Thread: Mixing

Consider H to be a interval and f: H->H be a cts fxn.

For nontrivial closed intervals, suppose that U and V are contained in H, we can find m such that V contained in f^{m}(U).


Show that the periodic pts are dense in H.

I need to show that every point of H is the limit of some
sequence of points in H. Not sure where to proceed.

Show that f is topologically transitive.

f:H->H is topologically transitive if for all open sets U,V (non empty) there exists x in U and n such that f^{n}(x) in V. ie f^{n}(U) intersection V nonempty. Doesn't that just follow from assumption?

Suppose p is a repelling fixed point of a map f . Then there is some nbhd Nϵ (p) so that for any x not equal to p and x ∈ Nϵ (p), there is some n* so that that for all n ≥ n*, f^{n}(x) not in Nϵ(p).
Let d = ϵ. If k = n* we have

|f^{k}(x) -f^{k}(p)| = |f^{k}(x) -(p)| = >= ϵ = d

therefore has sensitive dependence.