Tag Archives: positive entropy

Herman’s example: minimal smooth diffeo with positive entropy

SUN Peng explained Herman’s example in great details.

Let M_1=SL(2,\mathbb{R})/\Gamma be a compact manifold with normalized measure \sigma and consider the map F_\alpha:\mathbb{T}\times M_1\to\mathbb{T}\times M_1, (\theta,\beta)\mapsto(\theta+\alpha,A_\theta\cdot \beta) where A_\theta=Rotation(\theta)\circ \mathrm{diag}\{\lambda,\lambda^{-1}\}. Here \lambda>1 is fixed.

His presentation focused on three parts:

1. positive entropy. He compared the top Lyapunov exponent of F_\alpha with the fiber exponent when we view F_\alpha as a cocycle over \mathbb{T}, which is shown to be larger than or equal to \log\frac{\lambda+\lambda^{-1}}{2}. Note D_{\theta,\beta}F_\alpha=(Id,\mathrm{Ad}_{A_\theta}) and \|\mathrm{Ad}_{A_\theta}\|\ge \|A_\theta\|^2.

2. minimality. He first considered an open set I\times V and its iterates under F^q_{p/q}. 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 p/q to some close irrationals he covered the whole manifold with finite iterates of I\times V. Then a G_\delta-argument gives the minimality of residual rotation number.

3. plenty of invariant measures. He conjugated fiber-wise map F_\alpha over \{\theta\}\times M_1 to some time-t_\theta map of the geodesic flow on M_1. So EVERY invariant measure of the geodesic gives rise to an invariant measure of F_\alpha. Then F_\alpha is NOT uniquely ergodic.