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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 Bulletta 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.

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.

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 Bulletta 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: 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.

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 Bulletta 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.

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.

Then it is proved 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).

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.

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 Bulletta 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: 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.