2022, 18: 103-130. doi: 10.3934/jmd.2022005

Ergodicity, mixing, Ratner's properties and disjointness for classical flows: On the research of Corinna Ulcigrai

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopin street 12/18, 87-100 Toruń, Poland

Received  September 28, 2021 Revised  October 11, 2021 Published  April 2022

Fund Project: Research supported by Narodowe Centrum Nauki grant UMO-2019/33/B/ST1/00364

We present and discuss C. Ulcigrai's results concerning mixing properties of locally Hamiltonian flows, spectral properties of smooth time changes of horocycle flows together with their Möbius orthogonality and the ergodicity problems of directional flows in the wind tree model of Ehrenfest.

Citation: Mariusz Lemańczyk. Ergodicity, mixing, Ratner's properties and disjointness for classical flows: On the research of Corinna Ulcigrai. Journal of Modern Dynamics, 2022, 18: 103-130. doi: 10.3934/jmd.2022005
References:
[1]

H. El AbdalaouiJ. Kułaga-PrzymusM. Lemańczyk and T. de la Rue, Möbius disjointness for models of an ergodic system and beyond, Israel J. Math., 228 (2018), 707-751.  doi: 10.1007/s11856-018-1784-z.

[2]

H. El AbdalaouiM. Lemańczyk and T. de la Rue, Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals, Int. Math. Res. Not. IMRN, 14 (2017), 4350-4368. 

[3]

V. I. Arnol'd, Topological and ergodic properties of closed 1-forms with incommensurable periods, Funct. Anal. Appl., 25 (1991), 81-90.  doi: 10.1007/BF01079587.

[4]

A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation flows, Ann. of Math., 165 (2007), 637-664.  doi: 10.4007/annals.2007.165.637.

[5]

A. AvilaG. Forni and C. Ulcigrai, Mixing for time-changes of Heisenberg nilflows, J. Differential Geom., 89 (2011), 369-410. 

[6]

A. Avila, G. Forni, D. Ravotti and C. Ulcigrai, Mixing for smooth time-changes of general nilflows, Adv. Math., 385 (2021), 107759, 65 pp. doi: 10.1016/j.aim.2021.107759.

[7]

A. Avila and P. Hubert, Recurrence for the wind-tree model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 1-11.  doi: 10.1016/j.anihpc.2017.11.006.

[8]

J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Moebius from horocycle flows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., 28, Springer, New York, 2013, 67–83. doi: 10.1007/978-1-4614-4075-8_5.

[9]

A. Bufetov and G. Forni, Limit theorems for horocycle flows, Ann. Sci. Éc. Norm. Supér, 47 (2014), 851-903. 

[10]

J. Chaika and A. Eskin, Every flat surface is Birkhoff and Oseledets generic in almost every direction, J. Mod. Dyn., 9 (2015), 1-23.  doi: 10.3934/jmd.2015.9.1.

[11]

J. ChaikaK. FrączekA. Kanigowski and C. Ulcigrai, Singularity of the spectrum for smooth area-preserving flows in genus two and translation surfaces well approximated by cylinders, Comm. Math. Phys., 381 (2021), 1369-1407.  doi: 10.1007/s00220-020-03895-x.

[12]

J. Chaika and A. Wright, A smooth mixing flow on a surface with nondegenerate fixed points, J. Amer. Math. Soc., 32 (2019), 81-117.  doi: 10.1090/jams/911.

[13]

A. I. Danilenko and V. V. Ryzhikov, On self-similarities of ergodic flows, Proc. Lond. Math. Soc., 104 (2012), 431-454.  doi: 10.1112/plms/pdr032.

[14]

A. EskinM. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes Études Sci., 120 (2014), 207-333.  doi: 10.1007/s10240-013-0060-3.

[15]

B. FayadG. Forni and A. Kanigowski, Lebesgue spectrum of countable multiplicity for conservative flows on the torus, J. Amer. Math. Soc., 34 (2021), 747-813.  doi: 10.1090/jams/970.

[16]

B. Fayad and A. Kanigowski, Multiple mixing for a class of conservative surface flows, Invent. Math., 203 (2016), 555-614.  doi: 10.1007/s00222-015-0596-6.

[17]

S. FerencziJ. Kułaga-Przymus and M. Lemańczyk, Sarnak's conjecture: What's new, Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, 2213 (2018), 163-235. 

[18]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8.

