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.

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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.

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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:


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

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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).

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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.

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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<r. 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.

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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).
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    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_con 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.

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    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}.

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