## Tag Archives: dominated splitting

### 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.

### dominated nonuniformly hyperbolic

Bochi, Fayad, Pujals, A remark on conservative diffeomorphisms.

Let $r>1$, $\mathrm{Diff}^r_\mu(M)$ endow with $C^1$ topology (which is not a complete metric space). A map $f\in\mathrm{Diff}^r_\mu(M)$ is said to be dominated nonuniformly hyperbolic if there exist a dominated splitting $TM=E^+\oplus E^-$, where $E^+$ (resp. $E^-$) coincides a.e. with the sum of the Oseledets spaces corresponding to positive (resp. negative) Lyapunov exponents. Let $\mathrm{SE}^r$ be the stably ergodic ones in $\mathrm{Diff}^r(M)$. Clearly the DNH is an open property in $\mathrm{SE}^r(M)$ and may not be open in $\mathrm{Diff}^r(M)$..

Theorem. There is an open and dense set $\mathcal{R}^r\subset\mathrm{SE}^r$ such that each $f\in\mathcal{R}^r$ is dominated nonuniformly hyperbolic. In particular the stably DNH are $C^1$ dense in $\mathrm{SE}^r$.

In fact $\mathcal{R}^r$ is the set of $f \in \mathrm{SE}^r$ such that $f$ has a dominated splitting $TM=E^+\oplus E^-$ with $\lambda_p(f ) > 0 > \lambda_{p+1}(f )$, where $p = \dim E^+$.

It is not true that every stably ergodic diffeomorphism can be approximated by a partially hyperbolic system.

1. A stably ergodic (or stably transitive) diffeomorphism $f$ must have a dominated splitting. This is true because if it did not, they perturbed $f$ and create a periodic point whose derivative is the identity. Then, using the Pasting lemma (for which $r>1$ regularity is an essential hypothesis), one breaks transitivity.

2. They further perturbed $f$ such that the sum of the Lyapunov exponents `inside’ each of the bundles of the (finest) dominated splitting is non-zero.

3. They finally perturbed $f$ such that the Lyapunov exponents in each of the bundles become almost equal. (If one attempted to make the exponents exactly equal, there is no guarantee that the perturbation is $C^r$.) Since the sum of the exponents in each bundle varies continuously, they concluded there are no zero exponents.

The continuation of the finest dominated splitting is not necessarily the finest dominated splitting of the perturbed diffeomorphism. A dominated splitting is stably finest if it has a continuation which is the finest dominated splitting of every sufficiently close diffeomorphism. It is easy to see that diffeomorphisms with stably finest dominated splittings are (open and) dense among diffeomorphisms with a dominated splitting.

### Some definitions

5. Let $f:X\to X$ be a homeomorphism and $\mu$ be an invariant ergodic measure. Consider a bounded function $\phi$ with $\mu(\phi)=0$ and the induced cocycle $\phi_n$.
Then by Birkhoff ergodicity Theorem, we know $\frac{\phi_n}{n}\to0$, a.e.. A related statement is that, for a.e. $x$, $\displaystyle |\phi_n(x)|\le\|\phi\|_0$ infinitely often.

Proof. Let $a=\|\phi\|_0$ and $X_{n,a}=\{x:\phi_n(x)\in [-a,a]\}$, $X_a=\bigcap_{k\ge1}\bigcup_{n\ge k}X_{n,a}$. Clearly $X_a$ is invariant and hence has measure either zero or one. Suppose $\mu(X_a)=0$. Let $Y_{+}$ be the set of points with $\phi_n(x)\ge a$ for all large enough $n$. Similarly we define $Y_-$. Clearly they are disjoint and both are invariant. Then by the choice $a$, we see $\mu(Y_+\cup Y_-)=1$. By ergodic assumption, we can assume $\mu(Y_+)=1$. So $\displaystyle 0=\mu(\phi_n)\ge\mu(\liminf_{n\to\infty}\phi_n)\ge a>0$, which is absurd.

Moreover we have the following dichotomy:
– either $\phi$ is a coboundary: $\phi(x)=h(x)-h(fx)$, $\mu$-a.e.,

– or $\sup_{n\ge0}\phi_n(x)=+\infty$ and $\inf_{n\ge0}\phi_n(x)=-\infty$ $\mu$-a.e..

Proof. Let $h(x)=\sup_{n\ge1}\phi_n(x)$ (measurable). Let’s assume $E=\{x:h(x)<+\infty\}$ has positive measure. Clearly $E$ is invariant and hence full measure. In particular $h$ is well defined a.e.. Let $g(x)=h(x)-\sup_{n\ge2}\phi_n(x)$. Clearly $g(x)\ge0$.

Note that $\phi_{k+1}(x)=\phi_k(fx)+\phi(x)$ for all $k\ge1$. So $\sup_{k\ge1}\phi_{k+1}(x)=\sup_{k\ge1}\phi_k(fx)+\phi(x)$, or equivalently, $h(x)-g(x)=h(fx)+\phi(x)$, or $\phi(x)+g(x)=h(x)-h(fx)$. So we need to show $g(x)=0$, $\mu$-a.e.. A sufficient condition is $\sum_{n\ge0} g_n(x)<\infty$, a.e..

Note that for $\mu$-a.e. $x$, there exists $n_k\to\infty$ with $\phi_{n_k}(x)\in[-a,a]$ and $h(f^{n_k}x)$ bounded, and hence $g_{n_k}(x)+\phi_{n_k}(x)=h(x)-h(f^{n_k}x)$ stays bounded.

