## 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.