TY - JOUR
T1 - On the dual of Rauzy induction
AU - Inoue, Kae
AU - Nakada, Hitoshi
N1 - Funding Information:
The authors would like to thank Thomas A. Schmidt for his careful reading of an early version of the manuscript and for giving helpful comments. We would also like to thank the referee who gave plenty of suggestions for polishing up the manuscript. Partially supported by Grant-in Aid for Scientific research (No. 24340020) by Japan Society for the Promotion of Science.
PY - 2017/8/1
Y1 - 2017/8/1
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.
UR - http://www.scopus.com/inward/record.url?scp=84957808866&partnerID=8YFLogxK
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
VL - 37
SP - 1492
EP - 1536
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
SN - 0143-3857
IS - 5
ER -