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

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

—each point in is the barycenter of a unique probability measure supported on , or equivalently

—the dual space of is an space (in the dual ordering).

A simplex is said to be **Bauer** if is closed in .

Oppositely, is said to be **Poulsen** if is dense in . (Poulsen in 1961 proved the existence of such simplex.)

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

Moreover they proved that is characterized by

1) strong homogeneity property: any affine homeomorphism between two proper faces can be extended to an affine automorphism of ;

2) universal property: any metrizable simplex is affinely homeomorphic to a closed face of .

Let . By 2) we see contains some simple arc (say the end points and . By 1) there exists an affine homeomorphism which carries onto . Hence is a simple arc in with connecting and . In particular is path-connected.

They further showed that is homeomorphic to the Hilbert space .

By the specification property, we know that the set of invariant probability measures is Poulsen if is a transitive Anosov diffeomorphism or a subshift of finite type. In particular the set of ergodic measures, is path-connected.

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4. Let be a smooth manifold. Take an atlas for . Define by the signs of the Jacobi determinants of the transition maps . Then we glue the trivial bundles with the transition functions . The resulting real line bundle is called the orientation line bundle on .

A density of order on is a map that assigns to each a map with . In term of the transition functions, the bundle of densities of order is given by .

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

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3. Two different minimalities of a foliation over :

1. each leaf is dense in ;

2. each leaf is a minimal submanifold with respect to some *energy* over (for example the geodesic with respect to the Riemannian metric).

Let be the Cat map (linear Anosov) and be Smale’s DA-map with a source . Then the extended unstable foliation is not minimal.

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2. (Piotr) Let be totally transitive and periodic dense. Then is weakly mixing.

Proof. Let be nonempty open subsets. There exists with . As an open set, the intersection also contains a periodic point , say . Then for all . Since is also transitive, there exists such that . In particular for some .

(Donnay 1988) There exists some smooth metric on (respectively, ) such that (resp. ) has a transitive geodesic flow.

(Lehrer and Weiss ) If is ergodic and , then for any prime , there is a set latex with pairwise disjoint and .

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1. Let be a closed manifold and be a codim 1 Anosov flow.

(Verjovsky) If the dimension of 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 is solvable, then the flow must admit a global cross-section.

(proven by P. Armandariz when and 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 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 be a vector field on a closed manifold . If the flow generated by preserves a Hölder continuous volume , then can be -approximated by a vector field that generates a flow preserving a invariant volume .

(Hart 1983, Asaoka 2008) For any -foliation on a closed manifold , there exists a diffeomorphism such that is a subbundle of . Moreover, they can choose the diffeomorphism so that it is arbitrary -close to the identity map.