On analytic perturbations of a family of Feigenbaum-like equations
2011 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 374, no 2, 355-373 p.Article in journal (Refereed) Published
We prove existence of solutions (0, A.) of a family of Feigenbaum-like equations phi(x) = 1+is an element of/lambda phi(phi(lambda x))- is an element of x + tau(x), (0.1) where epsilon is a small real number and tau is analytic and small on some complex neighborhood of (-1, 1) and real-valued on R. The family (0.1) appears in the context of period-doubling renormalization for area-preserving maps (cf. Gaidashev and Koch (preprint) 171). Our proof is a development of ideas of H. Epstein (cf. Epstein (1986) PI, Epstein (1988) In Epstein (1989) ) adopted to deal with some significant complications that arise from the presence of the terms -epsilon X + tau(x) in Eq. (0.1). The method relies on a construction of novel a-priori bounds for unimodal functions which turn out to be very tight. We also obtain good bounds on the scaling parameter lambda. A byproduct of the method is a new proof of the existence of a Feigenbaum-Coullet-Tresser function.
Place, publisher, year, edition, pages
2011. Vol. 374, no 2, 355-373 p.
Unimodal maps, Period-doubling, Composition operators, Renormalization, Herglotz functions, A-priori bounds
IdentifiersURN: urn:nbn:se:uu:diva-135342DOI: 10.1016/j.jmaa.2010.06.047ISI: 000283965000002OAI: oai:DiVA.org:uu-135342DiVA: diva2:375477