  Homogeneous continued fraction algorithms are multidimensional
generalizations of the classical Euclidean algorithm, the dissipative map $$
(x_1,x_2) \in \mathbb{R}_+^2 \longmapsto \left\{\begin{array}{ll}
  (x_1 - x_2, x_2), & \mbox{if $x_1 \geq x_2$}
  (x_1, x_2 - x_1), & \mbox{otherwise.} \end{array} \right. $$ We focus on
those which act piecewise linearly on finitely many copies of positive cones
which we call Rauzy induction type algorithms.
  In particular, a variation Selmer algorithm belongs to this class. We prove
that Rauzy induction type algorithms, as well as Selmer algorithms, are ergodic
with respect to Lebesgue measure.
