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Math Help - Chaos Theory: period points

  1. #1
    Junior Member
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    Chaos Theory: period points

    Hi, I have this question

    Let f: R \rightarrow R (R is the set of real numbers) and let f^2 = f(f(x)) intersect the graph of the identity mapping id in just one point (call it p). How many points of period-1 does f have? In how many points does the graph of f intersect that of id?

    I know that the point p is a fixed point of f^2 and I know that if f has a fixed point then it is also a fixed point of f^2, so f can have at most one fixed point. But I just can't seem to get my head around this question.

    Please help
    Katy
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  2. #2
    MHF Contributor

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    Yes, if p is a point of period 1 for f, then it is a fixed point for f and so for f^2. Thus, if f^2 has only one fixed point, then so does f. The only thing you seem to have left undone is the number of points in which the graph of f intersects the graph of the identity function. But if p lies on both y= f(x) and y= x, then y= f(p)= p and p is a fixed point. Since f has only one fixed point, in how many points can y= f(x) intersect y= x?
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  3. #3
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    So it intersects only once? Is that right?

    So is there no way that there can be no fixed points for f and a fixed point for f^2?
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