## Foliations

Let $M$ be a closed manifold (mostly 3 dimension) and $\mathcal{F}$ a foliation on $M$. A leaf $F\in\mathcal{F}$ is closed if $\overline{F}=F$ (like a periodic orbit or a genus-$g$ surface). It is proper if $\overline{F}\backslash F$ is closed (like the graph of $y=\sin(1/x)$). It is recurrent if it is either closed (trivial proper) or non-proper (like a line on $\mathbb{T}^2$ with irrational slope). Let $\mathcal{C}$ be the part of closed leaves (note that $\mathcal{C}$ may not be closed, like the periodic orbits of Anosov flow). Let $\mathcal{P}$ be the part of proper leaves (nontrivial proper: not closed) . In particular all leaves outside $\mathcal{P}$ are recurrent.

The foliation $\mathcal{F}$ is said to be non-wandering if $\mathcal{P}$ has no interior. And $\mathcal{F}$ is said to be recurrent if $\mathcal{P}=\emptyset$, that is, every leaf of $\mathcal{F}$ is recurrent. Moreover, $\mathcal{F}$ is said to be almost periodic, if $\{\overline{F}:F\in\mathcal{F}\}$ forms a new decomposition of $M$ (either disjoint or coincide). Yokoyama observed the following proposition:

Proposition. A almost periodic foliation is recurrent.
Proof. Let $F\in\mathcal{F}$ be a non-closed element and $x\in\overline{F}\backslash F$. Clearly $F(x)\cap F=\emptyset$. Then almost periodicity implies that $\overline{F{x}}=\overline{F}$. So $\overline{F}\backslash F=\overline{F{x}}\backslash F\supset F(x)\neq\emptyset$. So every non-closed leaf is not proper and $\mathcal{F}$ is recurrent.

A foliation $\mathcal{F}$ is said to be R-closed if $\{(x,y)\in M\times M:y\in\overline{F(x)}\}$ is a closed subset.

2. A support function on a convex domain is the signed distance. Let $Q$ be a closed strictly convex domain around the origin. Then its support function is given by $h(\phi)=\sup\{x\cos\phi+y\sin\phi:(x,y)\in Q\}$. It is easy to see that the supreme is attained at a point on the boundary, whose oriented tangent line has angle $\phi$ with positive $y$-axis. Using this parameter the billiard system admit a coordinate $(\phi,\theta)$, where $\theta$ is the angle of the out-going vector with the tangent direction. In particular ${\bf v}(\phi,\theta)=e^{i(\theta+\phi+\pi/2)}=\langle -\sin(\theta+\phi),\cos(\theta+\phi) \rangle$.

Let $C$ be a closed piecewise-smooth convex curve around the origin, $\phi$ the angle of the tangent line at a point $(x,y)\in C$ with $y$-axis (serving as a parametrization, so $(x(\phi),y(\phi))$ and the line $L(\phi)$). Then the distance from $o$ to the tangent line $L(\phi)$ is $h(\phi)=x\cos\phi+y\sin\phi$.

Taking derivative with respect to $\phi$, we get $h'(\phi)=x'\cos\phi+y'\sin\phi-x\sin\phi+y\cos\phi$.

$latex$

1. Phase transitions in statistical mechanics. A phase transition occurs when a material changes its properties in a dramatic way. For example water, as it is cooled and turns into ice. Phase transitions are characterized by an order quantity (like density) that changes as a function of a parameter of the system (such as the temperature). The special value of the parameter at which the system changes its phase is the system’s critical point.

A bifurcation occurs in a dynamical systems, when a small/smooth change of the parameter values (the bifurcation parameters) of a system causes a sudden ‘qualitative’ or topological change in its behavior. For example the ‘period-doubling bifurcation’ of Logistic map, the saddle-node bifurcation.

Phase transition in dynamical systems

– the parameters $t$’s where the pressure $P(f,t\phi)$ fail to be $C^k$ for $k=0,1,\cdots,\infty,\omega$;

– the parameters $t$’s where the system $f_t:M^2\to M^2$ shifts from integrability to nonintegrability, from regular to chaotic.

Taken from J. Franks, Anosov diffeomorphisms, in the book ‘Global Analysis’, (1968) 61–93.

