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Slow Continued Fractions and Permutative Representations of ${\mathcal{O}}_{N}$

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  03 January 2020

Christopher Linden
Christopher Linden*
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, USA Email: clinden2@illinois.edu
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Abstract

Representations of the Cuntz algebra ${\mathcal{O}}_{N}$ are constructed from interval dynamical systems associated with slow continued fraction algorithms introduced by Giovanni Panti. Their irreducible decomposition formulas are characterized by using the modular group action on real numbers, as a generalization of results by Kawamura, Hayashi, and Lascu. Furthermore, a certain symmetry of such an interval dynamical system is interpreted as a covariant representation of the $C^{\ast }$-dynamical system of the “flip-flop” automorphism of ${\mathcal{O}}_{2}$.

Keywords

Cuntz algebracontinued fractionspermutatative representationprojective linear groupMöbius transformation

MSC classification

Secondary: 11J70: Continued fractions and generalizations 39B12: Iteration theory, iterative and composite equations
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 4 , December 2020 , pp. 787 - 801
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Copyright
© Canadian Mathematical Society 2020

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