It seems that $\{\phi_n(x):n\ge0\}$ is dense in $\mathbb{R}$ $\mu$-a.e. in the second alternative.

——————

1. Let $f:X\to X$ be a topological dynamical system. Birkhoff Pointwise Ergodic Theorem states that, there exists a full measure set $G\subset X$ ($\mu(G^c)=0$ for all $f$-invariant measure $\mu$) such that the limit $\nu_x=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\delta_{f^kx}$ exists and is $f$-invariant and ergodic for each $x\in G$. Then consider the measure (be careful) $\nu=\int_G \nu_x d\mu(x)$. It is an $f$-invariant probability measure with that:
$\nu(A)=\int_G\nu_x(A)d\mu(x)=\int_G\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\delta_{f^kx}(A)d\mu(x)$
$\:=\int_G\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\delta_{A}(f^kx)d\mu(x)=\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\int_G\delta_{A}(f^kx)d\mu(x)$
$=\mu(A)$ for each measurable subset $A\subset X$.

So we have $\mu=\nu=\int_G \nu_x d\mu(x)$, the Ergodic Decomposition of $\mu$.

2. A foliation is a decomposition of a manifold into submanifolds (foliation box $\mathbb{R}^u\times\mathbb{R}^c$). A lamination is a partial foliation: $\mathbb{R}^u\times K$ for some compact subset $K\subset\mathbb{R}^c$

There are various versions of absolute continuity transversal abs cts (implied by bdd jacobian). leaf abs cts.

3. Let $\pi:G(M,k)\to M$ be the $k$-dimensional Grassmannian bundle over $M$. For each $k$-dimensional linear subspace $V\subset T_xM$, we denote $[V]$ the corresponding element in $G(x,k)\subset G(M,k)$. The topology of $G(M,k)$ is determined by the distance function $D$ such that for all $V\in G(x,k),W\in G(y,k)$
$D([V], [W]) = \inf\{L(\gamma) + \angle_{x}(V, P_{\gamma}W)|\gamma:y\to x\text{ is smooth}\}$,
where $P_\gamma$ is the parallel translation along $\gamma$. Under this topology the projection $\pi:G(M,k)\to M$ is a continuous map.

4. Let $f:M\to M$ be a diffeomorphism and $\Lambda$ be a compact invariant subset. Then $f$ is shadowing on $\Lambda$ if for each $\delta>0$ there exists $\epsilon>0$ such that every $\epsilon-$pseudo orbit can be $\delta-$shadowed by a genuine orbit of $f$.

Generalized to weakly shadowing property: for each $\delta>0$ there exists $\epsilon>0$ such that the $\epsilon-$chain is contained in the $\delta-$ neighborhood of a genuine orbit: $\{x_n\}\subset B(\mathcal{O}(x),\delta)$.

$\Lambda$ is stably weak shadowing at $f$ if there exist a nbhd $U\supset\Lambda$ and $\mathcal{U}\ni f$ such that $\bigcap_{n\in\mathbb{Z}}g^nU$ is weak shadowing under $g$ for each $g\in\mathcal{U}$. A special case is $M$ itself is weak shadowing.

$f$ is tame if there is a neighborhood $\mathcal{U}$ of $f$ such that each $g\in\mathcal{U}$ has only finitely many chain recurrent classes. (there are other different defintions with the same name.)

Conjecture:: $f$ is stably weak shadowing if and only if $f$ is tame.

Note: this is true if $\dim(M)=2$ and is conjectured for general case by GAN, Shaobo.

It is proved by YANG, Dawei that if a transitive set $\Lambda$ is stably weakly shadowing at $f$, then $\Lambda$ admits a dominated splitting.

Ergodic maps form a $G_\delta$ subset. Let $f\in\mathrm{Diff}^r_m(M)$ and $\phi\in C(M,\mathbb{R})$. Then Birkhoff ergodic theorem says that $A_n(\phi,f)(x)\to \phi^\ast(x)$ almost everywhere and in $L^2$. In particular the limit $\|A_n(\phi,f)-\int \phi dm\|\to \|\phi^\ast-\int \phi dm\|$ always exists, just may not be zero. Those maps $f$ with $\phi^\ast(x)\equiv \int \phi dm$ are characterized by
$\displaystyle \bigcap_{l\ge 1}\bigcap_{N\ge 1}\bigcup_{n\ge N}\{f\in\mathrm{Diff}^r_m(M):\|A_n(\phi,f)-\int \phi dm\|<1/l\}$, which is a $G_\delta$ subset. Therefore the ergodic ones form a $G_\delta$ subset of $\mathrm{Diff}^r_m(M)$.

Metrically transitive maps form a $G_\delta$ subset. A map $f\in\mathrm{Diff}^r_m(M)$ is said to be $m$-transitive, if $m$-a.e. points have dense orbits. Such a property also called weakly ergodic. These maps also form a $G_\delta$ subset of $\mathrm{Diff}^r_m(M)$. To this end, note that the open sets $U$ with $m(\partial U)=0$ form a basis of the topology on $M$, and the mapping $f\mapsto m(\bigcup_{\mathbb{Z}}f^nU)$ is lower semi-continuous. Therefore the set of maps with $m(\bigcup_{\mathbb{Z}}f^nU)> 1-1/l$ form an open subset.