7 research outputs found
Interval exchanges, admissibility and branching Rauzy induction
We introduce a definition of admissibility for subintervals in interval
exchange transformations. Using this notion, we prove a property of the natural
codings of interval exchange transformations, namely that any derived set of a
regular interval exchange set is a regular interval exchange set with the same
number of intervals. Derivation is taken here with respect to return words. We
characterize the admissible intervals using a branching version of the Rauzy
induction. We also study the case of regular interval exchange transformations
defined over a quadratic field and show that the set of factors of such a
transformation is primitive morphic. The proof uses an extension of a result of
Boshernitzan and Carroll
Return words of linear involutions and fundamental groups
We investigate the natural codings of linear involutions. We deduce from the
geometric representation of linear involutions as Poincar\'e maps of measured
foliations a suitable definition of return words which yields that the set of
first return words to a given word is a symmetric basis of the free group on
the underlying alphabet . The set of first return words with respect to a
subgroup of finite index of the free group on is also proved to be a
symmetric basis of
Interval exchanges, admissibility and branching Rauzy induction
We introduce a definition of admissibility for subintervals in interval exchange transformations. Using this notion, we prove a property of the natural codings of interval exchange transformations, namely that any derived set of a regular interval exchange set is a regular interval exchange set with the same number of intervals. Derivation is taken here with respect to return words. We characterize the admissible intervals using a branching version of the Rauzy induction. We also study the case of regular interval exchange transformations defined over a quadratic field and show that the set of factors of such a transformation is primitive morphic. The proof uses an extension of a result of Boshernitzan and Carroll