6 research outputs found
Rational approximations, multidimensional continued fractions and lattice reduction
We first survey the current state of the art concerning the dynamical
properties of multidimensional continued fraction algorithms defined
dynamically as piecewise fractional maps and compare them with algorithms based
on lattice reduction. We discuss their convergence properties and the quality
of the rational approximation, and stress the interest for these algorithms to
be obtained by iterating dynamical systems. We then focus on an algorithm based
on the classical Jacobi--Perron algorithm involving the nearest integer part.
We describe its Markov properties and we suggest a possible procedure for
proving the existence of a finite ergodic invariant measure absolutely
continuous with respect to Lebesgue measure.Comment: 30 pages, 4 figure