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

Abstract

We study two-colour rotations of the unit circle that take to the point if and to if . The rotations depend on discrete parameters and a continuous parameter and we choose to be the golden ratio . We shall show that the have an invariance property: the induced maps or first-return maps for are again two-colour rotations with renormalized parameters , . Moreover, we find conditions under which the induced maps have the form , that is, the are isomorphic to their induced maps and thus have another property, namely, that of self-similarity. We describe the structure of the attractor of a rotation and prove that the restriction of a rotation to its attractor is isomorphic to a certain family of integral isomorphisms obtained by lifting the simple rotation of the circle . A corollary is the uniform distribution of the -orbits on the attractor . We find a connection between the measure of the attractor and the frequency distribution function of points in -orbits over closed intervals . Explicit formulae for the frequency are obtained in certain cases.