One-dimensional Fibonacci tilings and induced two-colour rotations of the circle

We study two-colour rotations $S_\varepsilon(a,b)$ of the unit circle that take $x\in[0,1)$ to the point $\langle x+a\tau\rangle$ if $x\in[0,\varepsilon)$ and to $\langle x+b\tau\rangle$ if $x\in[\varepsilon,1)$. The rotations $S_\varepsilon(a,b)$ depend on discrete parameters $a,b\in\mathbb Z$ and a continuous parameter $\varepsilon\in[0,1)$ and we choose $\tau$ to be the golden ratio $\frac{1+\sqrt5}2$. We shall show that the $S_\varepsilon(a,b)$ have an invariance property: the induced maps or first-return maps for $S_\varepsilon(a,b)$ are again two-colour rotations $S_{\varepsilon'}(a',b')$ with renormalized parameters $\varepsilon'\in[0,1)$, $a',b'\in\mathbb Z$. Moreover, we find conditions under which the induced maps $S_{\varepsilon'}(a',b')$ have the form $S_{\varepsilon'}(a,b)$, that is, the $S_\varepsilon(a,b)$ are isomorphic to their induced maps and thus have another property, namely, that of self-similarity. We describe the structure of the attractor $\operatorname{Att}(S_\varepsilon(a,b))$ of a rotation $S_\varepsilon(a,b)$ and prove that the restriction of a rotation to its attractor is isomorphic to a certain family of integral isomorphisms $T_\varepsilon$ obtained by lifting the simple rotation of the circle $S(x)=\langle x+\tau\rangle$. A corollary is the uniform distribution of the $S_\varepsilon(a,b)$-orbits on the attractor $\operatorname{Att}(S_\varepsilon(a,b))$. We find a connection between the measure of the attractor $\operatorname{Att}(S_\varepsilon(a,b))$ and the frequency distribution function $\nu_\varepsilon(\theta_1,\theta_2)$ of points in $S_\varepsilon(a,b)$-orbits over closed intervals $[\theta_1,\theta_2]\subset[0,1)$. Explicit formulae for the frequency $\nu_\varepsilon(\theta_1,\theta_2)$ are obtained in certain cases.