## Collected notes

8. Let $f:M\to M$ be a partially hyperbolic diffeomorphism, $\mathcal{F}^s$ and $\mathcal{F}^u$ be the stable and unstable foliations (both are invariant). If $E^c$ also integrates to an invariant, regular foliation $\mathcal{F}$, then $(f,\mathcal{F})$ is said to be normally hyperbolic.

Regular foliation is introduced by Pugh, Shub, Wilkinson in ‘Holder foliations, revisited’, saying that the leaves are $C^1$ and assembled $C^1$-continuously. It seems that every integral foliation is regular. If this is true, then every central foliation is automatically regular.

Let $(f,\mathcal{F})$ be normally hyperbolic. Then it is said to be plaque-expansive, if there exist a plaquation $\mathcal{P}$ of $\mathcal{F}$, and $\epsilon,\delta>0$, such that any two $\delta$-orbits respect $\mathcal{P}$ and $\epsilon$-shadowing each other must lie on the same plaques of $\mathcal{P}$.

– does normal hyperbolicity imply plaque-expansivity?

Positive if $\mathcal{F}$ is $C^1$; if $\mathcal{F}$ is uniformly compact; if $D^cf$ is an isometry, or if $\mathcal{F}^s$ and $\mathcal{F}^u$ are quasi-isometric. Unknown for general cases.

The map $f$ is said to be dynamically coherent, if there exist two invariant foliations $\mathcal{F}^{cs}$ and $\mathcal{F}^{cu}$. In particular such $f$ is normally hyperbolic.

– does plaque-expansivity imply dynamical coherence?

Positive if $E^c$ is $C^1$. In fact in this case $f$ is robustly dynamically coherent. It is pointed out after Theorem 2.8 of Burns, Wilkinson, ‘Dynamical coherence and center bunching’ that $C^1$ can be weakened to plaque-expansivity.

7. Let $f:M\to M$ be a partially hyperbolic diffeomorphism, $E$ be a $G_\delta$ subset of positive volume. Consider the function $\eta:E\to\mathbb{R}, x\mapsto m_{s,x}(W^s(x,\delta)\cap E)$. Then $\eta$ is measurable.

Proof. Let $E=\bigcap_n U_n$. It is clear that the map $E\to \mathcal{K}_M, x\mapsto \overline{W}^s(x,\delta)$ continuous. Then we see the restriction $E\to \mathcal{K}|_E, x\mapsto \overline{W}^s(x,\delta)\cap U_n$ is lower semicontinuous since $U_n$ is open and the stable foliation $W^s$ is continuous. Similarly we can show that the map $\eta_n:E\to\mathbb{R}, x\mapsto m_{s,x}(W^s(x,\delta)\cap U_n)$ is also lower semicontinuous. Finally the statement follows by noting that $\eta(x)=\inf_n \eta_n(x)$.

6. Takens proved the following embedding theorem in (Dynamical Systems and Turbulence, 1981):

Let $M^d$ be a compact manifold without boundary. For generic $f\in\mathrm{Diff}^2(M)$ and generic $\phi\in\mathrm{C}(M,\mathbb{R})$, the delay
$(\phi,f,2d+1):M\to\mathbb{R}^{2d+1}, x\mapsto(\phi(f^ix))_{i=0}^{2d}$
is an embedding.

In fact for $(\phi,f,2d+1)$ to be an immersion at a point $p\in M$, it suffices to ensure the co-vectors $d_p(\phi\circ f^i)$, $0\le i\le 2d$ span the co-tangent space $T_p^*M$, which holds open and densely.

5. Let $G$ be a locally compact topological group. The left translate of a right Haar measure is a right Haar measure. More precisely, if $\mu$ is a right Haar measure, then $\mu_g:A\mapsto\mu(g^{-1}A)$ is also a right Haar measure. Since it is unique up to a constant, $\mu_g=\Delta(g)\cdot\mu$. In fact this gives a group homomorphism $\Delta:G\to\mathbb{R}^*$ (as a multiplicative group), which is called the Haar modulus, modular function or modular character.

