  We study the strong convergence of certain multidimensional continued
fraction algorithms. In particular, in the two-dimensional case, we prove that
the second Lyapunov exponent of Selmer's algorithm is negative and bound it
away from zero. Moreover, we give heuristic results on several other continued
fraction algorithms. Our results indicate that all classical multidimensional
continued fraction algorithms cease to be strongly convergent for high
dimensions. The only exception seems to be the Arnoux-Rauzy algorithm which,
however, is defined only on a set of measure zero.
