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The centralizer of Komuro-expansive flows and expansive \({{\mathbb {R}}}^d\) actions

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Abstract

In this paper we study the centralizer of flows and \({{\mathbb {R}}}^d\)-actions on compact Riemannian manifolds. We prove that the centralizer of every \(C^\infty \) Komuro-expansive flow with non-resonant singularities is trivial, meaning it is the smallest possible, and deduce there exists an open and dense subset of geometric Lorenz attractors with trivial centralizer. We show that \({{\mathbb {R}}}^d\)-actions obtained as suspension of \({\mathbb {Z}}^d\)-actions are expansive if and only if the same holds for the \({\mathbb {Z}}^d\)-actions. We also show that homogeneous expansive \({{\mathbb {R}}}^d\)-actions have quasi-trivial centralizers, meaning that it consists of orbit invariant, continuous linear reparameterizations of the \({{\mathbb {R}}}^d\)-action. In particular, homogeneous Anosov \({{\mathbb {R}}}^d\)-actions have quasi-trivial centralizer.

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References

  1. Afraimovich, V.S., Bykov, V.V., Shilnikov, L.P.: On the appearence and structure of the Lorenz attractor. Dokl. Acad. Sci. USSR 234, 336–339 (1977)

    Google Scholar 

  2. Araújo, V., Pacífico, M.J.: Three-Dimensional Flows. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  3. Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer, New York (1991)

    Google Scholar 

  4. Artigue, A.: Kinematic expansive flows. Ergod. Theory Dyn. Syst. 36(2), 390–421 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bakker, L., Fisher, T.: Open sets of diffeomorphisms with trivial centralizer in the \(C^1\)-topology. Nonlinearity 27, 2869–2885 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Banyaga, A., de la Llave, R., Wayne, C.E.: Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem. J. Geom. Anal. 6(4), 613–649 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Barbot, T., Maquera, C.: Transitivity of codimension one Anosov actions of \({{\mathbb{R}}}^k\) on closed manifolds. Ergod. Theory Dyn. Syst. 31(1), 1–22 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bonatti, C., Crovisier, S., G., Vago, Wilkinson, A.: Local density of diffeomorphisms with large centralizers. Ann. Sci. ENS. Ser. 4 41(6), 925–954 (2008). (Fascicule)

    MathSciNet  MATH  Google Scholar 

  9. Bonatti, C., Crovisier, S., Wilkinson, A.: The \(C^1\) generic diffeomorphism has trivial centralizer. Publ. Math. 109(1), 185–244 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bonatti, C., Bohnet, D.: Partially hyperbolic diffeomorphisms with a uniformly compact center foliation: the quotient dynamics. Ergod. Theory Dyn., Syst (2015)

    MATH  Google Scholar 

  11. Boyle, M., Lind, D.: Expansive subdynamics. Trans. Am. Math. Soc. 349, 55–102 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bowen, R., Walters, P.: Expansive one-parameter flows. J. Differ. Equ. 12, 180–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  13. Burslem, L.: Centralizers of partially hyperbolic diffeomorphisms. Ergod. Theory Dync. Syst. 24, 55–87 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Einsiedler, M., Lind, D., Miles, R., Ward, T.: Expansive subdynamics of algebraic \({\mathbb{Z}}^d\)-actions. Ergod. Theory Dyn. Syst. 21(6), 1695–1729 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fisher, T.: Trivial centralizers for codimension-one attractors. Bull. Lond. Math. Soc. 41(1), 51–56 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guckenheimer, J., Williams, R.F.: Structural stability of Lorenz attractors. Publ. Math. IHES 50, 59–72 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hayashi, S.: Connecting invariant manifolds and the solution of the \(C^1\) stability and \({\varOmega }\)-stability conjectures for flows. Ann. Math. (2) 145(1), 81–137 (1997)

    Article  MathSciNet  Google Scholar 

  18. Hirsch, M., Pugh, C., Shub, M.: Invariant Manifolds, Lect. Notes in Math, vol. 583. Springer, New York (1977)

    Book  Google Scholar 

  19. Kato, K., Morimoto, M.: Topological stability of Anosov flows and their centralizers. Topology 12, 255–273 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  20. Keynes, H., Sears, M.: F-expansive transformation groups. Gen. Topol. Appl. 10(1), 67–85 (1979)

    Article  MATH  Google Scholar 

  21. Komuro, M.: Expansive Properties of Lorenz Attractors: the Theory of Dynamical Systems and its Applications to Nonlinear Problems, pp. 4–26. World Scientific, Singapore (1984)

