THE CENTRALIZER OF C-r-GENERIC DIFFEOMORPHISMS AT HYPERBOLIC BASIC SETS IS TRIVIAL

• Type: Article in International Scientific Journal
• Date: 2018

Authors

Jorge Rocha (Author) FCUP; Varandas, P (Author) FCUP

Journal

Title: Proceedings of the American Mathematical Society

Vol. 146

Pages: 247-260

ISSN: 0002-9939

Publisher: American Mathematical Society

Indexing

ISI Web of Knowledge - 6 Citations

Scopus - 6 Citations

Other information

Authenticus ID: P-00N-8RG

DOI: 10.1090/proc/13712

Abstract (EN): In the late nineties, Smale proposed a list of problems for the next century, and, among these, it was conjectured that for every r >= 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 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.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 14