  We prove that for a dense set of irrational numbers $\alpha$, the analytic
centraliser of the map $e^{2\pi i \alpha} z+ z^2$ near $0$ is trivial. We also
prove that some analytic circle diffeomorphisms in the Arnold family, with
irrational rotation numbers, have trivial centralisers. These provide the first
examples of such maps with trivial centralisers.
