## Category Archives: Geometry

### Symplectic and contact manifolds

Let $(M,\omega)$ be a symplectic manifold. It said to be exact if $\omega=d\lambda$ for some one-form $\lambda$ on $M$.

(1) If $\omega=d\lambda$ is exact, then there is a canonical isomorphism between the v.f. and 1-forms. In particular, there exists a v.f. $X$ such that $\lambda=i_X\omega$. Then we have $\lambda(X)=\omega(X,X)=0$, and $L_X\lambda=i_X d\lambda+d i_X\lambda=i_X\omega +0=\lambda$, and $L_X\omega=d i_X\omega=d\lambda=\omega$.

(2) Suppose there exists a vector field $X$ on $M$ such that its Lie-derivative $L_X\omega=\omega$ (notice the difference with $L_X\omega=0$). Then Cartan’s formula says that $\omega=i_X d\omega+ di_X\omega=d\lambda$, where $\lambda=i_X\omega$. So $\omega=d\lambda$ is exact, and $L_X\lambda=i_Xd\lambda+di_X\lambda=i_X\omega+0=\lambda$.

### Collections

10. Let $f_a:S^1\to S^1$, $a\in[0,1]$ be a strictly increasing family of homeomorphisms on the unit circle, $\rho(a)$ be the rotation number of $f_a$. Poincare observed that $\rho(a)=p/q$ if and only if $f_a$ admits some periodic points of period $q$. In this case $f_a^q$ admits fixed points.

Note that $a\mapsto \rho(a)$ is continuous, and non-decreasing. However, $\rho$ may not be strictly increasing. In fact, if $\rho(a_0)=p/q$ and $f^q\neq Id$, then $\rho$ is locked at $p/q$ for a closed interval $I_{p/q}\ni a_0$. More precisely, if $f^q(x) > x$ for some $x$, then $\rho(a)=p/q$ on $[a_0-\epsilon,a_0]$ for some $\epsilon > 0$; if $f^q(x) 0$; while $a_0\in \text{Int}(I_{p/q})$ if both happen.

Also oberve that if $r=\rho(a)\notin \mathbb{Q}$, then $I_r$ is a singelton. So assuming $f_a$ is not unipotent for each $a\in[0,1]$, the function $a\mapsto \rho(a)$ is a Devil’s staircase: it is constant on closed intervals $I_{p/q}$, whose union $\bigcup I_{p/q}$ is dense in $I$.

9. Let $X:M\to TM$ be a vector field on $M$, $\phi_t:M\to M$ be the flow induced by $X$ on $M$. That is, $\frac{d}{dt}\phi_t(x)=X(\phi_t(x))$. Then we take a curve $s\mapsto x_s\in M$, and consider the solutions $\phi_t(x_s)$. There are two ways to take derivative:

(1) $\displaystyle \frac{d}{dt}\phi_t(x_s)=X(\phi_t(x_s))$.

(2) $\displaystyle \frac{d}{ds}\phi_t(x_s)=D\phi_t(\frac{d}{ds}x_s))$, which induces the tangent flow $D\phi_t:TM\to TM$ of $\phi_t:M\to M$.

Combine these two derivatives together:

$\displaystyle \frac{d}{dt}D_x\phi_t(x_s')=\frac{d}{dt}\frac{d}{ds}\phi_t(x_s) =\frac{d}{ds}\frac{d}{dt}\phi_t(x_s)=\frac{d}{ds}X(\phi_t(x_s)) =D_{\phi_t(x)}X\circ D_x\phi_t(x_s').$

This gives rise to an equation $\displaystyle \frac{d}{dt}D_x\phi_t=D_{\phi_t(x)}X\circ D_x\phi_t.$

Formally, one can consider the differential equation along a solution $x(t)$:
$\displaystyle \frac{d}{dt}D(t)=D_{\phi_t(x)}X\circ D(t)$, $D(0)=Id$. Then $D(t)$ is called the linear Poincare map along $x(t)$. Suppose $x(T)=x(0)$. Then $D(T)$ determines if the periodic orbit is hyperbolic or elliptic. Note that the path $D(t)$, $0\le t\le T$ contains more information than the above characterization.

