Spectral properties of renormalization for area-preserving maps

Denis  Gaidashev; Tomas  Johnson

doi:10.3934/dcds.2016.36.3651

Article Contents

2016, Volume 36, Issue 7: 3651-3675. Doi: 10.3934/dcds.2016.36.3651

This issue Previous Article On blow-up criterion for the nonlinear Schrödinger equation Next Article Neumann homogenization via integro-differential operators

Spectral properties of renormalization for area-preserving maps

Denis Gaidashev^1, and
Tomas Johnson^2,

1.
Department of Mathematics, Uppsala University, Uppsala
2.
Fraunhofer-Chalmers Research Centre for Industrial Mathematics, SE-412 88 Gothenburg

Received: December 2014

Revised: November 2015

Published online: March 2016

Abstract / Introduction Related Papers Cited by

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.

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Mathematics Subject Classification: 37E20, 37E40.

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