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.
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What is the meaning of “cov”? – FShrike Aug 21 '22 at 22:47
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@FShrike It's the minimum cardinality of an open cover each of whose elements have diameter less than $\epsilon$ with respect to the Bowen metric $d_n^f$ (e.g. as mentioned here: https://math.stackexchange.com/q/4345039/169085). – Alp Uzman Aug 22 '22 at 04:32
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See e.g. https://www.math.stonybrook.edu/~jack/DYNOTES/dn7.pdf , p. 7-8 for further details. – Alp Uzman Aug 22 '22 at 04:35
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(To be more accurate perhaps I should drop "open".) – Alp Uzman Aug 22 '22 at 04:36
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@AlpUzman Thank you for the reference – FShrike Aug 22 '22 at 07:56
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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.
Henno Brandsma
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