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?

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