim, because apparently the lim of the sequence above converges to a finite number. So why do they use the limsup in their definition? I thought that the limsup may also diverge to infinity.
2026-02-24 08:20:56.1771921256
Question about the definition of topological entropy (book: Brin & Stuck)
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Probably because any sequence $x_n$ in $\mathbb R$ (so also $x_n = \log(\operatorname{cov}(n,\varepsilon,f)$) (for some fixed $f$, $\varepsilon$) has a well-defined $\limsup$ (which can be $+\infty$). So first you have a well-defined (extended) number and then you can discuss its other properties, like whether it's always finite etc.