Expressing σ as a binary shift map.

Question

Having shown that the only fixed point of $\sigma$ is $x=0$, I've now got the show that the fundamental period-$2$ points of σ are of the form

$x=0 . ababab \ldots$ where $a,b \in \{0,1\}$ and $a\neq b$

I've then got to express these points as fractions in lowest terms using the sum of infinite geometric series.

Would I need to use the definition of a Bernoulli shift map,

$σ(x)=2x \pmod 1$?

Please help!

Answer 1

If $x=0.\overline{ab}$ then consider $11x=100x-x$ (here I'm using binary notation). What can you deduce from this?