Abstract:
A sufficient condition for mixing for special flows is obtained. This condition is used for the study of smooth flows on surfaces. It is proved that on any compact surface of genus p⩾1, different from the Klein bottle, there exists a smooth mixing flow with a smooth invariant measure.
Bibliography: 6 titles.
Citation:
A. V. Kochergin, “On mixing in special flows over a shifting of segments and in smooth flows on surfaces”, Math. USSR-Sb., 25:3 (1975), 441–469
\Bibitem{Koc75}
\by A.~V.~Kochergin
\paper On mixing in special flows over a~shifting of segments and in smooth flows on surfaces
\jour Math. USSR-Sb.
\yr 1975
\vol 25
\issue 3
\pages 441--469
\mathnet{http://mi.mathnet.ru/eng/sm3402}
\crossref{https://doi.org/10.1070/SM1975v025n03ABEH002217}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=516507}
\zmath{https://zbmath.org/?q=an:0321.28012}
Linking options:
https://www.mathnet.ru/eng/sm3402
https://doi.org/10.1070/SM1975v025n03ABEH002217
https://www.mathnet.ru/eng/sm/v138/i3/p471
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Adam Kanigowski, Mariusz Lemańczyk, Encyclopedia of Complexity and Systems Science Series, Ergodic Theory, 2023, 109
Jon Chaika, Krzysztof Frączek, Adam Kanigowski, Corinna Ulcigrai, “Singularity of the Spectrum for Smooth Area-Preserving Flows in Genus Two and Translation Surfaces Well Approximated by Cylinders”, Commun. Math. Phys., 381:3 (2021), 1369
Adam Kanigowski, Mariusz Lemańczyk, Encyclopedia of Complexity and Systems Science, 2020, 1
Chaika J., Wright A., “A Smooth Mixing Flow on a Surface With Nondegenerate Fixed Points”, J. Am. Math. Soc., 32:1 (2019), 81–117
Kanigowski A. Kulaga-Przymus J. Ulcigrai C., “Multiple Mixing and Parabolic Divergence in Smooth Area-Preserving Flows on Higher Genus Surfaces”, J. Eur. Math. Soc., 21:12 (2019), 3797–3855
Conze J.-P., Lemanczyk M., “Centralizer and Liftable Centralizer of Special Flows Over Rotations”, Nonlinearity, 31:8 (2018), 3939–3972
Ravotti D., “Quantitative Mixing For Locally Hamiltonian Flows With Saddle Loops on Compact Surfaces”, Ann. Henri Poincare, 18:12 (2017), 3815–3861
Fayad B., Kanigowski A., “Multiple mixing for a class of conservative surface flows”, Invent. Math., 203:2 (2016), 555–614
Fraczek K., Ulcigrai C., “Ergodic Properties of Infinite Extensions of Area-Preserving Flows”, Math. Ann., 354:4 (2012), 1289–1367
Mariusz Lemańczyk, Mathematics of Complexity and Dynamical Systems, 2012, 1618
Ulcigrai C., “Absence of mixing in area-preserving flows on surfaces”, Ann of Math (2), 173:3 (2011), 1743–1778
Avila A., Forni G., Ulcigrai C., “Mixing for Time-Changes of Heisenberg Nilflows”, J. Differ. Geom., 89:3 (2011), 369–410
Fraczek K., Lemanczyk M., “Ratner's Property and Mild Mixing for Special Flows Over Two-Dimensional Rotations”, J Mod Dyn, 4:4 (2010), 609–635
Scheglov D., “Absence of Mixing for Smooth Flows on Genus Two Surfaces”, J. Mod. Dyn., 3:1 (2009), 13–34
Mariusz Lemańczyk, Encyclopedia of Complexity and Systems Science, 2009, 8554
A. V. Kochergin, “Nondegenerate Saddles and Nonmixing Transformations II”, Math. Notes, 81:1 (2007), 126–129
A. V. Kochergin, “Nondegenerate Saddle Points and the Absence of Mixing in Flows on Surfaces”, Proc. Steklov Inst. Math., 256 (2007), 238–252
Ulcigrai, C, “Mixing of asymmetric logarithmic suspension flows over interval exchange transformations”, Ergodic Theory and Dynamical Systems, 27 (2007), 991
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