The following is taken from a post by Terrence Tao on the Chowla conjecture and the Sarnak conjecture :
Given a bounded sequence ${f: {\bf N} \rightarrow {\bf C}}$, define the topological entropy of the sequence to be the least exponent ${\sigma}$ with the property that for any fixed ${\varepsilon > 0}$, and for ${m}$ going to infinity the set ${\{ (f(n+1),\ldots,f(n+m)): n \in {\bf N} \} \subset {\bf C}^m}$ of ${f}$ can be covered by ${O( \exp( \sigma m + o(m) ) )}$ balls of radius ${\varepsilon}$ (in the ${\ell^\infty}$ metric). (If ${f}$ arises from a minimal topological dynamical system ${(X,T)}$ by ${f(n) := F(T^n x)}$ and ${X}$ is generated by ${F}$ and its shifts, the above notion is equivalent to the usual notion of the topological entropy of a dynamical system.
But I have difficulty understanding why his (or maybe Sarnak's) definition of the topological entropy of a sequence can be identified with the usual notion of the topological entropy of a dynamical system:
Specifically, I would like to ask:
(1) What is the meaning of "If ${f}$ arises from a minimal topological dynamical system ${(X,T)}$ by ${f(n) := F(T^n x)}$ and ${X}$ is generated by ${F}$ and its shifts"? In particular, what is the meaning of "minimal" and “${X}$ is generated by ${F}$ and its shifts"? (I really want to know if $X$ is necessarily a compact metric space)
(2) Once (1) is clarified, I hope it is not hard to show that "the above notion is equivalent to the usual notion of the topological entropy of a dynamical system" (but let me know if it is not so obvious). Is "topological entropy of a dynamical system" he meant simply the situation of compact metric space, where topological entropy can be relatively easily defined? Or topological entropy for general non-metrizable spaces?
My question is mainly about the clarification in (1), once (1) is clear, I think (2) shouldn't be hard.
