Discontinuity-growth of interval-exchange maps
Christopher F. Novak - Department of Mathematics and Statistics, University of Michigan - Dearborn, 4901 Evergreen Rd., Dearborn,MI 48128, United States (email) Abstract:
For an interval-exchange map $f$, the number of discontinuities $d(f^n)$
either exhibits linear growth or is bounded independently of $n$.
This dichotomy is used to prove that the group $\mathcal{E}$ of interval-exchanges does not contain distortion elements, giving
examples of groups that do not act faithfully via interval-exchanges. As a further application of
this dichotomy, a classification of centralizers in $\mathcal{E}$ is given. This classification is used to show
that $\text{Aut}(\mathcal{E}) \cong \mathcal{E}$
Keywords: Interval-exchange transformation, group actions,
discontinuity-growth,
distortion elements, centralizers, automorphism group.
Received: November 2008; Revised: May 2009; Available Online: August 2009. |