The centralizer of $C^r$-generic diffeomorphisms at hyperbolic basic sets is trivial
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- by Jorge Rocha and Paulo Varandas PDF
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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.References
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Additional Information
- Jorge Rocha
- Affiliation: Departamento de Matemática, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
- MR Author ID: 254453
- 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
- MR Author ID: 857790
- Email: paulo.varandas@ufba.br
- 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
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3723137