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Math Help - a simple chaos problem

  1. #1
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    a simple chaos problem

    i just forget how it was worked out

    Choose any irrational number x in the interval (0,1). Construct a series {X(i)} as follows: X(0)=x and Xi=(2*X(i-1))mod1 for i=1,2,3... so that the whole series is contained in (0,1). The question is the series thus constructed dense in (0,1)? why?
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  2. #2
    Senior Member Tinyboss's Avatar
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    If you let x_0=.a_1a_2a_3\ldots be the binary expansion, then the sequence goes like

    x_1=.a_2a_3a_4\ldots
    x_2=.a_3a_4a_5\ldots
    x_3=.a_4a_5a_6\ldots

    ...and so forth. Choose x_0=.101001000100001000001..., which is aperiodic and nonterminating, so x is indeed irrational. Then x_n<x_0 for all n.
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  3. #3
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    Quote Originally Posted by Tinyboss View Post
    If you let x_0=.a_1a_2a_3\ldots be the binary expansion, then the sequence goes like

    x_1=.a_2a_3a_4\ldots
    x_2=.a_3a_4a_5\ldots
    x_3=.a_4a_5a_6\ldots

    ...and so forth. Choose x_0=.101001000100001000001..., which is aperiodic and nonterminating, so x is indeed irrational. Then x_n<x_0 for all n.
    well... indeed... thanks
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  4. #4
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    and one can prove that the set of points which have a dense path in (0,1) is dense in (0,1). but can anyone show whether this set has a positive or zero measure?
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