Definition of topological entropy

Question

What the meaning of the limit that appears in the definition of topological entropy?

Let $X$ a compact metric space and $f\colon X\to X$ a continuous function and a subset $K \subset X$. The topological entropy of $f$ with respect to $K$ is the number $h(f,K)=\lim_{\epsilon \to 0} r(\epsilon, K) = \lim_{\epsilon \to 0} s(\epsilon, K)$

Where

$$r(\epsilon, K)=\lim_{n \to \infty} \sup \frac{1}{n}\log r_{n} (\epsilon, K)$$

and

$$s(\epsilon, K)=\lim_{n \to \infty} \sup \frac{1}{n}\log s_{n} (\epsilon, K)$$

And finally $r_n(\epsilon, K)$ is the smallest cardinality of any set $F$ that $(n,\epsilon)$-spans $K$ and $s_n(\epsilon, K)$ the largest cardinality of $(n,\epsilon)$ separated $E \subset K$

Answer 1

Topological entropy basically counts the (exponential) number of distinct orbits generated by the dynamical system, up to some 'resolution' and orbit length.

The $\epsilon$ limit is basically taking your resolution to 0, and $n$ limit is taking the orbit length to infinity.