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Math Help - Mixing

  1. #1
    Junior Member
    Joined
    Mar 2009
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    41

    Question 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?
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  2. #2
    Junior Member
    Joined
    Mar 2009
    Posts
    41

    Question

    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.
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