Multidimensional continued fractions, dynamical renormalization and KAM theory

TL;DR

This paper introduces a new algorithm based on homogeneous space dynamics that improves simultaneous rational approximation of vectors, aiding in dynamical systems and small divisor problems.

Contribution

It presents a simple, effective algorithm for multidimensional continued fractions using flow dynamics on SL(2,Z) SL(2,R), enabling better approximations and applications in KAM theory.

Findings

Provides best possible approximations for irrational vectors

Constructs renormalization schemes for vector field linearization

Develops methods for invariant tori construction in Hamiltonian systems

Abstract

The disadvantage of `traditional' multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and Kleinbock-Margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space SL(2,Z)\SL(2,R) (the space of lattices of covolume one) that indeed yields best possible approximations to any irrational vector. The algorithm is ideally suited for a number of dynamical applications that involve small divisor problems. We explicitely construct renormalization schemes for (a) the linearization of vector fields on tori of arbitrary dimension and (b) the construction of invariant tori for Hamiltonian systems.