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Statistical properties of actions of periodic orbits

Chaos 10, 195 (2000); doi:10.1063/1.166485

Issue Date: March 2000

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Mitsusada M. Sano
Department of Fundamental Sciences, Faculty of Integrated Human Studies, Kyoto University, Sakyo-ku, Kyoto, 606-8501, Japan
We investigate statistical properties of unstable periodic orbits, especially actions for two simple linear maps (p-adic Baker map and sawtooth map). The action of periodic orbits for both maps is written in terms of symbolic dynamics. As a result, the expression of action for both maps becomes a Hamiltonian of one-dimensional spin systems with the exponential-type pair interaction. Numerical work is done for enumerating periodic orbits. It is shown that after symmetry reduction, the dyadic Baker map is close to generic systems, and the p-adic Baker map and sawtooth map with noninteger K are also close to generic systems. For the dyadic Baker map, the trace of the quantum time-evolution operator is semiclassically evaluated by employing the method of Phys. Rev. E 49, R963 (1994). Finally, using the result of this and with a mathematical tool, it is shown that, indeed, the actions of the periodic orbits for the dyadic Baker map with symmetry reduction obey the uniform distribution modulo 1 asymptotically as the period goes to infinity. ©2000 American Institute of Physics.
History: Received 17 August 1999; accepted 23 November 1999
Permalink: http://link.aip.org/link/?CHAOEH/10/195/1
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KEYWORDS and PACS

Keywords
PACS
  • 05.45.Mt
    Statistical physics, thermodynamics, and nonlinear dynamical systems Nonlinear dynamics and nonlinear dynamical systems Semiclassical chaos ("quantum chaos")
  • 03.65.Sq
    Quantum mechanics, field theories, and special relativity Quantum mechanics Semiclassical theories and applications
  • 05.30.-d
    Statistical physics, thermodynamics, and nonlinear dynamical systems Quantum statistical mechanics
  • 02.60.-x
    Mathematical methods in physics Numerical approximation and analysis
  • 02.70.-c
    Mathematical methods in physics Computational techniques
  • YEAR: 2000

PUBLICATION DATA

ISSN:
1054-1500 (print)   1089-7682 (online)
Publisher:
AIP is a member of CrossRef AIP

REFERENCES (37)
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