A smooth mixing flow on a surface with nondegenerate fixed points

Chaika, Jon; Wright, Alex

doi:10.1090/jams/911

A smooth mixing flow on a surface with nondegenerate fixed points
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by Jon Chaika and Alex Wright

J. Amer. Math. Soc. 32 (2019), 81-117

DOI: https://doi.org/10.1090/jams/911

Published electronically: October 9, 2018

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Abstract:

We construct a smooth, area preserving, mixing flow with finitely many nondegenerate fixed points and no saddle connections on a closed surface of genus $5$. This resolves a problem that has been open for four decades.

References

V. I. Arnol′d, Topological and ergodic properties of closed $1$-forms with incommensurable periods, Funktsional. Anal. i Prilozhen. 25 (1991), no. 2, 1–12, 96 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 2, 81–90. MR 1142204, DOI 10.1007/BF01079587
A. A. Blohin, Smooth ergodic flows on surfaces, Trudy Moskov. Mat. Obšč. 27 (1972), 113–128 (Russian). MR 0370656
Jean-Pierre Conze and Krzysztof Fra̧czek, Cocycles over interval exchange transformations and multivalued Hamiltonian flows, Adv. Math. 226 (2011), no. 5, 4373–4428. MR 2770454, DOI 10.1016/j.aim.2010.11.014
Yitwah Cheung, Pascal Hubert, and Howard Masur, Dichotomy for the Hausdorff dimension of the set of nonergodic directions, Invent. Math. 183 (2011), no. 2, 337–383. MR 2772084, DOI 10.1007/s00222-010-0279-2
Bassam Fayad, Smooth mixing flows with purely singular spectra, Duke Math. J. 132 (2006), no. 2, 371–391. MR 2219261, DOI 10.1215/S0012-7094-06-13225-8
Bassam Fayad, Giovanni Forni, and Adam Kanigowski, Lebesgue spectrum for area preserving flows on the two torus, preprint, arXiv 1609.03757 (2016).
Bassam Fayad and Adam Kanigowski, Multiple mixing for a class of conservative surface flows, Invent. Math. 203 (2016), no. 2, 555–614. MR 3455157, DOI 10.1007/s00222-015-0596-6
K. Frączek and M. Lemańczyk, On symmetric logarithm and some old examples in smooth ergodic theory, Fund. Math. 180 (2003), no. 3, 241–255. MR 2063628, DOI 10.4064/fm180-3-3
K. Frączek and M. Lemańczyk, A class of special flows over irrational rotations which is disjoint from mixing flows, Ergodic Theory Dynam. Systems 24 (2004), no. 4, 1083–1095. MR 2085391, DOI 10.1017/S0143385704000112
Krzysztof Frączek and Mariusz Lemańczyk, On disjointness properties of some smooth flows, Fund. Math. 185 (2005), no. 2, 117–142. MR 2163107, DOI 10.4064/fm185-2-2
K. Frączek and M. Lemańczyk, On mild mixing of special flows over irrational rotations under piecewise smooth functions, Ergodic Theory Dynam. Systems 26 (2006), no. 3, 719–738. MR 2237466, DOI 10.1017/S0143385706000046
K. Frączek and M. Lemańczyk, Smooth singular flows in dimension 2 with the minimal self-joining property, Monatsh. Math. 156 (2009), no. 1, 11–45. MR 2470104, DOI 10.1007/s00605-008-0564-y
Giovanni Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 (2002), no. 1, 1–103. MR 1888794, DOI 10.2307/3062150
Michael-Robert Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5–233 (French). MR 538680, DOI 10.1007/BF02684798
A. B. Katok, Spectral properties of dynamical systems with an integral invariant on the torus, Funkcional. Anal. i Priložen. 1 (1967), no. 4, 46–56 (Russian). MR 0237903
