Treffer: Rotation encoding and self-similarity phenomenon

The author introduces codings of rotations as a tool to study problems related to the uniform distribution of sequences \((n\alpha)\). Codings of rotations are obtained by coding the orbit on the circle \(\Pi_1\) cut into two intervals \([0,\beta[ \cup [\beta,1[\), of a point \(x\) under the action of an irrational rotation of angle \(\alpha\). A special case is well understood, that is, when the length of one interval equals the angle of rotation. Then, sequences are called Sturmian; among their remarkable properties, their language is completly determined by the continued fraction expansion of the angle of rotation \(\alpha\). In this paper, the author focuses on non degenerated sequences (\(\beta \not\in {\mathbb Z}+\alpha{\mathbb Z}\) and \(\alpha \not \in {\mathbb Q}\)), coding the so-called i.d.o.c exchanges of three intervals. By using an induction process, he proves that such sequences are obtained as a two-letters projection of the iteration of four specific three-letters substitutions. The order of the substitutions defines a two-dimensional continued fraction expansion of the parameters \((\alpha,\beta)\) of the rotation. The author proves that this expansion is ultimately periodic if and only if the parameters belong to a same quadratic field (Lagrange type theorem). In this case, the equilibrium function of the sequence is not bounded, implying that the language of some non-degenerated codings of rotation is irregular. This shows an intrinsic difference of behavior with the degenerated case \(\beta \in {\mathbb Z}+\alpha{\mathbb Z}\).