The negative slope algorithm and the dimension group of free rank 3
Koshiro Ishimura
TL;DR
This paper introduces a method to construct the dimension group of free rank 3 using the negative slope algorithm, extending previous work on rank 2 groups and 2-dimensional continued fractions.
Contribution
The paper develops a new approach to build the dimension group of free rank 3 based on the negative slope algorithm, generalizing Effros and Shen's rank 2 construction.
Findings
Constructed the dimension group of free rank 3
Extended the negative slope algorithm to higher rank
Linked 3-interval exchange transformations with dimension groups
Abstract
E. G. Effros and C-L. Shen constructed the dimension group of free rank 2 from the simple continued fraction algorithm. The notion of negative slope algorithm was introduced by S. Ferenczi, C. Holton, and L. Zamboni in their study of 3-interval exchange transformations. The negative slope algorithm is the 2-dimensional continued fraction algorithm. Then the author succeed to construct the dimensional group of free rank 3 by the similar method which E. G. Effros and C-L. Shen used.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Algorithms and Data Compression · Fuzzy Logic and Control Systems