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Exactness of the Euclidean algorithm and of the Rauzy induction on the space of interval exchange transformations

Published online by Cambridge University Press:  30 November 2011

TOMASZ MIERNOWSKI  and
ARNALDO NOGUEIRA
TOMASZ MIERNOWSKI
Affiliation:
Institut de Mathématiques de Luminy, Aix-Marseille Université, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France (email: [email protected], [email protected])
ARNALDO NOGUEIRA
Affiliation:
Institut de Mathématiques de Luminy, Aix-Marseille Université, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France (email: [email protected], [email protected])
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Abstract

The two-dimensional homogeneous Euclidean algorithm is the central motivation for the definition of the classical multidimensional continued fraction algorithms, such as Jacobi–Perron, Poincaré, 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 1 , February 2013 , pp. 221 - 246
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
[2]Avila, A. and Forni, G.. Weak mixing for interval exchange transformations and translation flows. Ann. of Math. (2) 165(2) (2007), 637–664.CrossRefGoogle Scholar
[3]Bruin, H. and Hawkins, J.. Exactness and maximal automorphic factors of unimodal interval maps. Ergod. Th. & Dynam. Sys. 21(4) (2001), 1009–1034.CrossRefGoogle Scholar
[4]Bufetov, A. I.. Decay of correlations for the Rauzy–Veech–Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of abelian differentials. J. Amer. Math. Soc. 19(3) (2006), 579–623 (electronic).CrossRefGoogle Scholar
[5]Danthony, C. and Nogueira, A.. Measured foliations on nonorientable surfaces. Ann. Sci. École Norm. Sup. (4) 23(3) (1990), 469–494.CrossRefGoogle Scholar
[6]Kerckhoff, S. P.. Simplicial systems for interval exchange maps and measured foliations. Ergod. Th. & Dynam. Sys. 5(2) (1985), 257–271.CrossRefGoogle Scholar
[7]Laurent, M. and Nogueira, A.. Approximation to points in the plane by SL(2,Z)-orbits. J. London. Math. Soc. (2) to appear. Preprint, 2010, arXiv:1004.1326.Google Scholar
[8]Masur, H.. Interval exchange transformations and measured foliations. Ann. of Math. (2) 115(1) (1982), 169–200.CrossRefGoogle Scholar
[9]Messaoudi, A., Nogueira, A. and Schweiger, F.. Ergodic properties of triangle partitions. Monatsh. Math. 157(3) (2009), 283–299.CrossRefGoogle Scholar
[10]Miernowski, T. and Nogueira, A.. Absorbing sets of homogeneous subtractive algorithms. Preprint, 2011, arXiv:1104.3762.Google Scholar
[11]Nogueira, A.. The three-dimensional Poincaré continued fraction algorithm. Israel J. Math. 90(1–3) (1995), 373–401.CrossRefGoogle Scholar
[12]Nogueira, A.. The Borel–Bernstein theorem for multidimensional continued fractions. J. Anal. Math. 85 (2001), 1–41.CrossRefGoogle Scholar
[13]Nogueira, A. and Rudolph, D.. Topological weak-mixing of interval exchange maps. Ergod. Th. & Dynam. Sys. 17(5) (1997), 1183–1209.CrossRefGoogle Scholar
[14]Rauzy, G.. Échanges d’intervalles et transformations induites. Acta Arith. 34(4) (1979), 315–328.CrossRefGoogle Scholar
[15]Rokhlin, V. A.. Exact endomorphisms of a Lebesgue space. Amer. Math. Soc. Transl. Ser. 2 39 (1964), 1–36.Google Scholar
[16]Schweiger, F.. Multidimensional Continued Fractions. Oxford Science Publications/Oxford University Press, Oxford, 2000.CrossRefGoogle Scholar
[17]Veech, W. A.. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115(1) (1982), 201–242.CrossRefGoogle Scholar
[18]Veech, W. A.. The metric theory of interval exchange transformations. III. The Sah–Arnoux–Fathi invariant. Amer. J. Math. 106(6) (1984), 1389–1422.CrossRefGoogle Scholar
[19]Zorich, A.. Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier (Grenoble) 46(2) (1996), 325–370.CrossRefGoogle Scholar