A group is unimodular if and only if the modular function is identically 1. Examples of unimodular groups are abelian groups (left is right), compact groups ($\mathrm{Im}(\Delta)$ is compact and hence trivial), discrete groups, semisimple Lie groups and connected nilpotent Lie groups.

An example of a non-unimodular group is $\{f(x)=ax+b|a>0,b\in\mathbb{R}\}$, the group of transformations of the form $x\mapsto ax+b$ on the real line. This example shows that a solvable Lie group need not be unimodular.

4. Von Neumann (1933) proved that every compact, locally Euclidean group is a Lie group.

Partial generalizations were then obtained by Pontryagin (for commutative $G$) and Chevalley (for solvable $G$). To go further the following notation is needed:

Let $G$ be a topological group. It is said to have small subgroups if each open neighborhood $U$ of the identity $e\in G$ contains a nontrivial subgroup of $G$.

For example the circle group has no small subgroups (in general a Lie group does not have small subgroups), while the $p$-adic integers $\mathbb{Z}_p$ as additive group has, because each neighborhood $U\ni e$ will contain the subgroups $p^k\mathbb{Z}_p$ for all large integers $k$.

In 1952, Gleason Groups without small subgroups proved that every finite dimensional, locally compact group without small subgroup is a Lie group.

Montgomery and Zippin, proved that every finite dimensional, locally connected group has no small subgroups.

In 1953, H. Yamabe obtained: A connected locally compact group $G$ is a projective limit of a sequence of Lie groups, and if $G$ has no small subgroups, then $G$ is a Lie group.

Hilbert–Smith conjecture: let $G$ be a locally compact group. If there is a topological manifold $M$ such that $G$ admits a continuous, faithful group action on $M$, then $G$ must be a Lie group.

An equivalent form of the conjecture is: $\mathbb{Z}_p$ has no faithful group action on any topological manifold.

3. Let $A=(1,1;0,1)$ be a upper-triangle matrix. Then it has an eigenvector $(1,0)$ with eigenvalue $1$ and all other nonzero vectors ‘fall’ to the line of $(1,0)$ (geomotric eigenvector).

Consider a specified perturbation $A_\epsilon=(1+\epsilon,1;0,1/(1+\epsilon))$. While keeping the eigenvector $(1,0)$ with eigenvalue $1+\epsilon$, $A_\epsilon$ gets a new eigenvector $(1,-\frac{\epsilon^2+2\epsilon}{1+\epsilon})$ with eigenvalue $1/(1+\epsilon)$: the new one is splitted from the old geomotric eigenvector.

2. Palis conjectured that every $f\in\mathrm{Diff}^r(M)$ can be $C^r$–approximated by diffeomorphisms that have finitely physical measures with their basin full measure on $M$.

1. Let $(X,T)$ be uniquely ergodic. It is well known that for each continuous function $\phi$, its Birkhoff average exists for every point $x\in X$ and the convergence is unform. This is no longer true for general cocycles. In fact Alex Furman proved following theorem in 1997:

Let $(X,T)$ be uniquely ergodic. Then for any continuous subadditive cocycle $\Phi=\{\phi_n:n\ge1\}$, $\limsup_{n\to\infty}\frac{\phi_n(x)}{n}\le\mu(\Phi)$ uniformly and everywhere.

Moreover for each $F_\sigma$ subset $E$ with $\mu(E)=0$, there exists cocycle $\Phi$ with $\limsup_{n\to\infty}\frac{\phi_n(x)}{n}<\mu(\Phi)$ for every point $x\in E$.

He also gave several necessary conditions to ensure the uniformness of cocycles. Finally he mentioned one question of Walters:

Does there exists a non-uniform matrix-cocycle on every nonatomic uniquely ergodic system?