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.
| Original language | English |
|---|---|
| Pages (from-to) | 1-45 |
| Number of pages | 45 |
| Journal | Ergodic Theory and Dynamical Systems |
| DOIs | |
| Publication status | Accepted/In press - 2016 Feb 11 |
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ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
Cite this
On the dual of Rauzy induction. / Inoue, Kae; NAKADA, HITOSHI.
In: Ergodic Theory and Dynamical Systems, 11.02.2016, p. 1-45.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - On the dual of Rauzy induction
AU - Inoue, Kae
AU - NAKADA, HITOSHI
PY - 2016/2/11
Y1 - 2016/2/11
N2 - 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.
AB - 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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UR - http://www.scopus.com/inward/citedby.url?scp=84957808866&partnerID=8YFLogxK
U2 - 10.1017/etds.2015.89
DO - 10.1017/etds.2015.89
M3 - Article
AN - SCOPUS:84957808866
SP - 1
EP - 45
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
SN - 0143-3857
ER -