Boundaries of rotation sets for homeomorphisms of the 𝑛-torus

Swanson, Richard; Walker, Russell

doi:10.1090/S0002-9939-96-03426-0

Boundaries of rotation sets for homeomorphisms of the $n$-torus
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by Richard Swanson and Russell Walker

Proc. Amer. Math. Soc. 124 (1996), 3247-3255

DOI: https://doi.org/10.1090/S0002-9939-96-03426-0

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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

Marcy Barge and Richard Swanson, Rotation shadowing properties of circle and annulus maps, Ergodic Theory Dynam. Systems 8 (1988), no. 4, 509–521. MR 980794, DOI 10.1017/S0143385700004661
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
John Franks, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc. 311 (1989), no. 1, 107–115. MR 958891, DOI 10.1090/S0002-9947-1989-0958891-1
John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
Michael Handel, The rotation set of a homeomorphism of the annulus is closed, Comm. Math. Phys. 127 (1990), no. 2, 339–349. MR 1037109, DOI 10.1007/BF02096762
Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8, Springer-Verlag, Berlin, 1987. Translated from the Portuguese by Silvio Levy. MR 889254, DOI 10.1007/978-3-642-70335-5
MichałMisiurewicz and Krystyna Ziemian, Rotation sets for maps of tori, J. London Math. Soc. (2) 40 (1989), no. 3, 490–506. MR 1053617, DOI 10.1112/jlms/s2-40.3.490
S. Newhouse, J. Palis, and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 57 (1983), 5–71. MR 699057, DOI 10.1007/BF02698773
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.