[19]

L. Flaminio and G. Forni, Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows, Electron. Res. Announc. Math. Sci., 26 (2019), 16-23.  doi: 10.3934/era.2019.26.002.

[20]

L. FlaminioK. FrączekJ. Kułaga-Przymus and M. Lemańczyk, Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds, Studia Math., 244 (2019), 43-97.  doi: 10.4064/sm170512-25-9.

[21]

G. Forni and C. Ulcigrai, Time-changes of horocycle flows, J. Mod. Dyn., 6 (2012), 251-273.  doi: 10.3934/jmd.2012.6.251.

[22]

K. Frączek and P. Hubert, Recurrence and non-ergodicity in generalized wind-tree models, Math. Nachr., 291 (2018), 1686-1711.  doi: 10.1002/mana.201600480.

[23]

K. FrączekJ. Kułaga and M. Lemańczyk, On the self-similarity problem for Gaussian-Kronecker flows, Proc. Amer. Math. Soc., 141 (2013), 4275-4291.  doi: 10.1090/S0002-9939-2013-11872-1.

[24]

K. Frączek and M. Lemańczyk, On mild mixing of special flows over irrational rotations under piecewise smooth functions, Ergodic Theory Dynam. Systems, 26 (2006), 719-738.  doi: 10.1017/S0143385706000046.

[25]

K. Frączek and M. Lemańczyk, On the self-similarity problem for ergodic flows, Proc. London Math. Soc., 99 (2009), 658-696.  doi: 10.1112/plms/pdp013.

[26]

K. Frączek and M. Schmoll, On ergodicity of foliations on $ {\mathbb{Z}}^d$-covers of half-translation surfaces and some applications to periodic systems of Eaton lenses, Comm. Math. Phys., 362 (2018), 609-657.  doi: 10.1007/s00220-018-3186-9.

[27]

K. Frączek and C. Ulcigrai, Non-ergodic $Z$-periodic billiards and infinite translation surfaces, Invent. Math., 197 (2014), 241-298.  doi: 10.1007/s00222-013-0482-z.

[28]

K. Frączek and C. Ulcigrai, Ergodic directions for billiards in a strip with periodically located obstacles, Comm. Math. Phys., 327 (2014), 643-663.  doi: 10.1007/s00220-014-2017-x.

[29]

E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.

[30]

P. W. Hooper, The invariant measures of some infinite interval exchange maps, Geom. Topol., 19 (2015), 1895-2038.  doi: 10.2140/gt.2015.19.1895.

[31]

B. Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum, Israel J. Math., 76 (1991), 289-298.  doi: 10.1007/BF02773866.

[32]

A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynam. Systems, 7 (1987), 531-557.  doi: 10.1017/S0143385700004193.

[33]

A. Kanigowski and J. Kułaga-Przymus, Ratner's property and mild mixing for smooth flows on surfaces, Ergodic Theory Dynam. Systems, 36 (2016), 2512-2537.  doi: 10.1017/etds.2015.35.

[34]

A. KanigowskiJ. Kułaga-Przymus and C. Ulcigrai, Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces, J. Eur. Math. Soc. (JEMS), 21 (2019), 3797-3855.  doi: 10.4171/JEMS/914.

[35]

A. Kanigowski and M. Lemańczyk, Flows with Ratner's property have discrete essential centralizer, Studia Math., 237 (2017), 185-194.  doi: 10.4064/sm8660-11-2016.

[36]

A. Kanigowski, M. Lemańczyk, C. Ulcigrai, On disjointness of some parabolic flows, arXiv: 1810.11576, 2018.

[37]

A. KanigowskiM. Lemańczyk and C. Ulcigrai, On disjointness properties of some parabolic flows, Invent. Math., 221 (2020), 1-111.  doi: 10.1007/s00222-019-00940-y.

[38]

I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225.  doi: 10.1007/BF01949145.

[39]

A. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR, 211 (1973), 775-778. 

[40]

A. Katok, Interval exchange transformations and some special flows are not mixing, Israel J. Math., 35 (1980), 301-310.  doi: 10.1007/BF02760655.

[41]

A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, Handbook of Dynamical Systems (eds. B. Hasselblatt and A. Katok), B1, Elsevier B. V., Amsterdam, 2006,649–743. doi: 10.1016/S1874-575X(06)80036-6.

[42]

M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31.  doi: 10.1007/BF01236981.

