## Polygonal invariant curves for a planar piecewise isometry

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- by Peter Ashwin and Arek Goetz PDF
- Trans. Amer. Math. Soc.
**358**(2006), 373-390 Request permission

## 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

- Roy Adler, Bruce Kitchens, and Charles Tresser,
*Dynamics of non-ergodic piecewise affine maps of the torus*, Ergodic Theory Dynam. Systems**21**(2001), no. 4, 959–999. MR**1849597**, DOI 10.1017/S0143385701001468 - 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. - Peter Ashwin and Xin-Chu Fu,
*Tangencies in invariant disc packings for certain planar piecewise isometries are rare*, Dyn. Syst.**16**(2001), no. 4, 333–345. MR**1870524**, DOI 10.1080/14689360110073650 - P. Ashwin and X.-C. Fu,
*On the geometry of orientation-preserving planar piecewise isometries*, J. Nonlinear Sci.**12**(2002), no. 3, 207–240. MR**1905204**, DOI 10.1007/s00332-002-0477-1 - P. Ashwin and A. Goetz,
*Invariant curves and explosions of periodic islands in systems of piecewise rotations*. (In preparation, 2004.) - Michael D. Boshernitzan,
*Rank two interval exchange transformations*, Ergodic Theory Dynam. Systems**8**(1988), no. 3, 379–394. MR**961737**, DOI 10.1017/S0143385700004521 - Jérôme Buzzi,
*Piecewise isometries have zero topological entropy*, Ergodic Theory Dynam. Systems**21**(2001), no. 5, 1371–1377. MR**1855837**, DOI 10.1017/S0143385701001651 - Arek Goetz,
*Perturbations of $8$-attractors and births of satellite systems*, Internat. J. Bifur. Chaos Appl. Sci. Engrg.**8**(1998), no. 10, 1937–1956. MR**1670619**, DOI 10.1142/S0218127498001613 - Arek Goetz,
*Dynamics of a piecewise rotation*, Discrete Contin. Dynam. Systems**4**(1998), no. 4, 593–608. MR**1641165**, DOI 10.3934/dcds.1998.4.593 - Arek Goetz,
*A self-similar example of a piecewise isometric attractor*, Dynamical systems (Luminy-Marseille, 1998) World Sci. Publ., River Edge, NJ, 2000, pp. 248–258. MR**1796163** - Arek Goetz and Guillaume Poggiaspalla,
*Rotations by $\pi /7$*, Nonlinearity**17**(2004), no. 5, 1787–1802. MR**2086151**, DOI 10.1088/0951-7715/17/5/013 - Eugene Gutkin and Nicolai Haydn,
*Topological entropy of generalized polygon exchanges*, Bull. Amer. Math. Soc. (N.S.)**32**(1995), no. 1, 50–56. MR**1273398**, DOI 10.1090/S0273-0979-1995-00555-0 - Eugene Gutkin and Nándor Simányi,
*Dual polygonal billiards and necklace dynamics*, Comm. Math. Phys.**143**(1992), no. 3, 431–449. MR**1145593** - Byungik Kahng,
*Dynamics of symplectic piecewise affine elliptic rotation maps on tori*, Ergodic Theory Dynam. Systems**22**(2002), no. 2, 483–505. MR**1898801**, DOI 10.1017/S0143385702000238 - K. L. Kouptsov, J. H. Lowenstein, and F. Vivaldi,
*Quadratic rational rotations of the torus and dual lattice maps*, Nonlinearity**15**(2002), no. 6, 1795–1842. MR**1938473**, DOI 10.1088/0951-7715/15/6/306 - J. H. Lowenstein, K. L. Kouptsov, and F. Vivaldi,
*Recursive tiling and geometry of piecewise rotations by $\pi /7$*, Nonlinearity**17**(2004), no. 2, 371–395. MR**2039048**, DOI 10.1088/0951-7715/17/2/001 - Anatole Katok and Boris Hasselblatt,
*Introduction to the modern theory of dynamical systems*, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR**1326374**, DOI 10.1017/CBO9780511809187 - 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. - J. H. Lowenstein and F. Vivaldi,
*Embedding dynamics for round-off errors near a periodic orbit*, Chaos**10**(2000), no. 4, 747–755. MR**1802663**, DOI 10.1063/1.1322027 - A. J. Scott, C. A. Holmes, and G. J. Milburn,
*Hamiltonian mappings and circle packing phase spaces*, Phys. D**155**(2001), no. 1-2, 34–50. MR**1837203**, DOI 10.1016/S0167-2789(01)00263-9 - S. Tabachnikov,
*On the dual billiard problem*, Adv. Math.**115**(1995), no. 2, 221–249. MR**1354670**, DOI 10.1006/aima.1995.1055

## 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
- 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. - © Copyright 2005 American Mathematical Society
- 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
- MathSciNet review: 2171238