# random experiment with two different functions on unit interval

Let $$X=[0,1]$$, and functions $$f(x)=x$$, $$g(x)=2x$$ mod $$1$$, and the probability of chosing $$f,g$$: $$\mu(f)=\mu(g)=\frac{1}{2}$$. Now if $$x$$ is the starting point, then what will be a general expression of state after $$n$$ iteration?

One of my friends says it will be $$X_n=(\eta_1+\eta_2+\dots+\eta_n)x$$ mod $$1$$, where $$\eta_i$$ are i.i.d taking values $$0,1$$but I don't understand his explanations.

This is some random experiment I understand but how the process will go on, I can not visualize. Each step of the random experiment one of the above function will play their role and with probability fifty-fifty, right? So If I consider a diagram according to the experiment or iteration, I will get some curve also? Thanks for describing me mathematically.

## 1 Answer

Your friend is wrong, but has the right idea. Let $$\eta_j = 1$$ if $$g$$ is picked at the $$j$$th iteration, and $$0$$ otherwise. (I.e. $$\eta_j$$ is the indicator variable.) Then

$$X_j = 2^{\eta_j} X_{j-1} \pmod 1$$

and so

$$X_n = 2^{\eta_n} 2^{\eta_{n-1}} \cdots 2^{\eta_2} 2^{\eta_1} X_0 = 2^{(\eta_n + \cdots + \eta_2 + \eta_1)} X_0 \pmod 1$$

As for visualization... If $$f$$ is picked then $$X$$ didn't change, but if $$g$$ is picked then $$X$$ doubles, and then "wraps around" $$\pmod 1$$ if the doubled value $$> 1$$. If you plot successive $$X_n$$ there are some flat stretches ($$f$$ being chosen) then some jumps which may look hard to comprehend.

For some starting values the series is easily described. E.g. starting at $$X_0=1/2$$ or indeed any $$2^{-k}$$ you keep doubling (whenever $$g$$ is chosen) until getting stuck at $$1=0$$. Alternatively starting at $$X_0 = 1/3$$ you just toggle between $$1/3$$ and $$2/3$$ (whenever $$g$$ is chosen). Somewhat similar things happen for a rational $$X_0$$ whose denominator is not a power of $$2$$ (and in particular the series eventually repeats). But for an irrational $$X_0$$ the series will go all over the place and will never repeat, and (I am not an expert on this last point) I believe in some technical sense may become "uniform" over the $$(0,1)$$ interval...?

• Thanksssssssssssssssssss very muchhhhhhhhhhhhhh I understood. – Marso Mar 28 at 8:38