An annulus diffeomorphism with non-Denjoy minimal sets

Turpin, Mark

doi:10.1090/S0002-9939-98-04364-0

An annulus diffeomorphism with non-Denjoy minimal sets
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Proc. Amer. Math. Soc. 126 (1998), 1851-1856 Request permission

Abstract:

We construct an annulus diffeomorphism with the property that a countably dense set of irrational rotation numbers are represented only by pseudocircles on which the diffeomorphism acts minimally but is not semi-conjugate to rigid rotation on the circle. This answers a question of Boyland about whether such behavior is possible only at the maximum or minimum of the rotation set.

References

Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Philip Boyland, The rotation set as a dynamical invariant, Twist mappings and their applications, IMA Vol. Math. Appl., vol. 44, Springer, New York, 1992, pp. 73–86. MR 1219350, DOI 10.1007/978-1-4613-9257-6_{4}
Robert L. Devaney, An introduction to chaotic dynamical systems, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986. MR 811850
Lawrence Fearnley, The pseudo-circle is unique, Trans. Amer. Math. Soc. 149 (1970), 45–64. MR 261559, DOI 10.1090/S0002-9947-1970-0261559-6
Michael Handel, A pathological area preserving $C^{\infty }$ diffeomorphism of the plane, Proc. Amer. Math. Soc. 86 (1982), no. 1, 163–168. MR 663889, DOI 10.1090/S0002-9939-1982-0663889-6
Jürgen K. Moser, Lectures on Hamiltonian systems, Mem. Amer. Math. Soc. 81 (1968), 60. MR 0230498
M. Turpin, The restricted rotation number of an example of Handel, preprint.