Analytic maps of parabolic and elliptic type with trivial centralisers

We prove that for a dense set of irrational numbers α\alphaα, the analytic centraliser of the map e2πiαz+z2e^{2\pi i \alpha} z+ z^2e2πiαz+z2 near 000 is trivial. We also prove that some analytic circle diffeomorphisms in the Arnol’d family, with irrational rotation numbers, have trivial centralisers. These provide the first examples of such maps with trivial centralisers