Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-23T15:43:15.551Z Has data issue: false hasContentIssue false
  • English
  • Français

On the dual of Rauzy induction

Published online by Cambridge University Press:  11 February 2016

KAE INOUE  and
HITOSHI NAKADA
KAE INOUE
Affiliation:
Faculty of Pharmacy, Keio University, Tokyo105-8512, Japan email inoue-ke@pha.keio.ac.jp
HITOSHI NAKADA
Affiliation:
Department of Mathematics, Keio University, Yokohama223-8522, Japan email nakada@math.keio.ac.jp
Get access Rights & Permissions [Opens in a new window]

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. Nonlinearity18 (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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 5 , August 2017 , pp. 1492 - 1536
Copyright
© Cambridge University Press, 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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), 505–525.CrossRefGoogle Scholar
Keane, M.. Interval exchange transformations. Math. Z. 141 (1975), 25–31.CrossRefGoogle Scholar
Keane, M.. Non-ergodic interval exchange transformations. Israel J. Math. 26(2) (1977), 188–196.CrossRefGoogle Scholar
Rauzy, G.. Echanges d’intervalles et transformations induites. Acta Arith. 34(4) (1979), 315–328 (in French).Google Scholar
Schweiger, F.. Ergodic Theory of Fibred Systems and Metric Number Theory. Oxford Science Publications, Clarendon, Oxford University Press, New York, 1995.Google Scholar
Veech, W. A.. Interval exchange transformations. J. Anal. Math. 33 (1978), 222–278.Google Scholar
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
Viana, M.. Ergodic theory of interval exchange maps. Rev. Mat. Complut. 19(1) (2006), 7–100.CrossRefGoogle Scholar
Viana, M.. Dynamics of interval exchange transformations and Teichmüller flows. Preliminary manuscript available from http://w3.impa.br/∼viana/out/ietf.pdf.Google Scholar
Yoccoz, J.-C.. Continued Fraction Algorithms for Interval Exchange Maps: an Introduction (Frontiers in Number Theory, Physics, and Geometry. I) . Springer, Berlin, 2006, pp. 401–435.Google Scholar
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. 1–69.Google Scholar
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