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.