9 - Topological Entropy

Summary

Given a compact metric space (X, d) and a map ƒ : X →X (not necessarily continuous) one asks: How many orbits does the dynamical system ƒ : X → X have? Of course, there are as many orbits as there are points, because every point has its own orbit. However, in many cases, these orbits behave in the same way. For example, take ƒ : S¹ – S¹ a rigid circle rotation (recall Definition 8.1.2) and x, y ∈ S¹. If x and y are ε apart, then ƒ(x) and ƒ(y) are also ε apart, and for every So the orbits of x and y behave in essentially the same way.

How many essentially different orbits does a dynamical system have? This depends on what we think is “essentially different”; however, it is reasonable to say that orb(a:) and orb(y) are at least ε apart if there exists i such that If “essentially different” means “at least ε apart,” then a rigid circle rotation has “1/ε essentially different orbits.” A dynamical system with sensitive dependence, on the other hand, has infinitely many essentially different orbits: In every neighborhood of every x, there is a y that eventually gets e apart from x, so x and y have essentially different orbits.

If x and y are very close together, it could take many iterations before (or if) we find. It is therefore more useful to compute how many essentially different orbits there are up to the nth iteration.