Exactness of the Euclidean algorithm and of the Rauzy induction on the   space of interval exchange transfomations
  The two-dimensional homogeneous Euclidean algorithm is the central motivation for the definition of the classical multidimensional continued fraction algorithms, as Jacobi-Perron, Poincar\'e, Brun and Selmer algorithms. The Rauzy induction, a generalization of the Euclidean algorithm, is a key tool in the study of interval exchange transformations. Both maps are known to be dissipative and ergodic with respect to Lebesgue measure. Here we prove that they are exact. 
Tomasz Miernowski and Arnaldo Nogueira
math.DS