I'm trying to solve an exercise of dynamical systems which says
Given $f:S^1\rightarrow S^1$ with a lifting F of f where $F(x+1)=F(x)+n$ with $n>2$ then f has periodic points of all periods.
I was thinking about using that if it has a point of period 3 then it has periodic points of all periods. I'm facing two problems here, in one hand Sharkovsky theorem as I know it applies for $f:I\rightarrow I$ where $I$ is a closed interval in $R$. I think that should not be a problem since I can think of $S^1$ as $R/Z$ and the theorem applies the same way. The other problem is that I'm not sure how to find the point of period 3.
Can someone give some hints about this?