[43]

S. KerckhoffH. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math., 124 (1986), 293-311.  doi: 10.2307/1971280.

[44]

A. V. Kočhergin, The absence of mixing in special flows over a rotation of the circle and in flows on a two-dimensional torus, Sov. Math. Dokl., 13 (1972), 949-952. 

[45]

A. V. Kočhergin, Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces, Mat. Sb. (N.S.), 96(138) (1975), 471-502. 

[46]

A. V. Kočhergin, Nondegenerate saddle points and the absence of mixing in flows on surfaces, Proc. Steklov Inst. Math., 256 (2007), 238-252.  doi: 10.1134/S0081543807010130.

[47]

A. V. Kočhergin, Non-degenerate fixed points and mixing in flows on a 2-torus, Sb. Mat., 194 (2003), 1195-1224.  doi: 10.1070/SM2003v194n08ABEH000762.

[48]

A. G. Kušhnirenko, Spectral properties of certain dynamical systems with polynomial dispersal, Vestnik Moskov. Univ. Ser. I Mat. Meh., 29 (1974), 101-108. 

[49]

B. Marcus, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math., 105 (1977), 81-105.  doi: 10.2307/1971026.

[50]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math., 115 (1982), 169-200.  doi: 10.2307/1971341.

[51]

K. Matomäki and M. Radziwiłł, Multiplicative functions in short intervals, Ann. of Math., 183 (2016), 1015-1056.  doi: 10.4007/annals.2016.183.3.6.

[52]

K. MatomäkiM. Radziwił and T. Tao, An averaged form of Chowla's conjecture, Algebra Number Theory, 9 (2015), 2167-2196.  doi: 10.2140/ant.2015.9.2167.

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S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk, 37 (1982), 3-49. 

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M. Ratner, Rigid reparametrizations and cohomology for horocycle flows, Invent. Math., 88 (1987), 341-374.  doi: 10.1007/BF01388912.

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D. Ravotti, Quantitative mixing for locally Hamiltonian flows with saddle loops on compact surfaces, Ann. Henri Poincaré, 18 (2017), 3815-3861.  doi: 10.1007/s00023-017-0619-5.

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P. Sarnak, Three lectures on the Möbius function, randomness and dynamics., Available from: http://publications.ias.edu/sarnak/.

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L. D. Simonelli, Absolutely continuous spectrum for parabolic flows/maps, Discrete Contin. Dyn. Syst., 38 (2018), 263-292.  doi: 10.3934/dcds.2018013.

[62]

Y. G. Sina${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over i} }}}$ and K. M. Khanin, Mixing of some classes of special flows over rotations of the circle, Funct. Anal. Appl., 26 (1992), 155-169.  doi: 10.1007/BF01075628.

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R. Tiedra de Aldecoa, Spectral analysis of time changes of horocycle flows, J. Mod. Dyn., 6 (2012), 275-285.  doi: 10.3934/jmd.2012.6.275.

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C. Ulcigrai, Mixing of asymmetric logarithmic suspension flows over interval exchange transformations, Ergodic Theory Dynam. Systems, 27 (2007), 991-1035.  doi: 10.1017/S0143385706000836.

[65]

C. Ulcigrai, Weak mixing for logarithmic flows over interval exchange transformations, J. Mod. Dyn., 3 (2009), 35-49.  doi: 10.3934/jmd.2009.3.35.

[66]

C. Ulcigrai, Absence of mixing in area-preserving flows on surfaces, Ann. of Math., 173 (2011), 1743-1778.  doi: 10.4007/annals.2011.173.3.10.

[67]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., 115 (1982), 201-242.  doi: 10.2307/1971391.

show all references

References:
[1]

H. El AbdalaouiJ. Kułaga-PrzymusM. Lemańczyk and T. de la Rue, Möbius disjointness for models of an ergodic system and beyond, Israel J. Math., 228 (2018), 707-751.  doi: 10.1007/s11856-018-1784-z.

[2]

H. El AbdalaouiM. Lemańczyk and T. de la Rue, Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals, Int. Math. Res. Not. IMRN, 14 (2017), 4350-4368. 

[3]

V. I. Arnol'd, Topological and ergodic properties of closed 1-forms with incommensurable periods, Funct. Anal. Appl., 25 (1991), 81-90.  doi: 10.1007/BF01079587.

[4]

A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation flows, Ann. of Math., 165 (2007), 637-664.  doi: 10.4007/annals.2007.165.637.