### Notes-09-14

4. Borel–Cantelli Lemma(s). Let $(X,\mathcal{X},\mu)$ be a probability space. Then

If $\sum_n \mu(A_n)<\infty$, then $\mu(x\in A_n \text{ infinitely often})=0$.

If $A_n$ are independent and $\sum_n \mu(A_n)=\infty$, then for $\mu$-a.e. $x$, $\frac{1}{\mu(A_1)+\cdots+\mu(A_n)}\cdot|\{1\le k\le n:x\in A_k\}|\to 1$.

The dynamical version often involves the orbits of points, instead of the static points. In particular, let $T$ be a measure-preserving map on $(X,\mathcal{X},\mu)$. Then

$\{A_n\}$ is said to be a Borel–Cantelli sequence with respect to $(T,\mu)$ if $\mu(T^n x\in A_n \text{ infinitely often})=1$;

$\{A_n\}$ is said to be a strong Borel–Cantelli sequence if $\frac{1}{\mu(A_1)+\cdots+\mu(A_n)}\cdot|\{1\le k\le n:T^k x\in A_k\}|\to 1$ for $\mu$-a.e. $x$.

3. Let $H(q,p,t)$ be a Hamiltonian function, $S(q,t)$ be the generating function in the sense that $\frac{\partial S}{\partial q_i}=p_i$. Then the Hamilton–Jacobi equation is a first-order, non-linear partial differential equation

$H + \frac{\partial S}{\partial t}=0$.

Note that the total derivative $\frac{dS}{dt}=\sum_i\frac{\partial S}{\partial q_i}\dot q_i+\frac{\partial S}{\partial t}=\sum_i p_i\dot q_i-H=L$. Therefore, $S=\int L$ is the classical action function (up to an undetermined constant).

2. Let $\gamma_s(t)$ be a family of geodesic on a Riemannian manifold $M$. Then $J(t)=\frac{\partial }{\partial s}|_{s=0} \gamma_s(t)$ defines a vector field along $\gamma(t)=\gamma_0(t)$, which is called a Jacobi field. $J(t)$ describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic $\gamma$.

Alternatively, A vector field $J(t)$ along a geodesic $\gamma$ is said to be a Jacobi field, if it satisfies the Jacobi equation:

$\frac{D^2}{dt^2}J(t)+R(J(t),\dot\gamma(t))\dot\gamma(t)=0,$

where $D$ denotes the covariant derivative with respect to the Levi-Civita connection, and $R$ the Riemann curvature tensor on $M$.

### Collections again

8. Definition. Given a family of maps $T_\epsilon:X\to X$ with corresponding invariant densities $\phi_\epsilon$. Then $T_0$ is said to be acim-stable if $lim_{\epsilon\to 0}T_\epsilon=T_0$ implies $lim_{\epsilon\to 0}\phi_\epsilon=\phi_0$.
The limits are taken with respect to properly chosen metrics on the space of maps and densities, respectively.

Functions of the bounded variation are continuous except at a most countable number of points, at which they have two one-sided limits.

7. Let $\mathcal{H}=(\mathbb{R}^3,\ast)$ be the 3D Heisenberg group, with $(a,b,c)\ast(x,y,z)=(a+x,b+y,c+z+ay)$. Let $\Gamma=\langle\alpha,\beta,\gamma|\alpha\ast\beta=\beta\ast\alpha\ast\gamma,\alpha\ast\gamma=\gamma\ast\alpha,\beta\ast\gamma=\gamma\ast\beta\rangle$ be a cocompact discrete subgroup (for example $\mathbb{Z}^3=\langle \mathbf{i},\mathbf{j},\mathbf{k}\rangle$). Then $M=\mathcal{H}/\Gamma$ is a 3D nilmanifold. A general non-toral
three-dimensional nilmanifold is also of this form. Suppose we have a homomorphism $h:\mathcal{H}\to\mathbb{R}$, which is of the form $(x,y,z)\mapsto ax+by$, which induces a 2D-foliation, say $\mathcal{F}_h$ on $\mathcal{H}$ and on $M$.

