Chaos in Topological Foliations

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We call a foliation (M,F) on a manifold M chaotic if it is topologically transitive and the union of closed leaves is dense in M. A foliated manifold M is not assumed to be compact. The chaotic foliations can be considered as multidimensional generalization of chaotic dynamical systems in the sense of Devaney. For foliations covered by fibrations we prove that a foliation is chaotic if and only if its global holonomy group is chaotic. We introduce the concept of the integrable Ehresmann connection for a foliation as a natural generalization of the integrable Ehresmann connection for smooth foliations. A description of the global structure of foliations with integrable Ehresmann connection and a criterion for the chaotic behavior of such foliations are obtained. Applying the method of suspension, a new countable family of pairwise nonisomorphic chaotic foliations of codimension two on 3-dimensional closed and nonclosed manifolds is constructed.

About the authors

N. I. Zhukova

HSE University

Nizhny Novgorod, Russia

G. S. Levin

HSE University

Nizhny Novgorod, Russia

N. S. Tonysheva

HSE University

Author for correspondence.
Nizhny Novgorod, Russia


  1. Жукова Н.И. Глобальные аттракторы полных конформных слоений // Мат. сб.- 2012.- 203, № 3.- C. 79-106.
  2. Жукова Н.И., Рогожина Е.А. Классификация компактных лоренцевых 2-орбифолдов с некомпактной полной группой изометрий// Сиб. мат. ж.- 2012.- 53, № 6.- C. 1292-1309.
  3. Жукова Н.И., Чебочко Н.Г. Структура лоренцевых слоений коразмерности два// Изв. вузов. Сер. Мат.-2020.-64, № 11.- C. 87-92.
  4. Жукова Н.И., Шеина К.И. Структура слоений с интегрируемой связностью Эресмана// Уфимск. мат. ж. -2022.- 14, № 1.-С. 23-40.
  5. Шапиро Я.Л. О приводимых римановых многообразиях в целом // Изв. вузов. Сер. Мат.- 1972.- № 6. -C. 78-85.
  6. Шапиро Я.Л. О двулистной структуре на приводимом римановом многообразии// Изв. вузов. Сер. Мат.-1972.-№ 12.- C. 102-110.
  7. Шапиро Я.Л., Жукова Н.И. О простых двуслоениях// Изв. вузов. Сер. Мат.- 1976.- № 4.-C. 95-104.
  8. Assaf D. IV, Gadbois S. Definition of chaos// Am. Math. Monthly.- 1992.- 99, № 9.-C. 865.
  9. Banks J., Brooks J., Cairns G., Davis G., Stacey P. On Devaney’s definition of chaos// Am. Math. Monthly.- 1992.-99, № 4.- С. 332-334.
  10. Bazaikin Y.V., Galaev A.S., Zhukova N.I. Chaos in Cartan foliations// Chaos.-2020.-30, № 10.- C. 1-9.
  11. Blumenthal R.A., Hebda J.J. Ehresmann connection for foliations// Indiana Univ. Math. J. -1984.- 33, № 4. -C. 597-611.
  12. Cairns G., Davis G., Elton E., Kolganova A., Perversi P. Chaotic group actions// Enseign. Math.- 1995.-41.-C. 123-133.
  13. Cairns G., Kolganova A., Nielsen A. Topological transitivity and mixing notions for group actions// Rocky Mountain J. Math.- 2007.- 37, № 2.-C. 371-397.
  14. Candel A., Conlon L. Foliations I.- Providence: Amer. Math. Soc., 2000.
  15. Churchill R.C. On defining chaos in absent of time// В сб.: «Deterministic Chaos in General Relativity».- 1994.-C. 107-112.
  16. Devaney R.L. An Introduction to Chaotic Dynamical Systems. -Menlo Park, etc.: The Benjamin/ Cummings Publishing Co., Inc., 1986.
  17. Grosse-Erdmann K.-G., Manguillot A.P. Linear Chaos.-London: Springer, 2011.
  18. Kashiwabara S. The decomposition of differential manifolds and its applications// Tohoku Math. J.- 1959.-11.-C. 43-53.
  19. Kervaire M.A. A manifold which does not admit any differentiable structure// Comment. Math. Helv.- 1960.-34, № 1.- C. 257-270.
  20. Kobayashi S., Nomizu K. Foundations of Differential Geometry. I. -New York-London: Interscience Publishers, 1963.
  21. Manolescu C. Four-dimensional topology.-∼cm5/4D.pdf.
  22. Polo F. Sensitive dependence on initial conditions and chaotic group actions// Proc. Am. Math. Soc.- 2010.-138, № 8.-C. 2815-2826.
  23. Reeb G. Sur la theorie generale des systemes dynamiques// Ann. Inst. Fourier (Grenoble). -1955.- 6.- C. 89-115.
  24. Suda T. Poincare maps and suspension flows: A categorical remarks// ArXive.- 2021.- 2107.06567 [math.DS].
  25. Tamura I. Topology of Foliations: An Introduction.- Providence: AMS, 1992.
  26. Thurston W.P. Three-Dimensional Geometry and Topology.- Prinston: Prinston Univ. Press, 1997.
  27. Vaisman I. On some spaces which are covered by a product space// Ann. Inst. Fourier (Grenoble).- 1977.-27, № 1.- C. 107-134.
  28. Zhukova N.I., Chubarov G.V. Aspects of the qualitative theory of suspended foliations// J. Differ. Equ. Appl. - 2003.- 9, № 3-4.-C. 393-405.
  29. Zhukova N.I., Chubarov G.V. Structure of graphs of suspended foliations// J. Math. Sci. (N.Y.) - 2022.- 261, № 3.-C. 410-425.

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