SUN Peng explained Herman’s example in great details.

Let be a compact manifold with normalized measure and consider the map where . Here is fixed.

His presentation focused on three parts:

1. positive entropy. He compared the top Lyapunov exponent of with the fiber exponent when we view as a cocycle over , which is shown to be larger than or equal to . Note and .

2. minimality. He first considered an open set and its iterates under . After getting a hyperbolic fiber-system, he show the boundary behavior is parabolic and minimal with respect to that fiber. Then perturbing the rotationn number to some close irrationals he covered the whole manifold with finite iterates of . Then a -argument gives the minimality of residual rotation number.

3. plenty of invariant measures. He conjugated fiber-wise map over to some time- map of the geodesic flow on . So EVERY invariant measure of the geodesic gives rise to an invariant measure of . Then is NOT uniquely ergodic.