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We gain tight rigorous bounds on the renormalisation fixed point function forperiod doubling in families of unimodal maps with degree 2 critical point. Bywriting the relevant eigenproblems in a modified nonlinear form, we use thesebounds, together with a contraction mapping argument, to gain tight bounds onthe essential eigenvalues and eigenfunctions of the linearised renormalisationoperator at the fixed point and also those of the operator encoding theuniversal scaling of added uncorrelated noise. We gain bounds on thecorresponding power series coefficients and universal constants accurate toover 400 significant figures, confirming and (in the case of noise) extendingthe accuracy of previous numerical estimates, by using multi-precision intervalarithmetic with rigorous directed rounding to implement operations on a spaceof analytic functions.

Rigorous computer-assisted bounds on renormalisation fixed point functions, eigenfunctions, and universal constants