[5]

A. AvilaG. Forni and C. Ulcigrai, Mixing for time-changes of Heisenberg nilflows, J. Differential Geom., 89 (2011), 369-410. 

[6]

A. Avila, G. Forni, D. Ravotti and C. Ulcigrai, Mixing for smooth time-changes of general nilflows, Adv. Math., 385 (2021), 107759, 65 pp. doi: 10.1016/j.aim.2021.107759.

[7]

A. Avila and P. Hubert, Recurrence for the wind-tree model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 1-11.  doi: 10.1016/j.anihpc.2017.11.006.

[8]

J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Moebius from horocycle flows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., 28, Springer, New York, 2013, 67–83. doi: 10.1007/978-1-4614-4075-8_5.

[9]

A. Bufetov and G. Forni, Limit theorems for horocycle flows, Ann. Sci. Éc. Norm. Supér, 47 (2014), 851-903. 

[10]

J. Chaika and A. Eskin, Every flat surface is Birkhoff and Oseledets generic in almost every direction, J. Mod. Dyn., 9 (2015), 1-23.  doi: 10.3934/jmd.2015.9.1.

[11]

J. ChaikaK. FrączekA. Kanigowski and C. Ulcigrai, Singularity of the spectrum for smooth area-preserving flows in genus two and translation surfaces well approximated by cylinders, Comm. Math. Phys., 381 (2021), 1369-1407.  doi: 10.1007/s00220-020-03895-x.

[12]

J. Chaika and A. Wright, A smooth mixing flow on a surface with nondegenerate fixed points, J. Amer. Math. Soc., 32 (2019), 81-117.  doi: 10.1090/jams/911.

[13]

A. I. Danilenko and V. V. Ryzhikov, On self-similarities of ergodic flows, Proc. Lond. Math. Soc., 104 (2012), 431-454.  doi: 10.1112/plms/pdr032.

[14]

A. EskinM. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes Études Sci., 120 (2014), 207-333.  doi: 10.1007/s10240-013-0060-3.

[15]

B. FayadG. Forni and A. Kanigowski, Lebesgue spectrum of countable multiplicity for conservative flows on the torus, J. Amer. Math. Soc., 34 (2021), 747-813.  doi: 10.1090/jams/970.

[16]

B. Fayad and A. Kanigowski, Multiple mixing for a class of conservative surface flows, Invent. Math., 203 (2016), 555-614.  doi: 10.1007/s00222-015-0596-6.

[17]

S. FerencziJ. Kułaga-Przymus and M. Lemańczyk, Sarnak's conjecture: What's new, Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, 2213 (2018), 163-235. 

[18]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8.

[19]

L. Flaminio and G. Forni, Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows, Electron. Res. Announc. Math. Sci., 26 (2019), 16-23.  doi: 10.3934/era.2019.26.002.

[20]

L. FlaminioK. FrączekJ. Kułaga-Przymus and M. Lemańczyk, Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds, Studia Math., 244 (2019), 43-97.  doi: 10.4064/sm170512-25-9.

[21]

G. Forni and C. Ulcigrai, Time-changes of horocycle flows, J. Mod. Dyn., 6 (2012), 251-273.  doi: 10.3934/jmd.2012.6.251.

[22]

K. Frączek and P. Hubert, Recurrence and non-ergodicity in generalized wind-tree models, Math. Nachr., 291 (2018), 1686-1711.  doi: 10.1002/mana.201600480.

[23]

K. FrączekJ. Kułaga and M. Lemańczyk, On the self-similarity problem for Gaussian-Kronecker flows, Proc. Amer. Math. Soc., 141 (2013), 4275-4291.  doi: 10.1090/S0002-9939-2013-11872-1.

[24]

K. Frączek and M. Lemańczyk, On mild mixing of special flows over irrational rotations under piecewise smooth functions, Ergodic Theory Dynam. Systems, 26 (2006), 719-738.  doi: 10.1017/S0143385706000046.

[25]

K. Frączek and M. Lemańczyk, On the self-similarity problem for ergodic flows, Proc. London Math. Soc., 99 (2009), 658-696.  doi: 10.1112/plms/pdp013.

[26]

K. Frączek and M. Schmoll, On ergodicity of foliations on $ {\mathbb{Z}}^d$-covers of half-translation surfaces and some applications to periodic systems of Eaton lenses, Comm. Math. Phys., 362 (2018), 609-657.  doi: 10.1007/s00220-018-3186-9.