Question: given $f:M\to M$, for what $g:N\to N$ does there exist a nontrivial $h:N\to M$ such that $h\circ g=f\circ h$? Franks proved that for some diffeo, it reduces to a homotopy problem. So the definition:

A diffeo $f:M\to M$ is a $\pi_1$-diffeo, if given any homeo $g:K\to K$ on a CW-complex $K$ with a continuous map $h:K\to M$ such that $h_\ast\circ g_\ast=f_\ast\circ h_ast$ from $\pi_1(K)\to \pi_1(M)$, there exists a unique base-point preserving map $\hat h:K\to M$ homotopic to $h$ such that $\hat h\circ g=f\circ\hat h$.

(covering version)

Examples of Anosov: hyperbolic toral automorphisms, hyperbolic nil-manifold automorphisms, hyperbolic infra-nilmanifold automorphisms (and their endomorphisms)

Theorem 2.2. Every hyperbolic infra-nilmanifold automorphism is a $\pi_1$-diffeo.

Problem: are all Anosov diffeo $\pi_1$?

Two diffeos $f:M\to M$ and $g:N\to N$ are $\pi_1$-conjugate if there exists an isomorphism $\phi:\pi_1(N)\to \pi_1(M)$, such that $\phi\circ g_\ast=f_\ast\circ \phi$ from $\pi_1(N)\to \pi_1(M)$.

So two $\pi_1$ diffeos are topological conjugate iff they are $\pi_1$-conjugate.

Theorem 3.6. Suppose $\pi_1(M)$ is torsion-free and $f$ is $\pi_1$ on $M$.
a. if $\pi_1(M)$ is virtually nilpotent, then $f$ is topologically conjugate to a hyperbolic infra-nilmanifold automorphism.
b. if $\pi_1(M)$ is nilpotent, then $f$ is topologically conjugate to a hyperbolic nilmanifold automorphism.
a. if $\pi_1(M)$ is abelian, then $f$ is topologically conjugate to a hyperbolic toral automorphism.

Theorem 6.3. Every transitive codim=1 Anosov is a hyperbolic toral automorphism. Two such diffeo are topological conjugate iff they are $\pi_1$-conjugate.

Theorem 8.2. If $\pi_1(M)$ is virtually nilpotent and $f$ is expanding, then $f$ is topologically conjugate to a hyperbolic infra-nilmanifold endomorphism.
A key step is that if $f:M\to M$ is expanding, then $\pi_1(M)$ has polynomial growth.

## Billiards

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 $Q$.

## Physical entropy production

A simple fact: let $T:X\to X$ be a homeomorphism preserving the measure-class of $\omega$, $J(T^k,x)$ be the Jacobian of $T^k$ at $x$. Then for any sequence $a(k)\to\infty$ with $\sum_k \frac{1}{a(k)}<+\infty$ we have $\limsup_k\frac{J(T^k,x)}{a(k)}=0$ for $\omega$-a.e. $x\in X$. For example $a(k)=e^{t k}$ with $t>0$ or $a(k)=k^{\alpha}$ with $\alpha>1$. This is a direct corollary of Borel-Cantelli property:

Consider the set $E_{k,\delta}=\{x\in X: \frac{J(T^k,x)}{a(k)}\ge\delta\}$. It is easy to see $\omega(E_{k,\delta})\le \frac{1}{\delta\cdot a(k)}$ and hence $\sum\omega(E_{k,\delta})\le \sum\frac{1}{\delta\cdot a(k)}<+\infty$. So $\omega(x\in E_{k,\delta} \text{infinitely often})=0$ for all $\delta>0$

Notes from papers by Jaksic, Pillet, Rey-Bellet, Ruelle and Young.

4. Let $\nu\ll \mu$ be two probability measures on $X$ and $\phi=\frac{d\nu}{d\mu}$ be its density. Then $1=\nu(X)=\mu(\phi)$. Moreover $1/\phi$ is well-defined with respect to $\nu$ and $\nu(1/\phi)=\mu(1)=1$, too.

The relative entropy can be defined as $E(\nu|\mu)=\nu(\log\phi)$ when $\nu\ll\mu$, $+\infty$ otherwise.
Note that $E(\nu|\mu)=\nu(-\log\frac{1}{\phi})\ge-\log\nu(1/\phi)=0$. So the relative entropy is nonnegative.

Convexity: assume $\nu_1,\nu_2\ll\mu$ and $p+q=1$, then $\phi=p\phi_1+q\phi_2$ and $E(p\nu_1+q\nu_2|\mu)\le p\nu_1(\log\phi_1)+q\nu_2(\log\phi_2)$.

