Smooth mixing flows with purely singular spectra. (English) Zbl 1099.37018

Summary: We give a geometric criterion that implies a singular maximal spectral type for a dynamical system on a Riemannian manifold. The criterion, which is based on the existence of fairly rich but localized periodic approximations, is compatible with mixing. Indeed, we check it for an ad hoc class of smooth mixing flows on \(\mathbb{T}^3\) obtained from linear flows by time change and thus providing natural examples of mixing smooth diffeomorphisms and flows with purely singular spectra.


37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37A25 Ergodicity, mixing, rates of mixing
Full Text: DOI


[1] B. R. Fayad, Analytic mixing reparametrizations of irrational flows , Ergodic Theory Dynam. Systems 22 (2002), 437–468. · Zbl 1136.37307
[2] -, Weak mixing for reparameterized linear flows on the torus , Ergodic Theory Dynam. Systems 22 (2002), 187–201. · Zbl 1001.37006
[3] B. Fayad, A. Katok, and A. Windsor, Mixed spectrum reparameterizations of linear flows on \(\TT^2\) , Mosc. Math. J. 1 (2001), 521–537. · Zbl 1123.37300
[4] B. Fayad and A. Windsor, A dichotomy between discrete and continuous spectrum for a class of special flows over rotations , · Zbl 1109.37004
[5] M. Guenais and F. Parreau, Valeurs propres de transformations liées aux rotations irrationnelles et aux fonctions en escalier , preprint, 2005.
[6] P. R. Halmos, Lectures on Ergodic Theory , Chelsea, New York, 1960. · Zbl 0117.10502
[7] M.-R. Herman, Exemples de flots hamiltoniens dont aucune perturbation en topologie \(C^\infty\) n’a d’orbites périodiques sur un ouvert de surfaces d’énergies , C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 989–994. · Zbl 0759.58016
[8] B. Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum , Israel J. Math. 76 (1991), 289–298. · Zbl 0790.28010
[9] A. B. Katok, Spectral properties of dynamical systems with an integral invariant on the torus (in Russian), Funkcional. Anal. i Priložen. 1 , no. 4 (1967), 46–56. · Zbl 0172.12102
[10] -, Time change, monotone equivalence, and standard dynamical systems (in Russian), Dokl. Akad. Nauk SSSR 223 (1975), 789–792.
[11] -, Combinatorial Constructions in Ergodic Theory and Dynamics , Univ. Lecture Ser. 30 , Amer. Math. Soc., Providence, 2003. · Zbl 1030.37001
[12] A. B. Katok and A. M. Stepin, Approximations in ergodic theory (in Russian), Uspekhi Mat. Nauk 22 , no. 5 (1967), 81–106. · Zbl 0172.07202
[13] A. V. KočErgin, The absence of mixing in special flows over a rotation of the circle and in flows on a two-dimensional torus (in Russian), Dokl. Akad. Nauk SSSR 205 (1972), 515–518. · Zbl 0262.28015
[14] -, Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces , Math. USSR-Sb. 25 (1975), 441–469. · Zbl 0326.28030
[15] A. N. Kolmogorov, On dynamical systems with an integral invariant on the torus (in Russian), Doklady Akad. Nauk SSSR (N.S.) 93 (1953), 763–766. · Zbl 0052.31904
[16] -, “Théorie générale des systèmes dynamiques et mécanique classique” in Proceedings of the International Congress of Mathematicians, Vol. 1 (Amsterdam, 1954) , Noordhoff, Groningen, Netherlands, 1957, 315–333.
[17] J. Neveu, Bases mathématiques du calcul des probabilités , 2nd ed., Masson, Paris, 1970. · Zbl 0203.49901
[18] D. S. Ornstein and M. Smorodinsky, Continuous speed changes for flows , Israel J. Math. 31 (1978), 161–168. · Zbl 0399.58003
[19] M. D. ŠKlover, Classical dynamical systems on the torus with continuous spectrum (in Russian), Izv. Vysš. Učebn. Zaved. Matematika 1967 , no. 10, 113–124.
[20] J.-C. Yoccoz, “Centralisateurs et conjugaison différentiable des difféomorphismes du cercle” in Petits diviseurs en dimension \(1\), Astérisque 231 , Soc. Math. France, Montrouge, 1995, 89–242.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.