## Tag Archives: minimal diffeo

### Some remarks about dominated splitting property

Denote $\mathcal{T}$ the set of transitive diffeos, $\mathcal{DS}$ the set of diffeo’s with Global Dominated Splittings (GDS for short), $\mathcal{M}$ the set of minimal diffeos.

It is proved that

$\mathcal{DS}\bigcap \mathcal{M}=\emptyset$: diffeo with GDS can’t be minimal (here).

$\mathcal{T}^o\subset \mathcal{DS}$: robustly transitive diffeo always admits some GDS (here).

So $\mathcal{T}^o\bigcap \mathcal{M}=\emptyset$, although $\mathcal{T}\supset \mathcal{M}$: the special property (minimality) can’t happen in the interior of the general property (transitivity).

A minor change of the proof shows that a diffeomorphism with GDS can’t be uniquely ergodic, either. So we have the following conservative version:

$\mathcal{DS}\bigcap \mathcal{UE}=\emptyset$: diffeos with GDS can’t be uniquely ergodic.

$\mathcal{E}^o\subset \mathcal{DS}$: stably ergodic diffeos always admits some GDS (here).

So $\mathcal{E}^o\bigcap \mathcal{UE}=\emptyset$, although $\mathcal{E}\supset \mathcal{UE}$.

Remark. It is a little bit tricky to define $\mathcal{E}^o$. The most natural definition may lead to an emptyset. One well-accepted definition is: $f\in\mathcal{E}^o$ if there exists a $C^1$ neighborhood $f\in\mathcal{U}\subset\mathrm{Diff}^1_m(M)$, such that every $g\in \mathcal{U}\cap \mathrm{Diff}^2_m(M)$ is ergodic. All volume-preserving Anosov satisfies the later definition, and this is the context of Pugh-Shub Stable Ergodicity Conjecture.

Remark. There is an open dense subset $\mathcal{R}\subset \mathcal{E}^o$, such that every $f\in \mathcal{R}$ is nonuniformly Anosov (here)

Remark. Let $(M,\omega)$ be a symplectic manifold with $\dim M\ge 4$, $\mathrm{PH}^{2}_{\omega}(M,2)$ be the set of $C^2$ symplectic partially hyperbolic maps with $\dim (E^c)=2$.
Then consider $f\in\mathcal{E}^o\cap \mathrm{PH}^{2}_{\omega}(M,2)$ and $\lambda^c_1(f,\omega)\ge \lambda^c_2(f,\omega)$ be the two central Lyapunov exponents. If the dominated splitting is not refined by the partially hyperbolic splitting, then it must split the central bundle, and $\lambda^c_1(f,\omega)> \lambda^c_2(f,\omega)$: $f$ is nonuniformly Anosov.

### 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.