Lecture 1 - Stable ergodicity on surfaces The map A(x,y) = (2x+y, x+y) on the torus is ergodic. That in itself is not overly difficult to prove. Whats more interesting is that the system is stably ergodic: every area-preserving C^2 diffeomorphism close to A is also ergodic. However, it is an open question if every C^1 diffeomorphism near A is ergodic. Ill give an outline of the proof of stable ergodicity for these types of systems and show why the C^1 case is so different from the C^2 case. Ill also talk about how, in some sense, every stably ergodic system on a surface has to be a linear example like the map A defined above.