# A problem in trying to show that a system is ergodic

If $$f:[0,1]\to [0,1]$$ is given by

$$f(x)= \begin{cases} 2x & \mbox{ if } x\in [0,1/3)\\ & \\ 2x-\frac{2}{3} & \mbox{ if } x\in [1/3,1/2)\\ 2x-\frac{1}{3} & \mbox{ if } x\in [1/2,2/3)\\ & \\ 2x-1 & \mbox{ if } x\in [2/3,1] \end{cases}$$ show that $$f$$ is Ergodic with respect of Lebesgue measure.

Idea: The technique is to show that given an invariant set of positive measure, this set will have full measure.

A very common process is to partition the domain, choose a point of density in the invariant set, use the the Lebesgue Density Theorem and the bounded distortion property

\begin{align*} \dfrac{m(f^k(E_1))}{m(f^k(E_2))}=\dfrac{m(E_1)}{m(E_2)} \end{align*}.

below is the iteration graph $$f^5 (x)$$. We see that the image size of each subinterval is $$2/3$$. The main problem that for some interval $$E_1$$, $$f^k(E_1)$$ is not $$(0,1)$$ integer "At first". I think that when we iterate $$n$$ times for very large $$n$$, we will have a graph virtually without breaks, with the linear arms. This would justify that the measure of each invariant set of positive measure is $$1$$. But it is only my intuition, I could not justify formally. Can anyone give a tip?

• This sounds rather like a homework question... – Anthony Quas Mar 30 at 3:15