Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Polygonal invariant curves for a planar piecewise isometry


Authors: Peter Ashwin and Arek Goetz
Journal: Trans. Amer. Math. Soc. 358 (2006), 373-390
MSC (2000): Primary 37B10, 37E15; Secondary 11R11, 20C20, 68W30
DOI: https://doi.org/10.1090/S0002-9947-05-03670-6
Published electronically: March 31, 2005
MathSciNet review: 2171238
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate a remarkable new planar piecewise isometry whose generating map is a permutation of four cones. For this system we prove the coexistence of an infinite number of periodic components and an uncountable number of transitive components. The union of all periodic components is an invariant pentagon with unequal sides. Transitive components are invariant curves on which the dynamics are conjugate to a transitive interval exchange. The restriction of the map to the invariant pentagonal region is the first known piecewise isometric system for which there exist an infinite number of periodic components but the only aperiodic points are on the boundary of the region. The proofs are based on exact calculations in a rational cyclotomic field. We use the system to shed some light on a conjecture that PWIs can possess transitive invariant curves that are not smooth.


References [Enhancements On Off] (What's this?)

  • 1. R.L. Adler, B. Kitchens and C. Tresser,
    Dynamics of non-ergodic piecewise affine maps of the torus.
    Ergod. Th. & Dynam. Sys. 21 (2001) 959-999. MR 1849597 (2002f:37075)
  • 2. P. Ashwin,
    Non-smooth invariant circles in digital overflow oscillations.
    Proceedings of the 4th Int. Workshop on Nonlinear Dynamics of Electronic Systems, Sevilla (1996) 417-422.
  • 3. P. Ashwin and X.-C. Fu,
    Tangencies in invariant circle packings for certain planar piecewise isometries are rare.
    Dynamical Systems 16, 4 (2001) 333-345. MR 1870524 (2002k:37071)
  • 4. P. Ashwin and X.-C. Fu,
    On the geometry of orientation preserving planar piecewise isometries.
    J. Nonlinear Sci. 12 (2002) 207-240. MR 1905204 (2003e:37053)
  • 5. P. Ashwin and A. Goetz,
    Invariant curves and explosions of periodic islands in systems of piecewise rotations.
    (In preparation, 2004.)
  • 6. Michael Boshernitzan,
    Rank two interval exchange transformations.
    Ergodic Theory and Dynamical Systems 8 (1988) 379-394. MR 0961737 (90c:28024)
  • 7. J. Buzzi,
    Piecewise isometries have zero topological entropy.
    Ergod. Th. Dyn. Sys. 21 (2001) 1371-1377.MR 1855837 (2002f:37029)
  • 8. A. Goetz,
    Perturbation of $8$-attractors and births of satellite systems.
    Intl. J. Bifn. Chaos 8 (1998) 1937-1956.MR 1670619 (2000b:37038)
  • 9. A. Goetz,
    Dynamics of a piecewise rotation.
    Continuous and Discrete Dynamical Systems 4 (1998) 593-608.MR 1641165 (2000f:37009)
  • 10. A. Goetz,
    A self-similar example of a piecewise isometric attractor.
    Dynamical systems (Luminy-Marseille, 1998), 248-258, World Sci. Publishing, River Edge, NJ (2000). MR 1796163 (2001k:37060)
  • 11. A. Goetz and Gauillaume Poggiaspalla,
    Rotations by $\pi/7$.
    Nonlinearity 17(5) (2004) 1787-1802. MR 2086151
  • 12. E. Gutkin and H. Haydn,
    Topological entropy of generalized polygon exchanges.
    Bull. Amer. Math. Soc. 32 (1995) 50-57. MR 1273398 (95c:58118)
  • 13. E. Gutkin and N. Simányi, Dual Billiards and necklace dynamics.
    Communications in Mathematical Physics 143 (1992) 431-449.MR 1145593 (92k:58139)
  • 14. B. Kahng,
    Dynamics of symplectic piecewise affine elliptic rotation maps on tori.
    Ergod. Th. & Dynam. Sys. 2 (2002) 483-505.MR 1898801 (2003d:37078)
  • 15. K.L. Kouptsov and J. H. Lowenstein and F. Vivaldi,
    Quadratic rational rotations of the torus and dual lattice maps.
    Nonlinearity 15 (2002) 1795-1842.MR 1938473
  • 16. Kapustov and Lowenstein and Vivaldi,
    Recursive tiling and geometry of piecewise rotations by $\pi/7$.
    Nonlinearity 17 (2004) 371-395.MR 2039048
  • 17. A. Katok and B. Hasselblat,
    Introduction to Modern Theory of Dynamical Systems,
    Cambridge University Press, Cambridge 1995.MR 1326374 (96c:58055)
  • 18. L. Kocarev, C.W. Wu and L.O. Chua,
    Complex behaviour in Digital filters with overflow nonlinearity: analytical results.
    IEEE Trans CAS-II 43 (1996) 234-246.
  • 19. J. H. Lowenstein and F. Vivaldi,
    Embedding dynamics for round-off errors near a periodic orbit.
    Chaos 10 (2000) 747-755. MR 1802663 (2001j:37074)
  • 20. A.J. Scott, C.A. Holmes and G.J. Milburn,
    Hamiltonian mappings and circle packing phase spaces.
    Physica D 155 (2001) 34-50.MR 1837203 (2002e:37082)
  • 21. S. Tabachnikov,
    On the dual billiard problem.
    Adv. Math. 115 (1995) 221-249.MR 1354670 (96g:58154)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37B10, 37E15, 11R11, 20C20, 68W30

Retrieve articles in all journals with MSC (2000): 37B10, 37E15, 11R11, 20C20, 68W30


Additional Information

Peter Ashwin
Affiliation: Department of Mathematical Sciences, University of Exeter, Exeter EX4 4QE, United Kingdom

Arek Goetz
Affiliation: Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, California 94132

DOI: https://doi.org/10.1090/S0002-9947-05-03670-6
Keywords: Piecewise isometry, interval exchange transformation, cells
Received by editor(s): November 22, 2002
Received by editor(s) in revised form: March 22, 2004
Published electronically: March 31, 2005
Additional Notes: The work on this article commenced during the second author’s visit to Exeter sponsored by the LMS. The second author was partially supported by NSF research grant DMS 0103882, and the San Francisco State Presidential Research Leave. We thank Michael Boshernitzan for interesting conversations and helpful suggestions. Symbolic computations were aided by Mathematica routines, some of which were developed in connection with a project by Goetz and Poggiaspalla.
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society