Theorem. Every Reebless foliation on $M$ is almost aligned with some $\mathcal{F}_h$.
Plante for $C^2$, Hammerlindl and Potrie for $C^{1,0}$.

Theorem. Every partially hyperbolic system on $M$ is accessible.
J. Rodriguez Hertz, F. Rodriguez Hertz and R. Ures (convervative), Hammerlindl and Potrie (general)

6. Let $r\ge 1$ and $S:\mathbb{R}^{r+1}\to\mathbb{R}$ be a $C^2$ function. Consider the solutions $x:\mathbb{Z}\to \mathbb{R}$ of the recurrence relation:
($\ast$) $\displaystyle R(x_{i-r},\cdots,x_{i+r}):=\sum_j \partial_{x_i}S(x_j,\cdots,x_{j+r})=0,$ for all $i\in\mathbb{Z}$.
Note that ($\ast$) is actually a finite sum of $r+1$ terms over $j=i-r,\cdots,i$. It is the derivative of formal series $W(x)=\sum_j S(x_j,\cdots,x_{j+r})$ with respect to $\partial_{x_i}$.
Example. Billiards, or generally twist maps, where $r=1$ and $S$ is the generating function, the solution gives the configuration of an orbit.

There are some conditions:
Periodicity. $S(x+1)=S(x)$. So $S$ descends to a map on $\mathbb{R}^{\mathbb{Z}}/\mathbb{Z}$.
Monotone. $\displaystyle\partial_{x_i,x_k}S(x_j,\cdot,x_{j+r})\le 0$ for all $j$ and all $i\neq k$, and $\displaystyle\partial_{x_j,x_{j+1}}S(x_j,\cdot,x_{j+r}) < 0$ for all $j$.
Coercivity. $S$ is bounded from below and there exists $k$ such that $S(x_j,\cdots,x_{j+r})\to\infty$ as $|x_k-x_{k+1}|\to\infty$.
Under these conditions the ($\ast$) is called a monotone variational recurrence relation.

A sequence $x$ is said to be a global minimizer, if $W(x)\le W(x+v)$ (understand as over all intervals) for all sequences $v$. Clearly a global minimizer solves ($\ast$). The collection of global minimizers is also closed under coordinately convergence.

For a real number $a$, a sequence $x$ with $x_0=a$ is called an $a$-minimizer, if it is minimizes among all $y$‘s with $y_0=a$.
Ana-minimizers in general need not be solutions to ($\ast$).
Given a rational $p/q$, we consider the operator $\tau_{p,q}$ (shift $p$ and subtract $q$) and Birkhoff orbits of rotational number $p/q$ prime sum $W_{p,q}=S(x_0,\cdots, x_{r})+\cdots+S(x_{p-1},\cdots,x_{p-1+r})$ over the periodic ones $x=\tau_{p,q}(x)$.

Periodic Peierls barrier. Let $a$ be a real and $p,q$ be coprime. Then as
$\displaystyle P_{p,q}(a):= \min_{\tau_{p,q}x=x,x_0=a} W_{p,q}(x)-\min_{\tau_{p,q}x=x}W_{p,q}(x)$.

It is easy to see that
There exists a periodic minimizer $x\in M_{p,q}$ with $x_0 =a$ if and only if $P_{p,q}(a)=0$.

$M_{p,q}$ gives an invariant curve if and only if $P_{p,q}(\cdot)\equiv 0$.

Then the Peierls barrier at a general frequency is defined as $P_{\omega}(a)=\lim_{p/q\to\omega}P_{p,q}(a)$ when the limit exists (see Mramor and Rink, arxiv:1308.3073).

### Billiards

9. Victor Ivrii conjecture. Let $Q$ be a strictly convex domain, $F$ be the billiard map on the phase space $\Omega=\partial Q\times(0,\pi)$. Let $\omega$ be the Lebesgue measure of $\Omega$, and $\ell$ be the Lebesgue measure on $Q$.

Conjecture 1. $\omega(\text{Per}(F))=0$ for all $Q$ with $C^\infty$ boundaries.

Remark. This is about a general domain $Q$, not a generic domain.

