Ergodic Theory and Dynamical Systems

Minimal non-ergodic directions on genus-2 translation surfaces

a1 Department of Mathematics, Northwestern University, 2033 Sheridan Avenue, Evanston, IL 60208-2730, USA (e-mail:
a2 Department of Mathematics, University of Illinois at Chicago, 851 South Morgan, Chicago, IL 60607-7045, USA (e-mail:

Article author query
cheung y   [Google Scholar] 
masur h   [Google Scholar] 


It is well known that on any Veech surface, the dynamics in any minimal direction is uniquely ergodic. In this paper it is shown that for any genus-2 translation surface which is not a Veech surface there are uncountably many minimal but not uniquely ergodic directions. The theorem can be applied to certain billiard tables.

(Published Online March 17 2006)
(Received February 14 2005)
(Revised July 7 2005)