Doubling map

The doubling map on the unit circle isIt is chaotic in Devaney's sense.

The doubling map preserves normalized Lebesgue measure on the unit circle. Each point has exactly two preimages, $x/2$ and $(x+1)/2$ modulo one, and each inverse branch scales lengths by $1/2$.

If $x=0.b_1{b}_2{b}_3\dots$ in base two, thenBinary cylinders make transitivity and density of periodic points transparent: concatenate prescribed finite blocks for transitivity and repeat a finite block for a periodic point.

The fixed points of $F^n$ areThus $F^n$ has $2^n-1$ fixed points.

For $n=2^k$, every proper period dividing $n$ divides $n/2$. Hence the number of points of exact period $2^k$ is