As mentioned in the headline I am considering the map $f(x_n)=x_{{n+1} [\mod 4]}$. It looks a little bit similar to the dyadic (bernoulli) map but instead of $\mod 1$ we have $\mod 4$ which means that in the binary representation after each step we move the coma $4$ steps to the right (am I correct?)
I also have $4$ points $x_0<x_1<x_2<x_3$. These are the members of a $4$-cycle with $f(x_n)$. Now I want to prove the existence of a fixed point, a 2 cycle and 3 cycle of $f(x_n)$. I do not see why it has to follow from the fact that there exists a 4 cycle that there are also other cycles and a fixed point. It might be possible to prove it using the intermediate value theorem. Do yo have any idea?