Thread: 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?

If you let be the binary expansion, then the sequence goes like





...and so forth. Choose , which is aperiodic and nonterminating, so is indeed irrational. Then for all n.

Originally Posted by Tinyboss

If you let be the binary expansion, then the sequence goes like





...and so forth. Choose , which is aperiodic and nonterminating, so is indeed irrational. Then for all n.

well... indeed... thanks

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?