BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Andrew Burbanks (University of Portsmouth\, UK) DTSTART:20210629T140000Z DTEND:20210629T150000Z DTSTAMP:20260423T005823Z UID:CRM-CAMP/50 DESCRIPTION:Title: Computer-assisted proofs for renormalisation fixed-points and eigenfunct ions for period-doubling universality in maps of the interval\nby Andr ew Burbanks (University of Portsmouth\, UK) as part of CRM CAMP (Computer- Assisted Mathematical Proofs) in Nonlinear Analysis\n\n\nAbstract\nWe prov e the existence of a fixed point to the renormalisation operator for perio d doubling in maps of even degree at the critical point. We work with a mo dified operator that encodes the action of the renormalisation operator on even functions. Building on previous work\, our proof uses rigorous compu ter-assisted means to bound operations in a space of analytic functions an d hence to show that a quasi-Newton operator for the fixed-point problem i s a contraction map on a suitable ball.\n\nWe bound the spectrum of the Fr echet derivative of the renormalisation operator at the fixed point\, esta blishing the hyperbolic structure\, in which the presence of a single esse ntial expanding eigenvalue explains the universal asymptotically self-simi lar bifurcation structure observed in the iterations of families of maps l ying in the relevant universality class.\n\nBy recasting the eigenproblem for the Frechet derivative in a particular nonlinear form\, we again use t he 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.\n\nOur computations use multi-precision interval arit hmetic with rigorous directed rounding modes to bound tightly the coeffici ents of the relevant power series and their high-order terms\, and the cor responding universal constants.\n LOCATION:https://researchseminars.org/talk/CRM-CAMP/50/ END:VEVENT END:VCALENDAR