On the singularization of the two-dimensional Jacobi-Perron algorithm. (English) Zbl 1220.11094

Schratzberger, B.

Exp. Math. 16, No. 4, 441-454 (2007).

Summary: We present a constructive method to convert the (two-dimensional) Jacobi-Perron evolution of \((x_1,x_2) \in [0,1]^2\) into the corresponding evolution of Podsypanin, and vice versa. A similar approach allows us to extend the result to Brun’s algorithm. The method, based on the techniques of singularization and insertion, is built up in steps. From experiments, we assume that each step terminates after finitely many states almost everywhere.

MSC:

11K55	Metric theory of other algorithms and expansions; measure and Hausdorff dimension
11K50	Metric theory of continued fractions

Keywords:

Jacobi-Perron algorithm; Podsypanin algorithm; singularization; \(S\)-expansions