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We first survey the current state of the art concerning the dynamicalproperties of multidimensional continued fraction algorithms defineddynamically as piecewise fractional maps and compare them with algorithms basedon lattice reduction. We discuss their convergence properties and the qualityof the rational approximation, and stress the interest for these algorithms tobe obtained by iterating dynamical systems. We then focus on an algorithm basedon the classical Jacobi--Perron algorithm involving the nearest integer part.We describe its Markov properties and we suggest a possible procedure forproving the existence of a finite ergodic invariant measure absolutelycontinuous with respect to Lebesgue measure.

Rational approximations, multidimensional continued fractions and lattice reduction