Devaney chaos

A map on a metric space is chaotic in Devaney's sense when it is topologically transitive, its periodic points are dense, and it has sensitive dependence on initial conditions.

A map $F:X\to X$ is topologically transitive when, for every pair of nonempty open sets $U,V$, some iterate satisfies $F^n(U)\cap V\ne \varnothing$.

Periodic points are dense when every nonempty open set contains a point fixed by some positive iterate.

There is a constant $\delta >0$ such that every neighbourhood of every point contains a second point whose orbit eventually separates from the first by more than $\delta$.

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