Classical homogeneous multidimensional continued fraction algorithms are ergodic
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2013
- DOI:
- 10.48550/arXiv.1302.5008
- arXiv:
- arXiv:1302.5008
- Bibcode:
- 2013arXiv1302.5008C
- Keywords:
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- Mathematics - Dynamical Systems;
- 11K55;
- 28D99