Period Doubling Renormalization for Area-Preserving Maps and Mild Computer Assistance in Contraction Mapping Principle

It has been observed that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of {\fR}^2. A renormalization approach has been used in a "hard" computer-assisted proof of existence of an area-preserving map with orbits of all binary periods in Eckmann et al (1984). As it is the case with all non-trivial universality problems in non-dissipative systems in dimensions more than one, no analytic proof of this period doubling universality exists to date. In this paper we attempt to reduce computer assistance in the argument, and present a mild computer aided proof of the analyticity and compactness of the renormalization operator in a neighborhood of a renormalization fixed point: that is a proof that does not use generalizations of interval arithmetics to functional spaces - but rather relies on interval arithmetics on real numbers only to estimate otherwise explicit expressions. The proof relies on several instance of the Contraction Mapping Principle, which is, again, verified via mild computer assistance

On Analytic Perturbations of a Family of Feigenbaum-like Equations

We prove existence of solutions $(\phi,\lambda)$ of a family of of Feigenbaum-like equations \label{family} \phi(x)={1+\eps \over \lambda} \phi(\phi(\lambda x)) -\eps x +\tau(x), where \eps is a small real number and $\tau$ is analytic and small on some complex neighborhood of $(-1,1)$ and real-valued on \fR. The family (\ref{family}) appears in the context of period-doubling renormalization for area-preserving maps (cf. \cite{GK}). Our proof is a development of ideas of H. Epstein (cf \cite{Eps1}, \cite{Eps2}, \cite{Eps3}) adopted to deal with some significant complications that arise from the presence of terms \eps x +\tau(x) in the equation (\ref{family}). The method relies on a construction of novel {\it 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

Coexistence of bounded and unbounded geometry for area-preserving maps

The geometry of the period doubling Cantor sets of strongly dissipative infinitely renormalizable H\'enon-like maps has been shown to be unbounded by M. Lyubich, M. Martens and A. de Carvalho, although the measure of unbounded "spots" in the Cantor set has been demonstrated to be zero. We show that an even more extreme situation takes places for infinitely renormalizable area-preserving H\'enon-like maps: bounded and unbounded geometries coexist with both phenomena occuring on subsets of positive measure in the Cantor sets

Renormalization and a-priori bounds for Leray self-similar solutions to the generalized mild Navier-Stokes equations

We demonstrate that the problem of existence of Leray self-similar blow up solutions in a generalized mild Navier-Stokes system with the fractional Laplacian $(-\Delta)^{\gamma/2}$ can be stated as a fixed point problem for a "renormalization" operator. We proceed to construct {\it a-priori} bounds, that is a renormalization invariant precompact set in an appropriate weighted $L^p$-space. As a consequence of a-priori bounds, we prove existence of renormalization fixed points for $d \ge 2$ and $d<\gamma <2 d+2$, and existence of non-trivial Leray self-similar mild solutions in $C^\infty([0,T),(H^k)^d \cap (L^p)^d)$, $k>0, p \ge 2$, whose $(L^p)^d$-norm becomes unbounded in finite time $T$