    Google Scholar 

  22. Kopell, N.: Commuting Diffeomorphisms. In: Global analysis, proceedings of symposia in pure mathematics XIV, Berkeley, CA, pp. 165–184 (1968)

  23. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  MATH  Google Scholar 

  24. Jungreis, D., Hirsch, M.: Rigidity of centralizers of Anosov flows. Int. J. Math. 2(1), 37–40 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lacerda, L.S.: Expansividade, especificação e sensibilidade em fluxos com singularidades. PhD Thesis. Instituto de Matemática, UFRJ, Rio de Janeiro (2010)

  26. Lee, J.M.: Introduction to Smooth Manifolds, 2nd edn. Springer, Berlin (2003)

    Book  Google Scholar 

  27. Maquera, C., Tahzibi, A.: Robustly transitive actions of \({\mathbb{R}}^2\) on compact three manifolds. Bull. Braz. Math. Soc. New Ser. 38(2), 189–201 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mañé, R.: A proof of the \(C^1\) stability conjecture. Inst. Hautes Études Sci. Publ. Math. 66, 161–210 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  29. Morales, C., Pacífico, M., Pujals, E.: Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers. Ann. Math. 160, 375–432 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Oka, M.: Expansive flows and their centralizers. Nagoya Math. J. 64, 1–15 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  31. Oka, M.: Expansiveness of real flows. Tsukuba J. Math. 14(1), 1–8 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  32. Palis Jr., J., de Melo, W.: Geometric Theory of Dynamical Systems: An Introduction. Springer, Berlin (1982)

    Book  MATH  Google Scholar 

  33. Palis Jr., J.: Vector fields generate few diffeomorphisms. Bull. Am. Math. Soc 80(3), 503–505 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  34. Palis, J., Yoccoz, J.-C.: Rigidity of centralizers of diffeomorphisms. Ann. Sci. École Norm. Sup. 4(22), 81–98 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pugh, C.: Real Mathematical Analysis, Undergraduate Texts in Mathematics. Springer, New York (2002)

    Book  Google Scholar 

  36. Rocha, J.: Rigidity of the \(C^1\)-centralizer of bidimensional diffeomorphisms. In: Dynamical systems (Santiago, 1990), Pitman Research Notes in Mathematics Series, vol. 285, pp. 211–229 . Longman Sci. Tech., Harlow (1993)

  37. Rocha, J.: A note on the \(C^0\)-centralizer of an open class of bidimensional Anosov diffeomorphisms. Ergod. Theory Dyn. Syst. 24, 55–87 (2005)

    Google Scholar 

  38. Sad, P.: Centralizers of vector fields. Topology 18(2), 97–104 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  39. Smale, S.: Dynamics retrospective: great problems, attempts that failed. Phys. D. 51, 267–273 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  40. Smale, S.: Mathematics: Frontiers and Perspectives, Mathematical Problems for the Next Century, pp. 271–294. American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  41. Sternberg, S.: Local contractions and a theorem of Poincaré. Am. J. Math. 79, 809–824 (1957)

    Article  MATH  Google Scholar 

  42. Walters, P.: Homeomorphisms with discrete centralizers and ergodic properties. Math. Syst. Theory 4(4), 322–326 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yoccoz, J.C.: Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Univ. Paris-Sud, Thèse (1985)

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Acknowledgements

The authors are deeply grateful to the anonymous referee for very careful reading of the manuscript and useful advices that helped to improve the manuscript. This work is part of the first author’s PhD thesis at UFBA. W.B. and P.V. were partially supported by BREUDS. J.R. and P.V. were partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds (FEDER), under the partnership agreement PT2020. W.B and P.V. and grateful to Universidade do Porto, where this work was developed, for the warm hospitality and excellent research conditions.

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Authors and Affiliations

  1. Universidade Federal do Espírito Santo, CEUNES, Rodovia Governador Mario Covas, Km 60, 29.932-900, São Mateus, Brazil

    Wescley Bonomo

  2. Departamento de Matemática, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007, Porto, Portugal

    Jorge Rocha

  3. Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110, Salvador, Brazil

    Paulo Varandas

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  1. Wescley Bonomo
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  2. Jorge Rocha
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  3. Paulo Varandas
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Correspondence to Paulo Varandas.

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Bonomo, W., Rocha, J. & Varandas, P. The centralizer of Komuro-expansive flows and expansive \({{\mathbb {R}}}^d\) actions. Math. Z. 289, 1059–1088 (2018). https://doi.org/10.1007/s00209-017-1988-7

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  • DOI: https://doi.org/10.1007/s00209-017-1988-7

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