32 research outputs found
On Analytic Perturbations of a Family of Feigenbaum-like Equations
We prove existence of solutions 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
is analytic and small on some complex neighborhood of 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 .
A byproduct of the method is a new proof of the existence of a
Feigenbaum-Coullet-Tresser function
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
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 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
-space.
As a consequence of a-priori bounds, we prove existence of renormalization
fixed points for and , and existence of non-trivial
Leray self-similar mild solutions in ,
, whose -norm becomes unbounded in finite time