Definition. A point $q\in\partial Q$ is said to be an absolute looping point, if $\omega_q(\bigcup_{n\neq0}F^n\Omega_q)>0$. Let $L(Q)$ be the set of absolute looping points.

Conjecture 2. $\ell(L(Q))=0$ for all $Q$.

Question: When $L(Q)=\emptyset$?

8. In Boltzmann gas model, the identical round molecules are confined by a box. Sinai has replaced the box by periodic boundary conditions so that the molecules move on a flat torus.

On circular and elliptic billiard tables, for all $p\ge 3$, the $(p,q)$-periodic orbits forms a continuous family and hence all the trajectories have the same length.

An invariant noncontractible topological annulus, $A\subset\Omega$, whose interior contains no invariant circles, is a Birkhoff instability region. The dynamics in an instability region
has positive topological entropy. Hence Birkhoff conjecture implies
that any non-elliptical billiard has positive topological entropy.

How to construct a strictly convex $C^1$-smooth billiard table with metric positive entropy? b) How to construct a convex $C^2$-smooth billiard table with positive metric entropy?

Recall Bunimovich stadium is not $C^2$, and not strictly convex.

A periodic orbit of period $q$ corresponds to an (oriented) closed polygon with $q$ sides, inscribed in $Q$, and satisfying the condition on the angles it makes with the boundary. Birkhoff called these the harmonic polygons.

Then the maximal circumference of 2-orbit yields the diameter of $Q$. The minimax circumference of 2-orbit corresponds
to the width of $Q$.

### Area of the symmetric difference of two disks

This post goes back to high school: the area $\delta_d$ of the symmetric difference of two $d$-dimensional disks when one center is shifted a little bit. Let’s start with $d=1$. So we have two intervals $[-r,r]$ and $[x-r,x+r]$. It is easy to see the symmetric difference is of length $\delta_1(x)=2x$.

Then we move to $d=2$: two disks $L$ and $R$ of radius $r$ and center distance $x=2a. So the angle $\theta(x)$ satisfies $\cos\theta=\frac{a}{r}$.

The symmetric difference is the union of $R\backslash L$ and $L\backslash R$, which have the same area: $\displaystyle (\pi-\theta)r^2+2x\sqrt{r^2-x^2}-\theta r^2=2(\frac{\pi}{2}-\arccos\frac{x}{r})r^2+2x\sqrt{r^2-x^2}$. Note that the limit
$\displaystyle \lim_{x\to0}\frac{\text{area}(\triangle)}{2a}=\lim_{a\to0}2\left(\frac{r^2}{\sqrt{1-\frac{a^2}{r^2}}}\cdot\frac{1}{r}+\sqrt{r^2-a^2}\right)=4r$.
So $\delta_2(x)\sim 4rx$.

I didn’t try for $d\ge3$. Looks like it will start with a linear term $2d r^{d-1}x$.

—————–

Now let ${\bf r}(t)=(a\cos t,\sin t)$ be an ellipse with $a>1$, and ${\bf r}'(t)=(-a\sin t,\cos t)$ be the tangent vector at ${\bf r}(t)$. Let $\omega$ be the angle from ${\bf j}=(0,1)$ to ${\bf r}'(t)$.
Let $s(t)=\int_0^t |{\bf r}'(u)|du$ be the arc-length parameter and $K(s)=|{\bf l}''(s)|$ be the curvature at ${\bf l}(s)={\bf r}(t(s))$. Alternatively we have $\displaystyle K(t)=\frac{a}{|{\bf r}'(t)|^{3}}$.

The following explains the geometric meaning of curvature:

$\displaystyle K(s)=\frac{d\omega}{ds}$, or equivalently, $K(s)\cdot ds=d\omega$. $(\star)$.

Proof. Viewed as functions of $t$, it is easy to see that $(\star)$ is equivalent to $K(t)\cdot \frac{ds}{dt}=\frac{d\omega}{dt}$.

