## 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.

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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.

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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.

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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 $\bigcup_{0\le i\le p-1}T^iB\supset A$.

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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.