A. B. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR 211 (1973), 775–778 (Russian). MR 0331438
Anatole Katok, Interval exchange transformations and some special flows are not mixing, Israel J. Math. 35 (1980), no. 4, 301–310. MR 594335, DOI 10.1007/BF02760655
A. Ya. Khinchin, Continued fractions, University of Chicago Press, Chicago, Ill.-London, 1964. MR 0161833
Adam Kanigowski and Joanna Kułaga-Przymus, Ratner’s property and mild mixing for smooth flows on surfaces, Ergodic Theory Dynam. Systems 36 (2016), no. 8, 2512–2537. MR 3570023, DOI 10.1017/etds.2015.35
Adam Kanigowski, Joanna Kuł aga Przymus, and Corinna Ulcigrai, Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces, preprint, arXiv 1606.09189 (2016), J. Eur. Math. Soc. (JEMS) (to appear).
A. V. Kočergin, The absence of mixing in special flows over a rotation of the circle and in flows on a two-dimensional torus, Dokl. Akad. Nauk SSSR 205 (1972), 515–518 (Russian). MR 0306629
A. V. Kočergin, Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces, Mat. Sb. (N.S.) 96(138) (1975), 471–502, 504 (Russian). MR 0516507
A. V. Kočergin, Nondegenerate saddles, and the absence of mixing, Mat. Zametki 19 (1976), no. 3, 453–468 (Russian). MR 415681
A. V. Kochergin, A mixing special flow over a rotation of the circle with an almost Lipschitz function, Mat. Sb. 193 (2002), no. 3, 51–78 (Russian, with Russian summary); English transl., Sb. Math. 193 (2002), no. 3-4, 359–385. MR 1913598, DOI 10.1070/SM2002v193n03ABEH000636
A. V. Kochergin, Nondegenerate fixed points and mixing in flows on a two-dimensional torus, Mat. Sb. 194 (2003), no. 8, 83–112 (Russian, with Russian summary); English transl., Sb. Math. 194 (2003), no. 7-8, 1195–1224. MR 2034533, DOI 10.1070/SM2003v194n08ABEH000762
A. Kochergin, Well-approximable angles and mixing for flows on $\Bbb T^2$ with nonsingular fixed points, Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 113–121. MR 2119032, DOI 10.1090/S1079-6762-04-00136-2
A. V. Kochergin, Hölder time change and the mixing rate in a flow on a two-dimensional torus, Tr. Mat. Inst. Steklova 244 (2004), no. Din. Sist. i Smezhnye Vopr. Geom., 216–248 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 1(244) (2004), 201–232. MR 2075117
A. V. Kochergin, Nondegenerate fixed points and mixing in flows on a two-dimensional torus. II, Mat. Sb. 195 (2004), no. 3, 15–46 (Russian, with Russian summary); English transl., Sb. Math. 195 (2004), no. 3-4, 317–346. MR 2068956, DOI 10.1070/SM2004v195n03ABEH000807
A. V. Kochergin, Some generalizations of theorems on mixing flows with nondegenerate saddles on a two-dimensional torus, Mat. Sb. 195 (2004), no. 9, 19–36 (Russian, with Russian summary); English transl., Sb. Math. 195 (2004), no. 9-10, 1253–1270. MR 2122367, DOI 10.1070/SM2004v195n09ABEH000843
A. V. Kochergin, Nondegenerate saddles, and absence of mixing. II, Mat. Zametki 81 (2007), no. 1, 145–148 (Russian); English transl., Math. Notes 81 (2007), no. 1-2, 126–129. MR 2333872, DOI 10.1134/S0001434607010130
Andrey Kochergin, Causes of stretching of Birkhoff sums and mixing in flows on surfaces, Dynamics, ergodic theory, and geometry, Math. Sci. Res. Inst. Publ., vol. 54, Cambridge Univ. Press, Cambridge, 2007, pp. 129–144. MR 2369445, DOI 10.1017/CBO9780511755187.006
A. N. Kolmogorov, On dynamical systems with an integral invariant on the torus, Doklady Akad. Nauk SSSR (N.S.) 93 (1953), 763–766 (Russian). MR 0062892