Note that $\displaystyle \cos\omega=\frac{{\bf r}'(t)\cdot {\bf j}}{|{\bf r}'(t)|}=\frac{\cos t}{|{\bf r}'(t)|}$. Taking derivatives with respect to $t$, we get
$\displaystyle -\sin\omega\cdot\frac{d\omega}{dt}=-\frac{a^2\sin t}{|{\bf r}'(t)|^3}$. Then $(\star)$ is equivalent to

$\displaystyle \frac{a}{|{\bf r}'(t)|^{3}}\cdot |{\bf r}'(t)|=\frac{a^2\sin t}{\sin\omega\cdot |{\bf r}'(t)|^3}$, or
$\displaystyle \sin\omega\cdot |{\bf r}'(t)|=a\sin t$. Note that $\displaystyle \sin^2\omega=1-\cos^2\omega=1-\frac{\cos^2 t}{|{\bf r}'(t)|^2}$. Therefore $\displaystyle \sin^2\omega\cdot |{\bf r}'(t)|^2= |{\bf r}'(t)|^2-\cos^2 t=a^2\sin^2 t$, which completes the verification.

### Invariant subsets of ACIP of partially hyperbolic diffeomorphism

4. (Notes from the paper Stable ergodicity for partially hyperbolic attractors with negative central exponents)
Let $f\in\mathrm{Diff}^1(M)$ and $L$ be a partially hyperbolic attractor. Then there exists a $C^1$ neighborhood $\mathcal{U}\ni f$ such that every $g\in\mathcal{U}$ possesses a partially hyperbolic attractor $L_g$ near $L$. Moreover assume $f_n\in\mathrm{Diff}^2(M)\to f\in\mathrm{Diff}^2(M)$ with Gibbs u-states $\mu_n$ on $L_n$, then any weak limit is a Gibbs u-state on $L$.

Let $\mu$ be an ergodic Gibbs u-state with negative central Lyapunov exponents. Then there exist an open set $U$ such that $\mu(U\Delta B(\mu))=0$. The analog doesn’t hold for Gibbs u-states with positive central Lyapunov exponents, since the stable and unstable directions play different roles in dissipative systems.
Proof. We build a magnet $K$ over $A_r\cap F^u(x,\delta)$ with fiber $W^s(\cdot,r)$. Then every nearby point $y\in L$ with Birkhoff-regular plaque $F^u(y,2\delta)$, the intersection $F^u(y,2\delta)\cap K$ has positive leaf volume, and some point in there must be Birkhoff-regular, say $p\in W^s(q,r)$ for some $q\in A_r\cap F^u(x,\delta)$. Then Hopf test: for any $z\in F^u(y,2\delta)$, $\phi_-(z)=\phi_-(p)=\phi_+(p)=\phi_+(q)=\phi_-(q)=\phi_-(x).$ So all Birkhoff-regular plaques lie in the same ergodic omponent.

Moreover suppose $\mu$ is the unique Gibbs u-state of $(f,L)$. Then there exists a $C^2$ neighborhood $\mathcal{U}\ni f$ such that for every $g\in\mathcal{U}$, $(g,L_g)$ possesses a unique Gibbs u-state $\mu_g$. Moreover $\mu_g$ has only negative central Lyapunov exponents and $\mu_g\to \mu$ as $g\to f$. So we say $(f,L,\mu)$ is stably ergodic. Since all these measures are hyperbolic, further analysis shows that $(f,L,\mu)$ is indeed stably Bernoulli.

The key property they listed there is: for every $\delta>0$, there exists $r>0$ and $\epsilon>0$ depending continuously of $f$ such that

– for every regular point $x$ with $\chi(x)\cap[-\delta,\delta]=\emptyset$, the frequency of times $n$ such that the size of local Pesin manifolds at $f^nx$ is larger than $r$ is larger than $\epsilon$.

– Moreover, for every ergodic hyperbolic measure $\mu$ with $\chi(\mu)\cap[-\delta,\delta]=\emptyset$, theand hence the set $A_r$ of points with large Pesin manifolds has positive measure: by Kac’s formula, $\displaystyle \mu(A_r)=\int\frac{1}{n}\sum_{0\le k < n}1_{A_r}(x)d\mu\ge \epsilon$.

