Some short notes

5. Let E be a locally convex topological linear space, K\subset E be a compact convex subset of E and \partial_e K be the set of the extreme points of K. Let A(K) be the set of all affine continuous functions on K. Endowed with the supremum norm, A(K) is a Banach space.

Then K is said to be a (Choquet) simplex if

—each point in K is the barycenter of a unique probability measure supported on \partial_e K, or equivalently
—the dual space of A(K) is an L^1 space (in the dual ordering).

A simplex K is said to be Bauer if \partial_e K is closed in K.
Oppositely, K is said to be Poulsen if \partial_e K is dense in K. (Poulsen in 1961 proved the existence of such simplex.)

Lindenstrauss, Olsen and Sternfeld showed in 1978 here that given two Poulsen simplices P and Q, there is an affine homeomorphism h:P\to Q. In other words, there exists a unique Poulsen simplex (up to affine homeomorphisms), say \mathcal{P}.

Moreover they proved that \mathcal{P} is characterized by
1) strong homogeneity property: any affine homeomorphism between two proper faces can be extended to an affine automorphism of \mathcal{P};

2) universal property: any metrizable simplex is affinely homeomorphic to a closed face of \mathcal{P}.

Let x,y\in\partial_e \mathcal{P}. By 2) we see \mathcal{P} contains some simple arc I (say the end points a and b. By 1) there exists an affine homeomorphism h:\mathcal{P}\to \mathcal{P} which carries \{a,b\} onto \{x,y\}. Hence h(I) is a simple arc in \mathcal{P} with connecting x and y. In particular \partial_e \mathcal{P} is path-connected.

They further showed that \partial_e \mathcal{P} is homeomorphic to the Hilbert space \ell_2.

By the specification property, we know that the set \mathcal{M}(f) of invariant probability measures is Poulsen if f is a C^2 transitive Anosov diffeomorphism or a subshift of finite type. In particular the set of ergodic measures, \mathcal{M}^e(f) is path-connected.

4. Let M be a smooth manifold. Take an atlas \{\phi_i: U_i\to\mathbb{R}^d\} for M. Define \psi_{ij}: U_i\cap U_j\to \{ 1, -1\} by the signs of the Jacobi determinants of the transition maps \phi_j\circ \phi_i^{-1}. Then we glue the trivial bundles U_i\times \mathbb{R} with the transition functions \psi_{ij}. The resulting real line bundle is called the orientation line bundle \mathcal{L} on M.

A density of order \alpha on M is a map \sigma that assigns to each x\in M a map \sigma_x:\wedge^d(T_xM)\to\mathbb{R} with \sigma_x(\lambda\cdot  \vec{v})=|\lambda|^{\alpha}\cdot \sigma_x(\vec{v}). In term of the transition functions, the bundle of densities of order \alpha is given by \psi_{ij}=|\text{Jac}(\phi_j\circ \phi_i^{-1})|^{-\alpha}.

If \sigma is a density of order \alpha, then |\sigma| is also a density of order \alpha. A positive density of order 1 on M defines a positive measure on M that is equivalent to Lebesgue measure on coordinate charts.

3. Two different minimalities of a foliation \mathcal{F} over M:
1. each leaf F(x) is dense in M;
2. each leaf F(x) is a minimal submanifold with respect to some energy over TM (for example the geodesic with respect to the Riemannian metric).

Let A=(2,1;1,1):\mathbb{T}^2\to \mathbb{T}^2 be the Cat map (linear Anosov) and f:\mathbb{T}^2\to \mathbb{T}^2 be Smale’s DA-map with a source o\in\mathbb{T}^2. Then the extended unstable foliation \mathcal{F}^u_f is not minimal.

2. (Piotr) Let f:X\to X be totally transitive and periodic dense. Then f is weakly mixing.
Proof. Let A,B,U,V be nonempty open subsets. There exists n=n(f,A,U)\ge1 with f^{-n}A\cap U\neq\emptyset. As an open set, the intersection also contains a periodic point p, say f^kp=p. Then f^{-n-ik}A\cap U\neq\emptyset for all i\ge0. Since f^k is also transitive, there exists m=m(f^{k},f^{-n}B,V)\ge1 such that f^{-mk-n}B\cap V\neq\emptyset. In particular (f\times f)^{-l}(A\times B)\cap(U\times V)\neq\emptyset for some l=mk+n\ge1.

(Donnay 1988) There exists some smooth metric g on \mathbb{T}^2 (respectively, S^2) such that (\mathbb{T}^2,g) (resp. (S^2,g)) has a transitive geodesic flow.

(Lehrer and Weiss ) If T is ergodic and \mu(X\backslash A)>0, then for any prime p, there is a set latex B with \{T^iB:0\le i\le p-1\} pairwise disjoint and \bigcup_{0\le i\le p-1}T^iB\supset A.

1. Let M be a closed manifold and f_t:M\to M be a codim 1 Anosov flow.

(Verjovsky) If the dimension of M is at least 4, then every Anosov flow of codimension one is transitive.

(Franks and Williams) There exist non-transitive Anosov flows on some 3-manifolds.

Verjovsky conjecture (1): if the fundamental group \pi_1(M) is solvable, then the flow f_t must admit a global cross-section.

(proven by P. Armandariz when \dim M=3 and f is transitive and by J. Plante in the general case)

(Ghys, Codimension one Anosov flows and suspensions Page 60): The hypothesis on the fundamental group is necessary since the geodesic flow of a negatively curved compact surface provides an example of a codimension one Anosov flow with no global cross-section.

Verjovsky conjecture (2): if the dimension of M is at least 4, then any codimension-one Anosov flow is topologically equivalent to a suspension flow over an Anosov diffeomorphism of the torus.

(Asaoka 2008). Any transitive codimension-one Anosov flow is topologically conjugated to a smooth volume-preserving Anosov flow.

(Asaoka 2008) Let X_0 be a C^2 vector field on a C^\infty closed manifold M. If the flow generated by X_0 preserves a Hölder continuous volume m_0, then X_0 can be C^1-approximated by a C^\infty vector field X_k that generates a flow preserving a C^\infty invariant volume m_k\to m_0.

(Hart 1983, Asaoka 2008) For any C^r-foliation \mathcal{F} on a C^\infty closed manifold M, there exists a C^r diffeomorphism h:M\to M such that T(h\mathcal{F})=Dh(T\mathcal{F}) is a C^r subbundle of TM. Moreover, they can choose the diffeomorphism h so that it is arbitrary C^r-close to the identity map.

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