I am trying to dip my toes into random dynamical systems. I am trying to understand the following:
Consider a process of iterates $$T_0 ^k = T_0 \circ \ldots ^k \ldots\circ T_0,\ \ \ \ k\geq 1$$
of a smooth transformation $T_0: M$ ⟲ of a manifold onto itself.
The first part seems understandable. Say, take $T_0 = f_0 (x) = x^2 + 3$, and then just iterate starting at some value.
Now, I do not have a strong background in topology or differential geometry. Hopefully this will change in the future. I know the very basics of the subjects at best. So, I am not really understanding what a manifold has to do with any of this, and I have never seen that "circle arrow" symbol before in my life.
Is there a way anyone can explain what is going on here without going to in depth about manifolds? Or is it completely useless to look at this stuff right now and just come back when I have a stronger pure math background?
Thanks.
Best Answer
The circle arrow is standard in the area: it avoids writing $M$ once more, as in $M\to M$. But I never understand why people want to avoid one extra character.
(Smooth) manifolds are the spaces where we can define smooth maps, although locally (meaning: up to a diffeomorphism) they are always like $\mathbb R^n$ for some $n$. The problem is that many interesting properties only occur because of nonlocal properties. For example, if the manifold is compact, which is not a local property, then some quite interesting dynamics starts to occur.
So, although you don't specify your interests, unavoidably you will need eventually to learn about smooth manifolds a bit more, but only as a user, which only requires the basics.