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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
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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 

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