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

Analytic maps of parabolic and elliptic type with trivial centralisers