3. In the continued paper here fundamental domains have been found for many invariant subsets, in particular for the set of (Birkhoff) heteroclinic points $H_f(\mu,\nu)=B(\mu,f)\cap B(\nu,f^{-1})$ (see Theorem 3.2 there, where $\mu\neq \nu$). It is unknown if the argument can be carried out to the set of (Birkhoff) homoclinic points $H_f(\mu)=B(\mu,f)\cap B(\mu,f^{-1})$ (for general invariant but nonergodic measure $\mu$). Here is an example where there does exist a fundamental domain. Consider a flow on the plate $D$ with spiraling source $o$ in the center and two saddles $p,q$ at the corners.

The second picture is from here, and is called Bowen eye-like attractor. Suppose the dynamics is symmetric and $V_f(x)=\mu=\frac{\delta_p+\delta_q}{2}$ for every $x\in D^o\backslash\{o\}$, where $f$ is the time-1 map. Then it is easy to see that there exists a fundamental domain $E$ of $B(f,\mu)$. We can blow up the center, identify the corresponding boundaries of two copies and reverse the flow direction on the second copy. Then the subset $E$ turns out to be a fundamental domain of the set of (Birkhoff) homoclinic points $H_{\hat f}(\mu)$.

2. Let $f:M\to M$ be a $C^2$ partially hyperbolic diffeomorphism, $\mu$ be an Absolutely Continuous, Invariant Probability measure. That is, the density function $\phi=\frac{d\mu}{dm}$ is well defined in $L^1(m)$, and the set $E_\mu=\{x\in M:\phi(x)>0\}$ is well defined in the measure-class of $\mathcal{M}(m)$.

It is proved (Proposition 3, here) that $E_\mu$ is bi-essentially saturated (by a density argument). Similar argument shows that every invariant subset of $E_\mu$ is also bi-essentially saturated. At that time I thought the classical Hopf argument can only claim the bi-essential $\mu$-saturation of $E_\mu$, and Proposition 3 might be out of the range of Hopf argument. Now it seems this is not the case if we combine some results in Gibbs $u$-measures, which states, for example, the conditional measures $\mu_{W^u(x)}$ of $\mu$ with respect to the unstable foliation $\mathcal{W}^u$ is not only abs. cont., but also smooth: the canonical density (see here) $\rho^u_{\text{can}}(x,y)=\frac{d\mu_{W^u(x)}(y)}{dm_{W^u(x)}}$ is Holder, bounded and bounded away from zero, since ACIP is automatically a Gibbs $u$-measure.

So let $E$ be an invariant subset of $E_\mu$. Then Hopf argument implies that

• $\mu_{W^u(x)}(E\backslash W^u(x))=0$ for $\mu$-a.e. $x\in E$, or equivalently,
• $m_{W^u(x)}(E\backslash W^u(x))=0$ for $\mu$-a.e. $x\in E$ (by the previous observation), and moreover
• $m_{W^u(x)}(E\backslash W^u(x))=0$ for $m$-a.e. $x\in E$ (since $\mu\simeq m$ on $E_\mu$).
• Then a standard argument shows that $E$ is essentially $u$-saturated. Similarly ACIP is automatically a Gibbs $s$-measure and $E$ is essentially $s$-saturated. This shows that $E$ is bi-essentially saturated by Hopf argument and Gibbs theory.

1. Let $W$ be a plaque of the Pesin unstable manifold of $f$, and consider a function $\rho(x)$ with the property that $\displaystyle \frac{\rho(x)}{\rho(y)}=\prod_{k\ge1}\frac{J^u(f,f^{-k}y)}{J^u(f,f^{-k}x)}$ for all $x,y\in W$, and the normalizing condition $\int_W \rho\,dm_W=1$. Let $\mu=\rho m_W$ be the induced probability on $W$. It is conditionally invariant under $f$: Consider its pushforward $f\mu=\eta m_{fW}$. Then: $\mu(A)=(f\mu)(fA)=\int_{fA}\eta(y) dm_{fW}(y)=\int_{A}\eta(fx)\cdot J^u(f,x)dm_W(x)$ for any $A\subset W$. Hence $\rho(x)=\eta(fx)\cdot J^u(f,x)$. In particular $\displaystyle \frac{\eta(fx)}{\eta(fy)}=\frac{\rho(x)}{\rho(y)}\cdot\frac{J^u(f,y)}{J^u(f,x)}=\frac{\rho(fx)}{\rho(fy)}$.
Then by definition, both $\rho$ and $\eta$ induce probabilities and must coincide:
$f(\rho\cdot m_W)=(\rho\circ f)\cdot m_{fW}$. Such measures are called the leafwise u-Gibbs measures.

