Spectral Properties Of Renormalization For Area-Preserving Maps
2016 (English)In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 36, no 7, 3651-3675 p.Article in journal (Refereed) PublishedText
Area-preserving maps have been observed to undergo a universal period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser period doubling cascade in one-dimensional dynamics. A renormalization approach has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of existence of a conservative renormalization fixed point. Furthermore, it has been shown by Gaidashev, Johnson and Martens that infinitely renormalizable maps in a neighborhood of this fixed point admit invariant Cantor sets with vanishing Lyapunov exponents on which dynamics for any two maps is smoothly conjugate. This rigidity is a consequence of an interplay between the decay of geometry and the convergence rate of renormalization towards the fixed point. In this paper we prove a result which is crucial for a demonstration of rigidity: that an upper bound on this convergence rate of renormalizations of infinitely renormalizable maps is sufficiently small.
Place, publisher, year, edition, pages
2016. Vol. 36, no 7, 3651-3675 p.
Renormalization, area-preserving maps, period-doubling, hyperbolicty, computer-assited proof, rigidity
IdentifiersURN: urn:nbn:se:uu:diva-283745DOI: 10.3934/dcds.2016.36.3651ISI: 000371999400008OAI: oai:DiVA.org:uu-283745DiVA: diva2:919727