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Prog. Theor. Phys. Vol. 107 No. 6 (2002) pp. 1117-1145

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Dynamical Ordering of Non-Birkhoff Orbits and Topological Entropy in the Standard Mapping

Yoshihiro Yamaguchi1 and Kiyotaka Tanikawa2

1Teikyo Heisei University, Ichihara 290-0193, Japan
2National Astronomical Observatory, Mitaka 181-8588, Japan

(Received February 4, 2002)

Abstract:

The standard mapping is an analytical, reversible monotone twist mapping. The appearance ordering (i.e. the so-called dynamical ordering), of symmetric non-Birkhoff periodic orbits (SNBO) in the standard mapping is derived. Essential use is made of the reversibility. After the establishment of various properties of the symmetry axes under the mapping, two theorems connecting the dynamical ordering are proved. Then, the braids for SNBOs are constructed with the aid of techniques developed in braid group theory. A lower bound of the topological entropy of a system possessing an SNBO is obtained using the eigenvalue of the reduced Burau matrix representation of the braid constructed from the SNBO. The behavior of the topological entropy in the integrable limit is discussed.


URL : http://ptp.ipap.jp/link?PTP/107/1117/
DOI : 10.1143/PTP.107.1117

[ Full Text PDF : FREE ACCESS (1353K) ] Citation:

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Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 108 No. 2 (2002) pp. 231-240 :
    Non-Birkhoff Periodic Orbits in a Circle Map
    Yoshihiro Yamaguchi and Kiyotaka Tanikawa
  2. Progress of Theoretical Physics Vol. 109 No. 2 (2003) pp. 187-202 :
    Non-Birkhoff Orbits with 2n Turning Points in the Standard Map
    Kiyotaka Tanikawa and Yoshihiro Yamaguchi
  3. Progress of Theoretical Physics Vol. 111 No. 5 (2004) pp. 689-705 :
    Oscillatory Orbits in the Standard Mapping
    Yoshihiro Yamaguchi and Kiyotaka Tanikawa
  4. Progress of Theoretical Physics Vol. 113 No. 2 (2005) pp. 261-281 :
    Generalized Horseshoes in the Standard Mapping
    Kiyotoka Tanikawa and Yoshihiro Yamaguchi
  5. Progress of Theoretical Physics Vol. 113 No. 5 (2005) pp. 935-951 :
    Dynamical Ordering of Symmetric Periodic Orbits for the Area Preserving Hénon Map
    Yoshihiro Yamaguchi and Kiyotaka Tanikawa
  6. Progress of Theoretical Physics Vol. 114 No. 4 (2005) pp. 763-791 :
    Generalized Dynamical Ordering and Topological Entropy in the Hénon Map
    Yoshihiro Yamaguchi and Kiyotaka Tanikawa
  7. Progress of Theoretical Physics Vol. 117 No. 4 (2007) pp. 601-632 :
    Non-Birkhoff Periodic Orbits of Farey Type and Dynamical Ordering in the Standard Mapping
    Yoshihiro Yamaguchi and Kiyotaka Tanikawa
  8. Progress of Theoretical Physics Vol. 118 No. 4 (2007) pp. 675-699 :
    Non-Birkhoff Periodic Orbits of Farey Type and Dynamical Ordering in the Standard Mapping. II
    Yoshihiro Yamaguchi and Kiyotaka Tanikawa
  9. Progress of Theoretical Physics Vol. 121 No. 4 (2009) pp. 657-669 :
    Topological Entropy in a Parameter Range of the Standard Map
    Yoshihiro Yamaguchi and Kiyotaka Tanikawa

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