### Some distinguished meausres

This is a note taken from V. Kaimanovich’s paper Bowen-Margulis and Patterson measures on negatively curved compact manifolds.

Let $M$ be a simply connected negatively curved manifold (the exponential map $\exp_x:T_xM\to M$ turns out to be a diffeomorphism for every $x$ by Cartan-Hadamard Theorem). Two geodesic rays on $M$ are called asymptotic if they they have a bounded distance in the future. Denote by $\partial M$ the set of asymptotic classes of geodesic rays on $M$. Then $\overline{M}=M\sqcup\partial M$ is called the visibility compactification of $M$. Let $\partial^2 M=\partial M\times \partial^2 M\backslash \triangle$.

• For each pair $(x,\alpha)\in M\times \partial M$, there exists a unique geodesic ray $\gamma=\gamma_{x,\alpha}\in\alpha$ with $\gamma(0)=x$.
• For each pair $(\alpha,\beta)\in \partial^2 M$, there exists a unique complete geodesic $\gamma$ with $\gamma(R_+)\in\alpha$ and $\gamma(R_-)\in\beta$.
• Each unit vector $\xi\in SM$ corresponds to a complete geodesic $\gamma_\xi$, and hence two maps $\xi\mapsto \alpha(\gamma_\xi)$ and $\xi\mapsto \beta(\gamma_\xi)$. This induces a $\mathbb{R}$-fibration $SM\to \partial^2 M$, $\xi\mapsto (\alpha(\gamma_\xi),\alpha(\gamma_\xi))$.
• For each pair $(x,\alpha)$, we pick the geodesic ray $\gamma$ and define Busemann function $b_{\alpha,x}:y\in M\mapsto \lim_{t\to\infty}(d(x,\gamma(t))-t)$. The level sets of $b_{\alpha,x}$ are the horospheres on $M$ centered at $\alpha\in\partial M$.
• More generally, let $b_\alpha(y,z)=b_{\alpha,x}(y)-b_{\alpha,x}(z)$ (need to check it is independent of the choice of $x$ and $\gamma_{\alpha,x}$) (Remark). Moreover, $b_\alpha(y,z)$ gives the signed distance between the horospheres passing through $y$ and $z$ centered at $\alpha$.
Remark: Another view point is $b_\alpha(y,z)=b_{\alpha,z}(y)$. Clearly it is independent of $x$.
• Define another function $B_x:(\alpha,\beta)\in\partial ^2M \mapsto b_\alpha(x,y)+b_\beta(x,y)$, where $y$ lies on the geodesic $\gamma_{\alpha,\beta}$. (Check it is independent of the choices of $y$ on that geodesic). Geometrically, it measures the length of the segment cut out from $\gamma$ by the horospheres passing through $x$ and centered at $\alpha$ and $\beta$. Moreover we have $B_x(\alpha,\beta)-B_y(\alpha,\beta)=b_{\alpha}(x,y)+b_{\beta}(x,y)$.
• ### Short notes

8. (Alejandro) Let $f:X\to X$ be an arbitrary transitive homeomorphism and $u:X\to(0,1/4)$ be an arbitrary non-constant continuous function. Then, let’s define $c(x):=u(x)-u(fx)+1$, $x\in X$, and consider the suspension flow $f_t:X_c\to X_c$on $X_c$. Note that for each $x\in X$ and $t\in(0,1/4)$: $f_1(x,t+u(x))=(x,t+u(x)+1)=(x,t+u(fx)+c(x))=(fx,t+u(fx))$. So the compact set $\lbrace(x,t+u(x)):x\in X\rbrace$ is $f_1$-invariant for every $t\in(0,1/4)$, and $f_1$ is not transitive. Notice the function c is not constant because $f$ is transitive and $u$ is not constant itself.

### 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 $\mathcal{P}$.