Ergodic Theory and Dynamical Systems

Research Article

Veech’s dichotomy and the lattice property

JOHN SMILLIEa1 and BARAK WEISSa2

a1 Department of Mathematics, Cornell University, Ithaca, NY, USA (email: smillie@math.cornell.edu)

a2 Department of Mathematics, Ben Gurion University, Be’er Sheva, 84105, Israel (email: barakw@math.bgu.ac.il)

Abstract

Veech showed that if a translation surface has a stabilizer which is a lattice in SL(2,xs211D), then any direction for the corresponding constant slope flow is either completely periodic or uniquely ergodic. We show that the converse does not hold: there are translation surfaces that satisfy Veech’s dichotomy but for which the corresponding stabilizer subgroup is not a lattice. The construction relies on work of Hubert and Schmidt.

(Received July 27 2006)

(Revised January 22 2008)