A pseudorotation with quadratic irrational rotation number

Turpin, Mark

doi:10.1090/S0002-9939-2011-10886-4

A pseudorotation with quadratic irrational rotation number
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Proc. Amer. Math. Soc. 140 (2012), 227-233 Request permission

Abstract:

We show, by explicit construction, that for any quadratic irrational number $\alpha$, there exists a pseudorotation on an indecomposable cofrontier $\Lambda$ with $\alpha$ as its rotation number. Our construction builds on a family of examples of Brechner, Guay, and Mayer. They observe that in the pseudorotations they construct, irrational numbers of constant type are not realized as rotation numbers. Circle rotations can realize any rotation number, but this is to our knowledge the first example of a pseudorotation with a quadratic irrational rotation number, and hence an irrational of constant type.

References

B. L. Brechner, M. D. Guay, and J. C. Mayer, Rotational dynamics on cofrontiers, Continuum theory and dynamical systems (Arcata, CA, 1989) Contemp. Math., vol. 117, Amer. Math. Soc., Providence, RI, 1991, pp. 39–48. MR 1112801, DOI 10.1090/conm/117/1112801
W. T. Ingram and Howard Cook, A characterization of indecomposable compact continua, Topology Conference (Arizona State Univ., Tempe, Ariz., 1967) Arizona State Univ., Tempe, Ariz., 1968, pp. 168–169. MR 0253298
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
Ivan Niven, Irrational numbers, The Carus Mathematical Monographs, No. 11, Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, N.Y., 1956. MR 0080123, DOI 10.5948/9781614440116
Mark H. Turpin, The restricted rotation number of an example of Handel, Discrete Contin. Dynam. Systems Added Volume II (1998), 253–258. Dynamical systems and differential equations, Vol. II (Springfield, MO, 1996). MR 1722474