## Second collection

7. A closed geodesic in $(M,g)$ is a differentiable closed curve such that its lift to the universal cover $(\widetilde{M},\tilde{g})$ is a geodesic.

Theorem. Let $\alpha$ be a non-trivial free homotopy class in a compact Riemannian manifold $M$. Then there exists a closed geodesic $\gamma_0\in\alpha$ with $\text{L}(\gamma_0)=\inf\{\text{L}(\gamma)|\gamma\in\alpha\}$.

The set of eigenvalues with multiplicities of the Laplace–Beltrami operator acting on the manifold is called the spectrum of a manifold. Two manifolds are said to be isospectral if they have the same spectrum.

The multiplicity of a given length is the number of distinct free homotopy classes of geodesics that contain a closed geodesic of that length. The length spectrum, denoted by $L(M)$, is the set of lengths of smooth closed geodesics, counted with multiplicities. The marked length spectrum of $M$ is a function assigning each free homotopy class $\alpha$ the set of lengths of closed geodesics freely homotopic to $\alpha$.

Y. Colin de Verdiere showed that, for a given manifold $M$, there exists a generic subset $\mathcal{R}$, in the sense of Baire category, of the set of smooth Riemannian metrics on $M$, such that if $g\in \mathcal{R}$, the length spectrum of $(M,g)$ can be recovered from the Laplace spectrum. Moreover, the set $\mathcal{R}$ contains all metric with sectional curvature less than zero.

A closed geodesic $\gamma:S^1\to M$ is said to be a $1/k$–geodesic if $d(\gamma(t),\gamma(t+2\pi/k))=\frac{\text{L}(\gamma)}{k}$ for all $t\in S^1$. The $1/k$ length spectrum of $M$, denoted by $L_{1/k}(M)$, is the set of lengths of $1/k$–geodesics in $M$. The length spectrum does not necessarily persist under Gromov–Hausdorff convergence (see here). But the $1/k$ length spectrum does.

6. Let $(X,m)$ be a measure space and $f:X\to X$ be a measurable map. Then $(X,m,f)$ is said to be conservative if $m(E)=0$ for each measurable subset $E\subset X$ with mutually disjoint $\{f^{-n}E:n\ge0\}$.

Halmos–Sucheston theorem: If $(X,m,f)$ is conservative, so is $(X,m,f^k)$ for each $k\ge1$.

Proof: let $k\ge2$ be fixed and $E$ be a measurable subset with $\{f^{-kn}E:n\ge0\}$ being mutually disjoint.

Let $C=\bigcup_{n\ge0}f^{-kn}E$, $I(x)=\{0\le i\le k-1:x\in f^{-i}C\}$ and $E_j=\{x\in E:|I(x)|=j\}$ for $j=1,\cdots,k$.
Clearly $E_j$ are measurable (since $C$ is), mutually disjoint and $E=\bigcup_{j=1}^{k}E_j$.
Claim: $E_j\cap f^{-n}E_j=\emptyset$ for all $n\ge1$. Therefore $m(E_j)=0$ and $m(E)=0$, which completes the argument.

Proof of Claim: Suppose on the contrary that there exist $j\in\{1,\cdots,k\}$ and $n\ge1$ such that $E_j\cap f^{-n}E_j\neq\emptyset$. In other words, there exists $x\in E_j$ with $f^{n}x\in E_j$. By definition of $E_j$, there exist $0= n_1<\cdots< n_j\le k-1$ with $x\in f^{-n_i}C$ and $0= m_1<\cdots< m_j\le k-1$ with $x\in f^{-m_i-n}C$. Observe that:

*: $m_i+n\neq0(\text{mod} k)$ for all $i$ (since $E_j\subset E$, $m_i+n\ge n\ge1$).
**: $m_i+n(\text{mod} k)$ are all different (since $m_i$‘s are).

So $I(x)\supset\{0,m_i+n(\text{mod} k)\}$, contradicting $x\in E_j$. This completes the proof of Claim.

5. A compact Riemannian manifold $M$ is said to be uniformly secure if there is a number $n\in\mathbb{N}$ such that for any two points $x,y\in M$, the set of geodesics connecting them can be blocked by $n$ point obstacles. For example the standard sphere $S^2$ is not secure at all. A nontrivial example is the compact surface $M_g$ of genus $g\ge2$ of constant curvature -1.
Conjecture: uniform security implies flatness.

Wing Kai Ho proved this conjecture for non-simply connected, orientable Riemannian surfaces.

4. Let $(M,g)$ be a Riemannian manifold, $TM$ the tangent bundle and $TTM$ the double tangent bundle. The connection map $K:TTM\to TM$ is given by $K(\xi)=\nabla_{\dot{x}}v$, where $\xi=\frac{d}{dt}|_{t=0}(x(t),v(t))$. The canonical flip is an involution $j:(TTM,\pi_{TTM})\to (TTM,(\pi_{TM})_*)$.

3. Let $H:S^1\times M\to\mathbb{R}$ be a periodic Hamiltonian function and $\phi_t$ be the induced Hamiltonian flow. Then the $k$–speeder flow $\psi_t=\phi_{kt}$ is generated by $k\cdot H(kt,x)$.

2. A diffeo $f:M\to M$ is said to conservative if for any measurable subset $E$ with $\{f^k E:k\in\mathbb{Z}\}$ mutually disjoint, $m(E)=0$.

It is ergodic if for any measurable subset $E$ with $fE=E$, $m(E)=0\text{ or }1$. Clearly ergodicity implies conservativity. For Anosov diffeo, conservativity implies transitvie. Is there any special property of the SRB measure of $f$?

1. A factor $\pi:([0,1]^{\mathbb{Z}},\sigma)\to (Y,S)$ is said to be nontrivial if $Y$ is not a singleton.
Linderstrauss proved that for each nontrivial factor $\pi:([0,1]^{\mathbb{Z}},\sigma)\to (Y,S)$, there exists $k\ge1$ such that $([0,1]^{\mathbb{Z}},\sigma)$ is a factor of $(Y,S^k)$. In particular $h_{top}(Y,S)=h_{top}([0,1]^{\mathbb{Z}},\sigma)=\infty$.