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)


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.


Continue Fraction Rotation Number Modular Group Irrational Number Diophantine Approximation 
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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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