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The centralizer of $ C^r$-generic diffeomorphisms at hyperbolic basic sets is trivial


Authors: Jorge Rocha and Paulo Varandas
Journal: Proc. Amer. Math. Soc. 146 (2018), 247-260
MSC (2010): Primary 37D20, 37C20, 37C15; Secondary 37F15, 37C05
DOI: https://doi.org/10.1090/proc/13712
Published electronically: July 27, 2017
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Abstract: In the late nineties, Smale proposed a list of problems for the next century, and, among these, it was conjectured that for every $ r\ge 1$ a $ C^r$-generic diffeomorphism has trivial centralizer. Our contribution here is to prove the triviality of $ C^r$-centralizers on hyperbolic basic sets. In particular, $ C^r$-generic transitive Anosov diffeomorphisms have a trivial $ C^1$-centralizer. These results follow from a more general criterium for expansive homeomorphisms with the gluing orbit property. We also construct a linear Anosov diffeomorphism on $ \mathbb{T}^3$ with discrete, non-trivial centralizer and with elements that are not roots. Finally, we prove that all elements in the centralizer of an Anosov diffeomorphism preserve some of its maximal entropy measures, and use this to characterize the centralizer of linear Anosov diffeomorphisms on tori.


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Additional Information

Jorge Rocha
Affiliation: Departamento de Matemática, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
Email: jrocha@fc.up.pt

Paulo Varandas
Affiliation: Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil
Email: paulo.varandas@ufba.br

DOI: https://doi.org/10.1090/proc/13712
Keywords: Centralizers, Anosov diffeomorphisms, uniform hyperbolicity, generic properties
Received by editor(s): December 8, 2016
Received by editor(s) in revised form: February 20, 2017
Published electronically: July 27, 2017
Additional Notes: JR was partially supported by CMUP (UID/MAT/00144/2013) and PTDC/MAT-CAL/3884/2014, which are funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020.
Communicated by: Nimish Shah
Article copyright: © Copyright 2017 American Mathematical Society

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