Spectral properties of renormalization for area-preserving maps

TL;DR

This paper investigates the spectral properties of the renormalization operator for area-preserving maps, establishing bounds on convergence rates that are key to understanding the rigidity of their invariant structures.

Contribution

It proves an upper bound on the convergence rate of renormalizations for infinitely renormalizable area-preserving maps, advancing the understanding of their spectral and rigidity properties.

Findings

Bound on the convergence rate of renormalization is sufficiently small.

Supports the rigidity of invariant Cantor sets in area-preserving maps.

Enhances understanding of spectral properties of the renormalization operator.

Abstract

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.