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.

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    […] have defined ergodicity for dissipative systems (see the previous post). So one might ask: for transitive Anosov , is it ergodic with respect to […]

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