Denjoy's theorem on rotation number

Okay kiddo, let's talk about Denjoy's theorem on rotation number.

First, we need to understand what rotation number means. Imagine you have a circle and a point moving around it. The point makes a full turn around the circle and ends up in the same spot as before. The rotation number is the number of times the point goes around the circle divided by the number of times it goes in the same direction. Think of it like this - if you walk around the playground in a circle and end up where you began after taking 3 full turns while facing forward, your rotation number is 3/3 or 1.

Now, Denjoy's theorem is all about this concept of rotation number but in a more complex setting. It says that if we have a continuous function (which just means a function that doesn't jump around) that maps a circle onto itself in a way that preserves the orientation (which just means it doesn't flip the circle around), then the rotation number of any point on the circle will be a constant number.

What does this mean? It means that no matter where the point on the circle starts, it will always end up going around the circle the same number of times in the same direction. Think of it like if we had a merry-go-round with a person on it, and the merry-go-round is spinning in a certain direction. Even if the person starts on a different horse, they will always end up going around the same number of times in the same direction because the direction of the merry-go-round hasn't changed.

So, Denjoy's theorem basically tells us that if we have a certain kind of function that preserves the orientation of a circle and we look at the rotation number for any point on that circle, it will always be the same number no matter where that point is on the circle. Pretty cool, huh?