Abstract
We construct, over some minimal translations of the two torus, special flows under a differentiable ceiling function that combine the properties of mixing and rank one.
Similar content being viewed by others
References
Adams, T.M.: Smorodinsky’s conjecture on rank-one mixing. Proc. Am. Math. Soc. 126, 739–744 (1998)
Chacon, R.V.: Transformations having continuous spectrum. J. Math. Mech. 16, 399–415 (1966)
Fayad, B.: Weak mixing for reparametrized linear flows on the torus. Ergodic Theory Dyn. Syst. 22, 187–201 (2002)
Fayad, B.: Analytic mixing reparametrizations of irrational flows. Ergodic Theory Dyn. Syst. 22, 437–468 (2002)
Fayad, B.: Polynomial decay of correlations for a class of smooth flows on the two torus. Bull. Soc. Math. Fr. 129, 487–503 (2001)
Fayad, B.: Partially mixing and locally rank 1 smooth transformations and flows on the torus Td,d≥3. J. Lond. Math. Soc., II. Ser. 64, 637–654 (2001)
Kalikow, S.A.: Twofold mixing implies threefold mixing for rank one transformations. Ergodic Theory Dyn. Syst. 4, 237–259 (1984)
Katok, A.B.: Cocycles, cohomology and combinatorial constructions in ergodic theory. In: Smooth ergodic theory and its applications (Seattle, WA, 1999) 107–173. Proc. Symp. Pure Math. 69, In collaboration with E. A. Robinson, Jr.
Katok, A.B.: Spectral properties of dynamical systems with an integral invariant on the torus. Funct. Anal. Appl. 1, 46–56 (1967)
Katok, A.B., Stepin, A.M.: Approximations in ergodic theory. Usp. Mat. Nauk 22, 81–106 (1967)
Khanin, K.M., Sinai, Y.G.: Mixing for some classes of special flows over rotations of the circle. Funct. Anal. Appl. 26, 155–169 (1992)
King, J.L.: Joining-rank and the structure of finite rank mixing transformations. J. Anal. Math. 51, 182–227 (1988)
King, J.L., Thouvenot, J.P.: A canonical structure theorem for finite joining-rank maps. J. Anal. Math. 56, 211–230 (1991)
Kočergin, A.V.: Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces. Mat. Sb. 25, 471–502 (1975)
Kolmogorov, A.N.: On dynamical systems with an integral invariant on the torus. Dokl. Akad. Nauk, SSSR 93, 763–766 (1953)
Ornstein, D.S.: On the root problem in ergodic theory. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability Vol. II (1970/1971) 347–356. Probability theory. Univ. Cal. Press 1972
Ornstein, D.S.: Ergodic theory, randomness, and dynamical systems. New Haven, Conn.: Yale University Press 1974
Ryzhikov, V.V.: Mixing, rank and minimal self-joining of actions with invariant measure. Mat. Sb. 183, 133–160 (1992)
Shklover, M.D.: On dynamical systems on the torus with continuous spectrum. Izv. Vuzov 10, 113–124 (1967)
Thouvenot, J.P.: Some properties and applications of joinings in ergodic theory. In: Ergodic theory and its connections with harmonic analysis, Alexandria, 1993, 207–235. Cambridge: Cambridge Univ. Press 1995
Yoccoz, J.C.: Petits diviseurs en dimension 1. Astérisque (Appendix 1) (1982)
Zeitz, P.: The centralizer of a rank-one flow. Isr. J. Math. 84, 129–145 (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fayad, B. Rank one and mixing differentiable flows. Invent. math. 160, 305–340 (2005). https://doi.org/10.1007/s00222-004-0408-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-004-0408-x