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On the dual of Rauzy induction

  • KAE INOUE (a1) and HITOSHI NAKADA (a2)

Abstract

We investigate a certain dual relationship between piecewise rotations of a circle and interval exchange maps. In 2005, Cruz and da Rocha [A generalization of the Gauss map and some classical theorems on continued fractions. Nonlinearity 18 (2005), 505–525]  introduced a notion of ‘castles’ arising from piecewise rotations of a circle. We extend their idea and introduce a continuum version of castles, which we show to be equivalent to Veech’s zippered rectangles [Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115 (1982), 201–242]. We show that a fairly natural map defined on castles represents the inverse of the natural extension of the Rauzy map.

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[1] Cruz, S. D. and da Rocha, L. F. C.. A generalization of the Gauss map and some classical theorems on continued fractions. Nonlinearity 18 (2005), 505525.
[2] Keane, M.. Interval exchange transformations. Math. Z. 141 (1975), 2531.
[3] Keane, M.. Non-ergodic interval exchange transformations. Israel J. Math. 26(2) (1977), 188196.
[4] Rauzy, G.. Echanges d’intervalles et transformations induites. Acta Arith. 34(4) (1979), 315328 (in French).
[5] Schweiger, F.. Ergodic Theory of Fibred Systems and Metric Number Theory. Oxford Science Publications, Clarendon, Oxford University Press, New York, 1995.
[6] Veech, W. A.. Interval exchange transformations. J. Anal. Math. 33 (1978), 222278.
[7] Veech, W. A.. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115(1) (1982), 201242.
[8] Viana, M.. Ergodic theory of interval exchange maps. Rev. Mat. Complut. 19(1) (2006), 7100.
[9] Viana, M.. Dynamics of interval exchange transformations and Teichmüller flows. Preliminary manuscript available from http://w3.impa.br/∼viana/out/ietf.pdf.
[10] Yoccoz, J.-C.. Continued Fraction Algorithms for Interval Exchange Maps: an Introduction (Frontiers in Number Theory, Physics, and Geometry. I) . Springer, Berlin, 2006, pp. 401435.
[11] Yoccoz, J.-C.. Interval exchange maps and translation surfaces. Homogeneous Flows, Moduli Spaces and Arithmetic (Clay Mathematics Proceedings, 10) . American Mathematical Society, Providence, RI, 2010, pp. 169.
[12] Zorich, A.. Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier (Grenoble) 46(2) (1996), 325370.

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