arXiv:1302.5008math.DS
Classical homogeneous multidimensional continued fraction algorithms are ergodic
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Abstract
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.
We often write "otherwise" in mathematics: https://arxiv.org/html/1501.05438v2#:~:text=otherwise,-%2C https://arxiv.org/html/1302.5008v3#:~:text=otherwise https://arxiv.org/html/1501.05438v2#:~:text=otherwise,-%2E In fact, I think that's the most common alternative, not "else".
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