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Absence of mixing in area-preserving flows on surfaces. (English) Zbl 1251.37003
This paper solves an open question by A. Katok and J.-P. Thouvenot; see, for example, Section 6.3.2 of [“Spectral properties and combinatorial constructions in ergodic theory”, in: Handbook of dynamical systems. Volume 1B. Amsterdam: Elsevier. 649–743 (2006; Zbl 1130.37304)]. The statement of the main result is the following: Let \(S\) be a closed surface of genus \(g\geq 2\) and let \(\Phi:\mathbb{R}\times S\rightarrow S\) be a flow given by a multi-valued Hamiltonian associated to a smooth closed differential \(1\)-form \(\eta\). If \(\Phi\) has only simple saddles and no saddle loops homologous to zero then \(\Phi\) is not mixing for a typical such form \(\eta\).

MSC:
37A25 Ergodicity, mixing, rates of mixing
37E35 Flows on surfaces
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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[1] A. Avila and G. Forni, ”Weak mixing for interval exchange transformations and translation flows,” Ann. of Math., vol. 165, iss. 2, pp. 637-664, 2007. · Zbl 1136.37003
[2] A. Avila, S. Gouëzel, and J. Yoccoz, ”Exponential mixing for the Teichmüller flow,” Publ. Math. Inst. Hautes Études Sci., iss. 104, pp. 143-211, 2006. · Zbl 1263.37051
[3] V. I. Arnol\('\)d, ”Topological and ergodic properties of closed \(1\)-forms with incommensurable periods,” Funktsional. Anal. i Prilozhen., vol. 25, iss. 2, pp. 1-12, 96, 1991. · Zbl 0732.58001
[4] E. Calabi, ”An intrinsic characterization of harmonic one-forms,” in Global Analysis (Papers in Honor of K. Kodaira), Tokyo: Univ. Tokyo Press, 1969, pp. 101-117. · Zbl 0194.24701
[5] I. P. Cornfeld, S. V. Fomin, and Y. G. Sinaui, Ergodic Theory, New York: Springer-Verlag, 1982, vol. 245. · Zbl 0493.28007
[6] K. Frcaczek and M. Lemańczyk, ”On symmetric logarithm and some old examples in smooth ergodic theory,” Fund. Math., vol. 180, iss. 3, pp. 241-255, 2003. · Zbl 1047.37027
[7] K. Frcaczek and C. Ulcigrai, Ergodic properties of infinite extension of area-preserving flows.
[8] K. Frcaczek and M. Lemańczyk, ”On disjointness properties of some smooth flows,” Fund. Math., vol. 185, iss. 2, pp. 117-142, 2005. · Zbl 1093.37001
[9] G. Forni, ”Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,” Ann. of Math., vol. 155, iss. 1, pp. 1-103, 2002. · Zbl 1034.37003
[10] A. B. Katok, ”Invariant measures of flows on oriented surfaces,” Soviet Math. Dokl., vol. 14, pp. 1104-1108, 1973. · Zbl 0298.28013
[11] A. B. Katok, ”Interval exchange transformations and some special flows are not mixing,” Israel J. Math., vol. 35, iss. 4, pp. 301-310, 1980. · Zbl 0437.28009
[12] M. Keane, ”Interval exchange transformations,” Math. Z., vol. 141, pp. 25-31, 1975. · Zbl 0278.28010
[13] A. V. Kovcergin, ”On mixing in special flows over a rearrangement of segments and in smooth flows on surfaces,” Mat. Sb., vol. 96(138), pp. 471-502, 1975. · Zbl 0321.28012
[14] A. V. Kovcergin, ”Nonsingular saddle points and the absence of mixing,” Mat. Zametki, vol. 19, pp. 453-468, 1976. · Zbl 0344.28008
[15] A. V. Kovcergin, ”Nondegenerate fixed points and mixing in flows on a two-dimensional torus,” Mat. Sb., vol. 194, iss. 8, pp. 83-112, 2003. · Zbl 1077.37005
[16] A. V. Kovcergin, ”Some generalizations of theorems on mixing flows with nondegenerate saddles on a two-dimensional torus,” Mat. Sb., vol. 195, iss. 9, pp. 19-36, 2004. · Zbl 1082.37008
[17] A. V. Kovcergin, ”Nondegenerate fixed points and mixing in flows on a two-dimensional torus. II,” Mat. Sb., vol. 195, iss. 3, pp. 15-46, 2004. · Zbl 1077.37006
