What is the significance of Topological Entropy?

Question

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?

Answer 1

Perhaps the primary value is that it is related to other invariants outside of just topological dynamics, as expressed in the so-called Variational Principle, which says:

If $T : X \to X$ is a continuous self-map of a compact Hausdorff space $X$ then its topological entropy $h(T)$ is the equal to the supremum of the measure theoretic entropies $h_\mu(T)$, as $\mu$ varies over all probability measures on $X$ that are invariant under $T$ (meaning that $T^*(\mu)=\mu$).

Here's a Scholarpedia link which discusses these and many other matters regarding topological entropy.

Another nice thing about topological entropy is that in some restricted cases it gives very nice information about the number $P_n$ of periodic orbits of cardinality $n$ (see this other link on the same Scholarpedia page): $$h(T) = \limsup_{n \to \infty} \frac{\log P_n}{n} $$

Answer 2

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.