On Morimoto algorithm in diophantine approximation (Q1192433)

In the years 1926 to 1933 a series of papers by S. Morimoto was devoted to a continued fraction like algorithm which approximates the linear form \(\alpha x+\beta-y\). In the present paper a 2-dimensional map \(T\) is defined which produces the vertices of Morimoto's approximating polygon. The map \(T\) shows striking parallels to continued fractions: Let \(\alpha\) be a quadratic irrational then \(\beta\in\mathbb{Q}(\alpha)\) iff \((\alpha,\beta)\) is an eventually periodic point. Furthermore \(T\) is ergodic and admits an invariant measure. The main ingredients of the proofs are the notion of a reduced algebraic irrational (for the arithmetic part) and the construction of a natural extension (for the ergodic part).