Singularization of multidim. cont. fraction algorithms

The method of singularization has been introduced by C. Kraaikamp 1991 to improve some diophantine approximation properties of the regular one-dimensional continued fraction algorithm. A large family of semi- regular continued fraction algorithms, called S-expansions, can be related to each other via singularization and corresponding methods, such as insertion. Apart from the classification, the method allows to transfer certain diophantine and metrical (ergodicity, existence and explicit form of an invariant measure) properties from one algorithm to other algorithms of the same class. In Schratzberger 2004 and Schratzberger 2006d, for the first time, this method has been generalized into higher dimensions. In Schratzberger 2006b, 2006c, we were able to show that the two-dimensional Jacobi-Perron Algorithm, the Podsypanin Algorithm and the Brun Algorithm belong to the same class of S-expansions, i.e. can be transformed into each other by conversion processes based on the methods of singularization and insertion. We aim to investigate 1. The transferability of diophantine properties, such as the exponent of convergence, from one algorithm to other algorithms of the same class of (two-dimensional) S-expansions. 2. The transferability of metrical properties, in particular ergodicity, existence and explicit form of an invariant measure, from one algorithm to other algorithms of the same class of (two-dimensional) S-expansions. Due to the increased complexity of two-dimensional conversion processes, the transfer of these properties is not as obvious as in the one-dimensional case. Finally, we are interested in further generalizations of the method, both to other two-dimensional algorithms (e.g. Selmer Algorithm) and to dimensions higher than two.

One of the major goals of the project "Singularization of Multidimensional Continued Fraction Algorithms" was the extension of the method of singularization to a larger class of two-dimensional continued fraction algorithms. In particular, we were looking for a transformation from the Brun Algorithm into the Selmer Algorithm. Albeit enormous technical difficulties, with extensive usage of mathematical software packages such as Mathematica, we were finally successful in finding a constructive method to convert the one algorithm into the other a.e. Among others, the method includes over forty different matrix identities, and a detailed description of their exact application. As a result of a previous FWF project (P16964), a similar method had been established for the Podsypanin Algorithm and the Jacobi-Perron Algorithm. However, the final proof that this conversion process could successfully be applied to almost every initial x was still incomplete. We were able to finish this work. In particular, the paper implicitly includes a description of the probability structure of the Podsypanin Algorithm as a random system with complete connections. This structure is gradually transferred to the Jacobi-Perron Algorithm, to show convergence of the method a.e. That way now, the class of two dimensional S-expansions includes the Jacobi-Perron Algorithm, the Podsypanin Algorithm, the Brun Algorithm and the Selmer Algorithm. We further spent some time with the question of (fast) convergence of Lebesgue measure to the invariant Gauss measure, in case of the Multiplicative Brun Algorithm. In particular, we have been trying to transfer the method of Paul Lévy, thus clarifying the role of the dual algorithm in the method. However, due to technical difficulties, the problem has not been entirely solved yet. Together with Prof. Arnaldo Nogueira from the Institut de Mathématiques de Luminy, we have jointly been working on the question of recurrence, and hence weak convergence of the four-dimensional Poincaré Algorithm. The problem should in principle be solved, and be ready for publication soon. In the final period of the project, we have been investigating the effect of singularization processes on the quality of Diophantine approximations. In principle, the presumption that algorithms of the same class of two-dimensional S-expansions have the same approximation exponent seems to hold. However, the exact proof requires some future work.