  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.
