What is the significance of Topological Entropy? I'm taking a course on Dynamical systems and we learnt about topological entropy. I get that topological entropy is a quantity which measures how quickly distinct points separate asymptotically, and that it is a topological invariant (which makes it good). But how does knowing the topological entropy of a dynamical system help us understand the properties of a dynamical system better? For example, since positive entropy doesn't imply chaos and neither does chaos imply positive entropy as one would naively expect initially, entropy does not seem helpful in understanding chaotic behaviour.
So the question is, what insight does knowing the entropy give into a dynamical system?

A discrete-time dynamical system is a nonempty set $ X $ and a map $ f : X → X$,
together with some additional structure, such as measure-theoretic or topological
structure. The study of dynamical systems as a whole is primarily concerned with
the asymptotic behavior of such systems, that is how the system evolves after
repeated applications of $f$.
Topological entropy measures the evolution of distinguishable orbits over time,
thereby providing an idea of how complex the orbit structure of a system is. Entropy
distinguishes a dynamical system where points that are close together remain close
from a dynamical system in which groups of points move farther.
