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

TL;DR

This paper presents a mild computer-assisted proof of the analyticity and compactness of a renormalization operator for area-preserving maps, reducing reliance on extensive computer calculations in the study of period doubling universality.

Contribution

It introduces a novel approach that minimizes computer assistance by using interval arithmetics on real numbers, simplifying the proof of renormalization properties.

Findings

Proved analyticity of the renormalization operator.

Established compactness of the renormalization operator.

Verified contraction mappings with mild computer assistance.

Abstract

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.