Thread: a simple chaos problem

1. 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?

2. 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 for all n.

3. Originally Posted by Tinyboss
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 for all n.
well... indeed... thanks

4. 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?