8. Let be a partially hyperbolic diffeomorphism, and be the stable and unstable foliations (both are invariant). If also integrates to an invariant, regular foliation , then is said to be *normally hyperbolic*.

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

Let be normally hyperbolic. Then it is said to be *plaque-expansive*, if there exist a plaquation of , and , such that any two -orbits respect and -shadowing each other must lie on the same plaques of .

– does normal hyperbolicity imply plaque-expansivity?

Positive if is ; if is uniformly compact; if is an isometry, or if and are quasi-isometric. Unknown for general cases.

The map is said to be *dynamically coherent*, if there exist two invariant foliations and . In particular such is normally hyperbolic.

– does plaque-expansivity imply dynamical coherence?

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

7. Let be a partially hyperbolic diffeomorphism, be a subset of positive volume. Consider the function . Then is measurable.

*Proof.* Let . It is clear that the map continuous. Then we see the restriction is lower semicontinuous since is open and the stable foliation is continuous. Similarly we can show that the map is also lower semicontinuous. Finally the statement follows by noting that .

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

Let be a compact manifold without boundary. For generic and generic , the delay

is an embedding.

In fact for to be an immersion at a point , it suffices to ensure the co-vectors , span the co-tangent space , which holds open and densely.

5. Let be a locally compact topological group. The left translate of a right Haar measure is a right Haar measure. More precisely, if is a right Haar measure, then is also a right Haar measure. Since it is unique up to a constant, . In fact this gives a group homomorphism (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 ( is compact and hence trivial), discrete groups, semisimple Lie groups and connected nilpotent Lie groups.

An example of a non-unimodular group is , the group of transformations of the form 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 ) and Chevalley (for solvable ). To go further the following notation is needed:

Let be a topological group. It is said to have **small subgroups** if each open neighborhood of the identity contains a *nontrivial* subgroup of .

For example the circle group has no small subgroups (in general a Lie group does not have small subgroups), while the -adic integers as additive group has, because each neighborhood will contain the subgroups for all large integers .

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 is a projective limit of a sequence of Lie groups, and if has no small subgroups, then is a Lie group.

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

An equivalent form of the conjecture is: has no faithful group action on any topological manifold.

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

Consider a specified perturbation . While keeping the eigenvector with eigenvalue , gets a new eigenvector with eigenvalue : the new one is splitted from the old geomotric eigenvector.

2. Palis conjectured that every can be –approximated by diffeomorphisms that have finitely physical measures with their basin full measure on .

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

Let be uniquely ergodic. Then for any continuous subadditive cocycle , uniformly and everywhere.

Moreover for each subset with , there exists cocycle with for every point .

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?