[18] A. V. Kovcergin, ”Well-approximable angles and mixing for flows on \(\mathbb T^2\) with nonsingular fixed points,” Electron. Res. Announc. Amer. Math. Soc., vol. 10, pp. 113-121, 2004. · Zbl 1068.37027
[19] A. V. Kovcergin, ”Nondegenerate saddles and the absence of mixing in flows on surfaces,” Tr. Mat. Inst. Steklova, vol. 256, pp. 252-266, 2007. · Zbl 1153.37303
[20] A. B. Katok and J. -P. Thouvenot, ”Spectral properties and combinatorial constructions in ergodic theory,” in Handbook of Dynamical Systems, Vol. 1B, Amsterdam: Elsevier B.V., 2006, pp. 649-743. · Zbl 1130.37304
[21] M. Lemańczyk, ”Sur l’absence de mélange pour des flots spéciaux au-dessus d’une rotation irrationnelle,” Colloq. Math., vol. 84/85, pp. 29-41, 2000. · Zbl 0983.37004
[22] G. Levitt, ”Feuilletages des surfaces,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 32, iss. 2, pp. 179-217, 1982. · Zbl 0454.57015
[23] A. Mayer, ”Trajectories on the closed orientable surfaces,” Rec. Math. [Mat. Sbornik] N.S., vol. 12(54), pp. 71-84, 1943. · Zbl 0063.03856
[24] H. Masur, ”Interval exchange transformations and measured foliations,” Ann. of Math., vol. 115, iss. 1, pp. 169-200, 1982. · Zbl 0497.28012
[25] S. P. Novikov, ”The Hamiltonian formalism and a multivalued analogue of Morse theory,” Uspekhi Mat. Nauk, vol. 37, iss. 5, pp. 3-49, 248, 1982. · Zbl 0571.58011
[26] I. Nikolaev and E. Zhuzhoma, Flows on 2-Dimensional Manifolds. An Overview, New York: Springer-Verlag, 1999, vol. 1705. · Zbl 1022.37027
[27] G. Rauzy, ”Échanges d’intervalles et transformations induites,” Acta Arith., vol. 34, iss. 4, pp. 315-328, 1979. · Zbl 0414.28018
[28] D. Scheglov, ”Absence of mixing for smooth flows on genus two surfaces,” J. Mod. Dyn., vol. 3, iss. 1, pp. 13-34, 2009. · Zbl 1183.37080
[29] Y. G. Sinaui and K. M. Khanin, ”Mixing of some classes of special flows over rotations of the circle,” Funktsional. Anal. i Prilozhen., vol. 26, iss. 3, pp. 1-21, 1992. · Zbl 0797.58045
[30] Y. G. Sinai and C. Ulcigrai, ”A limit theorem for Birkhoff sums of non-integrable functions over rotations,” in Geometric and Probabilistic Structures in Dynamics, Providence, RI: Amer. Math. Soc., 2008, vol. 469, pp. 317-340. · Zbl 1154.37307
[31] C. Ulcigrai, Ergodic Properties of Some Area-Preserving Flows, ProQuest LLC, Ann Arbor, MI, 2007. · Zbl 1135.37004
[32] C. Ulcigrai, ”Mixing of asymmetric logarithmic suspension flows over interval exchange transformations,” Ergodic Theory Dynam. Systems, vol. 27, iss. 3, pp. 991-1035, 2007. · Zbl 1135.37004
[33] C. Ulcigrai, ”Weak mixing for logarithmic flows over interval exchange transformations,” J. Mod. Dyn., vol. 3, iss. 1, pp. 35-49, 2009. · Zbl 1183.37081
[34] W. A. Veech, ”Interval exchange transformations,” J. Analyse Math., vol. 33, pp. 222-272, 1978. · Zbl 0455.28006
[35] W. A. Veech, ”Gauss measures for transformations on the space of interval exchange maps,” Ann. of Math., vol. 115, iss. 1, pp. 201-242, 1982. · Zbl 0486.28014
[36] M. Viana, Dynamics of interval exchange maps and Teichmüller flows.
[37] J. Yoccoz, ”Continued fraction algorithms for interval exchange maps: an introduction,” in Frontiers in Number Theory, Physics, and Geometry. I, New York: Springer-Verlag, 2006, pp. 401-435. · Zbl 1127.28011
[38] A. Zorich, ”Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 46, iss. 2, pp. 325-370, 1996. · Zbl 0853.28007
[39] A. Zorich, ”Deviation for interval exchange transformations,” Ergodic Theory Dynam. Systems, vol. 17, iss. 6, pp. 1477-1499, 1997. · Zbl 0958.37002
[40] A. Zorich, ”How do the leaves of a closed \(1\)-form wind around a surface?,” in Pseudoperiodic Topology, Providence, RI: Amer. Math. Soc., 1999, vol. 197, pp. 135-178. · Zbl 0976.37012
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