Computer-assisted proofs for renormalisation fixed-points and eigenfunctions for period-doubling universality in maps of the interval

Abstract: We prove the existence of a fixed point to the renormalisation operator for period doubling in maps of even degree at the critical point. We work with a modified operator that encodes the action of the renormalisation operator on even functions. Building on previous work, our proof uses rigorous computer-assisted means to bound operations in a space of analytic functions and hence to show that a quasi-Newton operator for the fixed-point problem is a contraction map on a suitable ball.

We bound the spectrum of the Frechet derivative of the renormalisation operator at the fixed point, establishing the hyperbolic structure, in which the presence of a single essential expanding eigenvalue explains the universal asymptotically self-similar bifurcation structure observed in the iterations of families of maps lying in the relevant universality class.

By recasting the eigenproblem for the Frechet derivative in a particular nonlinear form, we again use the contraction mapping principle to gain rigorous bounds on eigenfunctions and their corresponding eigenvalues. In particular, we gain tight bounds on the eigenfunction corresponding to the essential expanding eigenvalue delta. We adapt the procedure to the eigenproblem for the scaling of added uncorrelated noise.

Our computations use multi-precision interval arithmetic with rigorous directed rounding modes to bound tightly the coefficients of the relevant power series and their high-order terms, and the corresponding universal constants.

analysis of PDEsclassical analysis and ODEsdynamical systemsfunctional analysisnumerical analysis

Audience: researchers in the discipline

CRM CAMP (Computer-Assisted Mathematical Proofs) in Nonlinear Analysis

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