Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Boundaries of rotation sets
for homeomorphisms of the $n$-torus


Authors: Richard Swanson and Russell Walker
Journal: Proc. Amer. Math. Soc. 124 (1996), 3247-3255
MSC (1991): Primary 58J22
DOI: https://doi.org/10.1090/S0002-9939-96-03426-0
MathSciNet review: 1328381
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We construct a $C^\omega $ diffeomorphism of the 3-torus whose rotation set is not closed. We prove that the rotation set of a homeomorphism of the $n$-torus contains the extreme points of its closed convex hull. Finally, we show that each pseudo-rotation set is closed for torus homeomorphisms.


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

  • [B-S] M. Barge and R. Swanson, Rotation shadowing properties of circle and annulus maps, Ergod. Thy. & Dynam. Sys. 8 (1988), 509--521. MR 90f:58107
  • [D-S] N. Dunford and N. Schwartz, Linear Operators, Part I, Pure and Applied Math. Vol. VII, John Wiley and Sons, New York, 1957. MR 22:8302
  • [F] J. Franks, Rotation vectors for torus homeomorphisms, Trans. Am. Math. Soc. vol. 311, No. 1, (1989), 107--115. MR 89k:58239
  • [G-H] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, Applied Mathematical Sciences 42, New York, 1983. MR 85f:58002
  • [H] M. Handel, The rotation set of a homeomorphism of the annulus is closed, Commun. Math. and Phys. 127 (1990), 339--349. MR 91a:58102
  • [M] R. Ma$\tilde n$é, Ergodic Theory and Differentiable Dynamics Springer-Verlag, New York, 1987. MR 88c:58040
  • [M-Z] M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori, J. London. Math. Soc. (2), vol. 40 (1989), 490--506. MR 91f:58052
  • [N-P-T] S. Newhouse, J. Palis, and F. Takens, Bifurcations and stability of families of diffeomorphisms, Publ. Math. IHES, vol. 57 (1983), 5--71. MR 84g:58080
  • [P] H. Poincarè, ``Étude particulière du tore'', Chap. XV in Sur les courbes définies par les equations différentielles. III, J. Math. Pures Appl. (4) 1 (1885), 220-244; reprinted in Oeuvres de Henri Poincaré. Tome I, Gauthier-Villars, Paris, 1928, pp. 137-158.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 58J22

Retrieve articles in all journals with MSC (1991): 58J22


Additional Information

Richard Swanson
Affiliation: Department of Mathematical Sciences, Montana State University, Bozeman, Montana 59717-0240
Email: dswanson@math.montana.edu

Russell Walker
Affiliation: Department of Mathematical Sciences, Montana State University, Bozeman, Montana 59717-0240
Email: walker@math.montana.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03426-0
Keywords: Rotation vectors on tori
Received by editor(s): April 4, 1995
Additional Notes: Research supported in part by NSF-OSR grant #9350546
Communicated by: Linda Keen
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society