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ISSN 1088-6826(e) ISSN 0002-9939(p)

Denjoy's theorem with exponents

Author(s): Alec Norton
Journal: Proc. Amer. Math. Soc. 127 (1999), 3111-3118.
MSC (1991): Primary 58F03; Secondary 28A80
Posted: April 23, 1999
MathSciNet review: 1600129
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Abstract: If is the (unique) minimal set for a $C^{1+\alpha }$ diffeomorphism of the circle without periodic orbits, $0<\alpha <1$ , then the upper box dimension of is at least $\alpha$ . The method of proof is to introduce the exponent $\alpha$ into the proof of Denjoy's theorem.

References:

1.: S. Bates and A. Norton, On sets of critical values in the real line, Duke Math. J. 83 (1996), 399-413. MR 97h:58019
2.: A.S. Besicovitch and S.J. Taylor, On the complementary intervals of a linear closed set of zero Lebesgue measure, J. London Math. Soc. 29 (1954), 449-459. MR 16:344d
3.: P. Bohl, Uber die hinsichtlich der unabhängigen variabeln periodische Differential gleichung erster Ordnung, Acta Math. 40 (1916), 321-336.
4.: A. Denjoy, Sur les courbes definies par les equations differentielles a la surface du tore, J. Math Pures et Appl. 11 (1932), 333-375.
5.: K.J. Falconer, Fractal Geometry, J. Wiley, New York, 1990. MR 92j:28008
6.: G.R. Hall, Bifurcation of an invariant attracting circle: a Denjoy attractor, Ergod. Th. & Dynam. Sys. 3 (1983), 87-118.
7.: J. Harrison, Denjoy fractals, Topology 28 (1989), 59-80. MR 90c:58148
8.: M.R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle a des rotations, Publ. Math. I.H.E.S. 49 (1979), 5-233.
9.: D. McDuff, $C^{1}$ -minimal subsets of the circle, Ann. Inst. Fourier, Grenoble 31 (1981), 177-193. MR 82h:58044
10.: P. McSwiggen, Diffeomorphisms of the $k$ -torus with wandering domains, Ergod. Th. & Dynam. Sys. 15 (1995), 1189-1205. MR 97a:58102
11.: A. Norton, Denjoy minimal sets are far from affine, to appear, Ergod. Th. & Dynam. Sys.
12.: A. Norton and D.P. Sullivan, Wandering domains and invariant conformal structures for mappings of the 2-torus, Ann. Acad. Sci. Fenn., Series A I Math. 21 (1996), 51-68. MR 96m:58151
13.: H. Poincaré, Mémoire sur les courbes définies par une équation différentielle I,II,III,IV, J. Math. Pures et Appl. (1881,82,85,86).
14.: E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970.
15.: D.P. Sullivan, Conformal Dynamical Systems, Geometric Dynamics, Lecture Notes in Math. 1007, Springer-Verlag, New York, 1983, pp. 636-650. MR 85m:58112

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