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A decoding problem in dynamics and in number theory.

Ralph M. Siegel1, Charles Tresser, George Zettler

  • 1Center for Molecular and Behavioral Neuroscience, Rutgers University, Newark, New Jersey 07102I.B.M., Yorktown Heights, New York 10598Columbia University, Department of Mathematics, New York, New York 10027.

Chaos (Woodbury, N.Y.)
|October 1, 1992
PubMed
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This study shows that orbit codes generated by circle homeomorphisms can determine the rotation number. Periodic codes can also be identified, with applications in dynamical systems and number theory.

Area of Science:

  • Dynamical Systems and Chaos Theory
  • Number Theory
  • Physiology

Background:

  • Circle homeomorphisms and their orbit coding have a 200-year history.
  • Kneading theory, a modern approach, is relevant for understanding oscillations in dynamical systems.
  • Previous work focused on specific cases (point/image arcs, rotations), with recent physiological applications.

Purpose of the Study:

  • To analyze orbit coding for general circle homeomorphisms.
  • To establish a connection between orbit codes and the rotation number.
  • To explore the decoding problem in both dynamical and number-theoretic contexts.

Main Methods:

  • Utilizing orbit coding derived from splitting a circle into two semi-open arcs.
  • Analyzing the properties of generated codes, including periodicity.

Related Experiment Videos

  • Establishing an arithmetic equivalence for decoding problems.

    • Main Results:

    • Any generated orbit code determines the rotation number of the homeomorphism (up to orientation), except in trivial cases.
    • Periodic codes can be identified as to whether they can be generated by a homeomorphism.
    • A direct equivalence is shown between the dynamical decoding problem and an arithmetic problem involving modular arithmetic.

    Conclusions:

    • Orbit coding provides a powerful tool for understanding circle homeomorphisms.
    • The rotation number can be deduced from orbit codes, offering insights into dynamical behavior.
    • The established equivalence simplifies decoding problems in both mathematics and its applications.