Dynamical systems proof - nonwandering set

[meteorologist1]

Hi, can anyone help me prove the following:

Show that the nonwandering set is closed and positively invariant.

I always have trouble working with sets because they're so abstract. If anyone can help me, that would be great. Thanks.

Definition of nonwandering set is here:
http://mathworld.wolfram.com/Nonwandering.html

 

[Hurkyl]

I don't recall the term "positively invariant".


To prove it's closed, you could pick an arbitrary convergent sequence of nonwandering points, and prove their limit must be nonwondering.

Or, you could try proving that the complement is open -- that each nonwandering point has a neighborhood of nonwandering points.

 

[meteorologist1]

Ok, I will try that.

For the term positively invariant:
A subset S of the domain D is an invariant set for the system if the orbit through a point of S remains in S for all t. If the orbit remains in S for t > 0, then S is said to be positively invariant.

If you like, you could also look at:
http://people.cs.uchicago.edu/~lebovitz/Eodesbook/ds.pdf

Thanks.

 

[xanaduet]

Hey

to show that a set W of non-wondering point is closed it is enough to show that it's complement is open; i.e. a set of "wondering" points is open.
First define "wondering" point ( negation of non-wondering point) : p is wondering (or isn't non-wondering) if there is a open interval K containing p such that any iteration of any point in K isn't in K.

To show that complement of W is open. Take a point p in a complement of W. P isn't non-wondering.
Hence there is an open interval K containing p s.t. every iterate of point in K ends up outside K.
But then every point of K isn't nonwondering as well (for any x in K that a nbhd of x contained in K and then all pts in thsi nbhd will end up outside K)
So K is contained in the complement of W. This proofs that the complement of W is open -> W is closed.

Take care

 

[meteorologist1]

Thank you very much, xanaduet.

 

[PBRMEASAP]

meteorologist1:
In the definition for positively invariant, when you say that the orbit through a point in S remains in S for t>0, does that mean that every iterate of that point stays in S for t>0? If so, I don't see why the nonwandering set is necessarily positively invariant. It seems like the orbit could leave S as long as it always comes back arbitrarily close to where it started.

edit: I suppose it's a little late for this now, but I think you can prove the nonwandering set is closed directly from the definition of a nonwandering point by showing that the set contains all its limit points (often a good way to show that a set is closed). That way you don't have to deal with complements or figuring out when a point isn't nonwandering.