First lecture: Birkhoff ergodic theorem
Second lecture: The spectrum of a measure preserving transformation. Note that we did not cover the Von Neumann ergodic theorem.
Background in ergodic theory 1 and 2. Also there is a more specific sheet on generic points.
Notes on special cases of Ratner's Theorem.
If you want that the horocycle flow on SL(2,R)/SL(2,Z) is ergodic see Proposition 3.1 of Einseidler's notes.
If you want that the geodesic flow on SL(2,R)/SL(2,Z) is ergodic see the attached note.
An outline of Dani's proof of measure and orbit classification for SL(2,R)/SL(2,Z)

Veech's 78 paper that many results are taken from.
Kerckhoff's paper whose proof we follow.
Some notes