“Recursive” Sequence Reaching Every Open Interval

[Don Jon]

Let x denote the position of a particle on the number line. From x, it can move to either the point a-a²+ax or to the point x-ax-a+a² for some fixed 0<a<1. Suppose the particle starts at the origin. Prove that any open interval that is a subset of the interval (a-1,a) contains a point that the particle can reach.

It’s fairly clear the particle can get arbitrarily close to the origin (by moving in one direction continuously and then suddenly swapping). Thus you can get arbitrarily close to a point that is reached by moving in one direction continuously. But I don’t know how to fill in the “holes” in between these points

 

[mfb]

Is this homework?

Does the particle do many consecutive steps, following these rules?

For a=0.5 it looks like you can approach any point in a binary way (reducing the distance by 2 with each additional step). I guess this can be generalized to other values of a.

 

[Don Jon]

Is this homework?

Does the particle do many consecutive steps, following these rules?

For a=0.5 it looks like you can approach any point in a binary way (reducing the distance by 2 with each additional step). I guess this can be generalized to other values of a.

Yes. The particle does do many consecutive steps. Could you be more explicit about the binary approach for a=0.5; I’m not quite understanding it. Thanks!

 

[willem2]

This is an iterated function system, altough a rather boring one in one dimension. Everything under "Properties" in the wikipedia article applies here. If I is the interval of interest (a-1,a) the main thing is to find out what f_1(I) ∪ f_2(I) is. (the union of the ranges of f_1 and f_2).

 

[willem2]

This is an iterated function system, altough a rather boring one in one dimension. Everything under "Properties" in the wikipedia article applies here. If I is the interval of interest (a-1,a) the main thing is to find out what f_1(I) ∪ f_2(I) is. (the union of the ranges of f_1 and f_2).