Horocycle and geodesic flows for $SL(2,R)/SL(2,Z)$:
-- min u.e. almost
-- something about symbolic dynamics and continued fractions?
Consulting the original notebook with Mark Pollicott, the missing line seems to say:
"min u.e. almost"
In my paper with Yitwah Cheung, we build a section to the horocycle flow on $SL(2, \mathbb R)/SL(2, \mathbb Z)$ (which Pierre Arnoux also knew about) that corresponds to what is known as the BCZ map, which looks something like an infinite substitution system. This return map is known to be weak mixing (unpublished work of Cheung and Quas), but there are probably many interesting symbolic question about it.