Invariant subsets of $z \mapsto z^2$

Question

Where can I find an explicit construction of closed invariant subsets of the map $z \mapsto z^2$ on the unit circle? Furstenberg mentions that there are continuum of such disjoint minimal sets but does not provide specific details.

Answer 1

For each $\alpha\in (0,1)\setminus \mathbb Q$ and $t\in \mathbb R$, consider the sequence $(a_n)_{n\ge 0}$ with $a_n=\lfloor \alpha n+t\rfloor - \lfloor \alpha (n-1)+t\rfloor$. This is called the Sturmian sequence with angle $\alpha$, an elements of $\{0,1\}^{\mathbb Z^+}$. Let $S_\alpha$ be the orbit closure under the shift of $(a_n)$. This is a closed invariant set under the shift (by definition). The frequency of 1's in blocks of elements of $S_\alpha$ converge uniformly to $\alpha$. Hence the $S_\alpha$ are disjoint.

Now you can turn these into invariant sets for the squaring map: $Z_\alpha=\{e^{2\pi i t}\colon t\in S_\alpha\}$ is an invariant set for the squaring map. These remain disjoint.

PS: Not sure if minimal was added later, or if I missed it, but the $Z_\alpha$ are minimal also.

Answer 2

Let's consider the semicircle $I_\theta=[\theta,\theta+1/2]$ for each $\theta\in\mathbb{T}$. Then there is a unique minimal ordered subset $K_\theta\subset I_\theta$, in the sense that the restriction of $\tau:x\mapsto 2x$ on $K_\theta$ is order-preserving. So the rotation number $\rho(\theta)=\rho(K_\theta,\tau)$ is well defined.

It is proved by Bullett and Sentenac that $\text{Im}(\rho)=\mathbb{T}$. In particular, for every rotation number $\rho\in\mathbb{T}$, there exists a minimal subset $K_\theta$ with rotation number $\rho$ (so they form a continuum family of minimal subset).

Example 1. For all $\frac{1}{6}<\theta<\frac{1}{3}$, we have $K_\theta=\{\frac{1}{3}, \frac{2}{3}\}\subset I_\theta$ and $\rho(\theta)=\frac{1}{2}$.

Example 2. For all $\frac{9}{62}<\theta<\frac{5}{31}$, we have $K_\theta=\{\frac{5}{31}, \frac{10}{31}, \frac{20}{31}, \frac{9}{31}, \frac{18}{31}\}\subset I_\theta$ and $\rho(\theta)=\frac{2}{5}$.

The unique invariant measure supported on $K_\theta$ is called a Sturmian measure. So I think this construction is related to Professor Anthony Quas's answer.