A. B. Katok, Ja. G. Sinaĭ, and A. M. Stepin, The theory of dynamical systems and general transformation groups with invariant measure, Mathematical analysis, Vol. 13 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1975, pp. 129–262. (errata insert) (Russian). MR 0584389
Anatole Katok and Jean-Paul Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 649–743. MR 2186251, DOI 10.1016/S1874-575X(06)80036-6
M. Lemańczyk, Sur l’absence de mélange pour des flots spéciaux au-dessus d’une rotation irrationnelle. part 1, Colloq. Math. 84/85 (2000), no. part 1, 29–41 (French, with English summary). Dedicated to the memory of Anzelm Iwanik. MR 1778837, DOI 10.4064/cm-84/85-1-29-41
Yuri Lima, CF1: uniform distribution and Denjoy–Koksma’s inequality, http://matheuscmss.wordpress.com/2012/02/16/cf1-uniform-distribution-and-denjoy-koksmas-inequality/.
Howard Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2) 115 (1982), no. 1, 169–200. MR 644018, DOI 10.2307/1971341
S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk 37 (1982), no. 5(227), 3–49, 248 (Russian). MR 676612
Alexander Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen, Abh. Math. Sem. Univ. Hamburg 1 (1922), no. 1, 77–98 (German). MR 3069389, DOI 10.1007/BF02940581
Davide Ravotti, Quantitative mixing for locally Hamiltonian flows with saddle loops on compact surfaces, Ann. Henri Poincaré 18 (2017), no. 12, 3815–3861. MR 3723342, DOI 10.1007/s00023-017-0619-5
E. A. Sataev, The number of invariant measures for flows on orientable surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 4, 860–878 (Russian). MR 0391184
Dmitri Scheglov, Absence of mixing for smooth flows on genus two surfaces, J. Mod. Dyn. 3 (2009), no. 1, 13–34. MR 2481330, DOI 10.3934/jmd.2009.3.13
Dmitri Scheglov, Absence of mixing for smooth flows on genus two surfaces, ProQuest LLC, Ann Arbor, MI, 2009, Thesis (Ph.D.)–The Pennsylvania State University.
Ya. G. Sinaĭ and K. M. Khanin, Mixing of some classes of special flows over rotations of the circle, Funktsional. Anal. i Prilozhen. 26 (1992), no. 3, 1–21 (Russian); English transl., Funct. Anal. Appl. 26 (1992), no. 3, 155–169. MR 1189019, DOI 10.1007/BF01075628
Rodrigo Treviño, On the ergodicity of flat surfaces of finite area, Geom. Funct. Anal. 24 (2014), no. 1, 360–386. MR 3177386, DOI 10.1007/s00039-014-0269-4
Corinna Ulcigrai, Mixing of asymmetric logarithmic suspension flows over interval exchange transformations, Ergodic Theory Dynam. Systems 27 (2007), no. 3, 991–1035. MR 2322189, DOI 10.1017/S0143385706000836
Corinna Ulcigrai, Weak mixing for logarithmic flows over interval exchange transformations, J. Mod. Dyn. 3 (2009), no. 1, 35–49. MR 2481331, DOI 10.3934/jmd.2009.3.35
Corinna Ulcigrai, Absence of mixing in area-preserving flows on surfaces, Ann. of Math. (2) 173 (2011), no. 3, 1743–1778. MR 2800723, DOI 10.4007/annals.2011.173.3.10
William A. Veech, Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem $\textrm {mod}\ 2$, Trans. Amer. Math. Soc. 140 (1969), 1–33. MR 240056, DOI 10.1090/S0002-9947-1969-0240056-X
William A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2) 115 (1982), no. 1, 201–242. MR 644019, DOI 10.2307/1971391
J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. of Math. (2) 33 (1932), no. 3, 587–642 (German). MR 1503078, DOI 10.2307/1968537