Convergence in Möbius Number Systems
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Alexandr Kazda
Abstract
The Möbius number systems use sequences of Möbius transformations to represent the extended real line or, equivalently, the unit complex circle. An infinite sequence of Möbius transformations represents a point x on the circle if and only if the transformations, in the limit, take the uniform measure on the circle to the Dirac measure centered at the point x. We present new characterizations of this convergence.
Moreover, we show how to improve a known result that guarantees the existence of Möbius number systems for some Möbius iterative systems.
As Möbius number systems use subshifts instead of the whole symbolic space, we can ask what is the language complexity of these subshifts. We offer (under some assumptions) a sufficient and necessary condition for a number system to be sofic.
© de Gruyter 2009
Articles in the same Issue
- Preface
- Symmetric CNS Trinomials
- On the β-Expansion of an Algebraic Number in an Algebraic Base β
- A Planar Integral Self-Affine Tile with Cantor Set Intersections with Its Neighbors
- Beta-Expansions with Negative Bases
- Convergence in Möbius Number Systems
- Factor Complexity of Infinite Words Associated with Non-Simple Parry Numbers
- On the Boundary of the Set of the Closure of Contractive Polynomials
- Christoffel Words and Markoff Triples
- Characterizations of Words with Many Periods
Articles in the same Issue
- Preface
- Symmetric CNS Trinomials
- On the β-Expansion of an Algebraic Number in an Algebraic Base β
- A Planar Integral Self-Affine Tile with Cantor Set Intersections with Its Neighbors
- Beta-Expansions with Negative Bases
- Convergence in Möbius Number Systems
- Factor Complexity of Infinite Words Associated with Non-Simple Parry Numbers
- On the Boundary of the Set of the Closure of Contractive Polynomials
- Christoffel Words and Markoff Triples
- Characterizations of Words with Many Periods
