Smooth type 𝐼𝐼𝐼 diffeomorphisms of manifolds

Hawkins, Jane

doi:10.1090/S0002-9947-1983-0688966-0

Smooth type $\textrm {III}$ diffeomorphisms of manifolds
HTML articles powered by AMS MathViewer

by Jane Hawkins

Trans. Amer. Math. Soc. 276 (1983), 625-643

DOI: https://doi.org/10.1090/S0002-9947-1983-0688966-0

PDF | Request permission

Abstract:

In this paper we prove that every smooth paracompact connected manifold of dimension $\geqslant 3$ admits a smooth type ${\text {III}}_\lambda$ diffeomorphism for every $0 \leqslant \lambda \leqslant 1$. (Herman proved the result for $\lambda = 1$ in [7].) The result follows from a theorem which gives sufficient conditions for the existence of smooth ergodic real line extensions of diffeomorphisms of manifolds.

References

Sur les courbes définies par les équations différentielles à la surface du tore

11

Yael Naim Dowker, Finite and $\sigma$-finite invariant measures, Ann. of Math. (2) 54 (1951), 595–608. MR 45193, DOI 10.2307/1969491
Nathaniel A. Friedman, Introduction to ergodic theory, Van Nostrand Reinhold Mathematical Studies, No. 29, Van Nostrand Reinhold Co., New York-Toronto-London, 1970. MR 0435350
Paul R. Halmos, Lectures on ergodic theory, Publications of the Mathematical Society of Japan, vol. 3, Mathematical Society of Japan, Tokyo, 1956. MR 0097489

Existence of smooth ergodic flows on smooth manifolds

8

Jane Hawkins and Klaus Schmidt, On $C^{2}$-diffeomorphisms of the circle which are of type $\textrm {III}_{1}$, Invent. Math. 66 (1982), no. 3, 511–518. MR 662606, DOI 10.1007/BF01389227

Construction de difféomorphismes ergodiques

Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations

49

Eberhard Hopf, Theory of measure and invariant integrals, Trans. Amer. Math. Soc. 34 (1932), no. 2, 373–393. MR 1501643, DOI 10.1090/S0002-9947-1932-1501643-6
Roger Jones and William Parry, Compact abelian group extensions of dynamical systems. II, Compositio Math. 25 (1972), 135–147. MR 338318
Y. Katznelson, Sigma-finite invariant measures for smooth mappings of the circle, J. Analyse Math. 31 (1977), 1–18. MR 486415, DOI 10.1007/BF02813295
John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
Wolfgang Krieger, On the Araki-Woods asymptotic ratio set and non-singular transformations of a measure space, Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970) Lecture Notes in Math., Vol. 160, Springer, Berlin, 1970, pp. 158–177. MR 0414823
Viktor Losert and Klaus Schmidt, A class of probability measures on groups arising from some problems in ergodic theory, Probability measures on groups (Proc. Fifth Conf., Oberwolfach, 1978) Lecture Notes in Math., vol. 706, Springer, Berlin, 1979, pp. 220–238. MR 536988
Donald S. Ornstein, On invariant measures, Bull. Amer. Math. Soc. 66 (1960), 297–300. MR 146350, DOI 10.1090/S0002-9904-1960-10478-8
Klaus Schmidt, Cocycles on ergodic transformation groups, Macmillan Lectures in Mathematics, Vol. 1, Macmillan Co. of India, Ltd., Delhi, 1977. MR 0578731
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1