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

Mathematische Zeitschrift volume 289, pages 1059–1088 (2018)Cite this article

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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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

Authors
  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

Keywords

  • Expansive flows
  • Trivial centralizers
  • \({{\mathbb {R}}}^d\)-actions
  • Singular-hyperbolicity
  • Geometric Lorenz attractors

Mathematics Subject Classification

  • 37C10
  • 37D20
  • 37D25
  • 37C85