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Geometry and Dynamics of Numbers Under Finite Resolution

  • Jacky Cresson
  • Jean-Nicolas Dénarié
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 550)

Abstract

We define a special set, called resolution space, which corresponds to real numbers obtained via the following resolution rule : every number greater than a given integer a is identified with ά. This space possesses a natural scaling structure and dynamics. We introduce several notions as locking and transient resonance zones, as well as unstable irrationals numbers. This space is the natural object coming in the 1/f frequency noise problem. Special numbers as Markoff’s irrationals are proved to play a specific role. This first criterion must be understood as a finite resolution in space for physical systems.

We then introduce an additional resolution criterion which allows only a finite construction of the previous space. Anatural notion of fuzzy zone is defined. This second criterion is interpreted as a finite time experiment in physics.

Keywords

Continue Fraction Rotation Number Modular Group Irrational Number Diophantine Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Jacky Cresson
    • 1
  • Jean-Nicolas Dénarié
    • 1
  1. 1.Equipe de Mathématiques de Besançon, CNRS-UMRUniversité de Franche ComtéFrance

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