[27]

K. Frączek and C. Ulcigrai, Non-ergodic $Z$-periodic billiards and infinite translation surfaces, Invent. Math., 197 (2014), 241-298.  doi: 10.1007/s00222-013-0482-z.

[28]

K. Frączek and C. Ulcigrai, Ergodic directions for billiards in a strip with periodically located obstacles, Comm. Math. Phys., 327 (2014), 643-663.  doi: 10.1007/s00220-014-2017-x.

[29]

E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.

[30]

P. W. Hooper, The invariant measures of some infinite interval exchange maps, Geom. Topol., 19 (2015), 1895-2038.  doi: 10.2140/gt.2015.19.1895.

[31]

B. Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum, Israel J. Math., 76 (1991), 289-298.  doi: 10.1007/BF02773866.

[32]

A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynam. Systems, 7 (1987), 531-557.  doi: 10.1017/S0143385700004193.

[33]

A. Kanigowski and J. Kułaga-Przymus, Ratner's property and mild mixing for smooth flows on surfaces, Ergodic Theory Dynam. Systems, 36 (2016), 2512-2537.  doi: 10.1017/etds.2015.35.

[34]

A. KanigowskiJ. Kułaga-Przymus and C. Ulcigrai, Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces, J. Eur. Math. Soc. (JEMS), 21 (2019), 3797-3855.  doi: 10.4171/JEMS/914.

[35]

A. Kanigowski and M. Lemańczyk, Flows with Ratner's property have discrete essential centralizer, Studia Math., 237 (2017), 185-194.  doi: 10.4064/sm8660-11-2016.

[36]

A. Kanigowski, M. Lemańczyk, C. Ulcigrai, On disjointness of some parabolic flows, arXiv: 1810.11576, 2018.

[37]

A. KanigowskiM. Lemańczyk and C. Ulcigrai, On disjointness properties of some parabolic flows, Invent. Math., 221 (2020), 1-111.  doi: 10.1007/s00222-019-00940-y.

[38]

I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225.  doi: 10.1007/BF01949145.

[39]

A. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR, 211 (1973), 775-778. 

[40]

A. Katok, Interval exchange transformations and some special flows are not mixing, Israel J. Math., 35 (1980), 301-310.  doi: 10.1007/BF02760655.

[41]

A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, Handbook of Dynamical Systems (eds. B. Hasselblatt and A. Katok), B1, Elsevier B. V., Amsterdam, 2006,649–743. doi: 10.1016/S1874-575X(06)80036-6.

[42]

M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31.  doi: 10.1007/BF01236981.

[43]

S. KerckhoffH. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math., 124 (1986), 293-311.  doi: 10.2307/1971280.

[44]

A. V. Kočhergin, The absence of mixing in special flows over a rotation of the circle and in flows on a two-dimensional torus, Sov. Math. Dokl., 13 (1972), 949-952. 

[45]

A. V. Kočhergin, Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces, Mat. Sb. (N.S.), 96(138) (1975), 471-502. 

[46]

A. V. Kočhergin, Nondegenerate saddle points and the absence of mixing in flows on surfaces, Proc. Steklov Inst. Math., 256 (2007), 238-252.  doi: 10.1134/S0081543807010130.

[47]

A. V. Kočhergin, Non-degenerate fixed points and mixing in flows on a 2-torus, Sb. Mat., 194 (2003), 1195-1224.  doi: 10.1070/SM2003v194n08ABEH000762.

[48]

A. G. Kušhnirenko, Spectral properties of certain dynamical systems with polynomial dispersal, Vestnik Moskov. Univ. Ser. I Mat. Meh., 29 (1974), 101-108. 

[49]

B. Marcus, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math., 105 (1977), 81-105.  doi: 10.2307/1971026.

[50]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math., 115 (1982), 169-200.  doi: 10.2307/1971341.

[51]

K. Matomäki and M. Radziwiłł, Multiplicative functions in short intervals, Ann. of Math., 183 (2016), 1015-1056.  doi: 10.4007/annals.2016.183.3.6.

[52]

K. MatomäkiM. Radziwił and T. Tao, An averaged form of Chowla's conjecture, Algebra Number Theory, 9 (2015), 2167-2196.  doi: 10.2140/ant.2015.9.2167.

[53]

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