## Some notations

7. Let $f$ be an Anosov diffeomorphism and $g\in\mathcal{U}(f)$ be close enough, which leads to a Holder continuous conjugate $h_g:M\to M$ with $g\circ h_g=h_g\circ f$. Ruelle found an explicit formula of $h_g$.

Let $f,g:M\to M$ be two homeomorphisms, $d(f,g)=\sup_M d(fx,gx)$, and $\mathcal{U}(f,\epsilon)=\{g \text{ homeo and }d(f,g)<\epsilon\}$. Let $g\in \mathcal{U}(f,\epsilon)$. Then the map $X_g:x\in M \mapsto \exp^{-1}_{fx}(gx)\in T_{fx}M$ gives a shifted-vector field on $M$, which induces a diffeomorhism $\mathcal{U}(f,\epsilon)\to \mathcal{X}(0_f,\epsilon), g\mapsto X_g$.
Let $f$ be a $C^r$ diffeomprhism. Then $\mathcal{X}^r(0_f,\epsilon)\to \mathcal{U}^r(f,\epsilon), g\mapsto X_g$ induces the local Banach structure and turns $\mathrm{Diff}^r(M)$ into a Banach manifold.

Let $X_g\circ f^{-1}=X_g^s+X_g^u$ be the decomposition of the correction $X_g\circ f^{-1}$ with respect to the hyperbolic splitting $TM= E_g^s\oplus E_g^u$. Then the derivative of $g\mapsto h_g$ in the direction of $X_g$ is given by the vector field $\displaystyle \sum_{n\ge 0}Dg^n X^s_g-\sum_{n\ge1}Dg^{-n}X^u_g$.

6. Let $M$ be a compact orientable surface of genus $g\ge1$, $s\ge1$ and let $\Sigma=\{p_1,\cdots,p_s\}$ be a subset of $M$. Let $\kappa= (\kappa_1,\cdots,\kappa_s)$ be a $s$-tuple of positive integers with $\sum (\kappa_i-1) =2g-2$.

A translation structure on $(M,\Sigma)$ of type $\kappa$ is an atlas on $M\backslash\Sigma$
for which the coordinate changes are translations, and such that each singularity $p_i$
has a neighborhood which is isomorphic to the $\kappa_i$-fold covering of a neighborhood
of $0$ in $\mathbb{R}^2\backslash\{0\}$.

The Teichmüller space $Q_{g,\kappa}= Q(M,\Sigma,\kappa)$ is the set of such structures modulo isotopy relative to $\Sigma$. It has a canonical structure of manifold.

## 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.

## Minimal but non-ergodic volume-preserving systems

In this post we will describe the example constructed by Furstenberg, a volume-preserving diffeomorphism $f\in\mathrm{Diff}^{\omega}_m(\mathbb{T}^2)$ which is minimal, but not ergodic. See also Parry’s book Topics in Ergodic Theory.

Let $\alpha$ be an irrational number and $R_\alpha:\mathbb{T}\to\mathbb{T}$ be the irrational rotation. Let $r:\mathbb{T}\to\mathbb{R}$ be a smooth function, which induces a skew-product $f:\mathbb{T}^2\to\mathbb{T}^2$, $(x,y)\mapsto (x+\alpha,y+r(x))$. Consider the following cohomological equation:

(*)      $\phi(x+\alpha)=e^{2\pi ik\cdot r(x)}\cdot\phi(x)$,
(@)      $\phi(x+\alpha)=k\cdot r(x)+\phi(x)$.

Remark 1. Viewed (@) as a real-valued equation, there is an obstruction for it to admit any solution since $r_0=\int r(x)dx >0$. But viewed as a $\mathbb{T}$-valued function, the obstruction is trivial for $k$ if $k\cdot r_0$ is an integer. (also nontrivial if $r_0$ is an irrational..)

Proposition 1. $f$ is not minimal, then the equation (*) has a continuous $S^1$-valued solution for some $k\ge1$.

Proposition 2. If the equation (*) has some measurable solution for some $k\neq 0$, then $f$ is not ergodic.

Remark 2. If $r(x)\equiv0.5$, then such $f$ is far from minimal and (@) has no real-valued solution for all $d\neq0$. But $k\cdot r_0$ is an integer for even number $k$ and the equation (*) for such $k$ admits (trivial) constant solutions.

Theorem. By a suitable choice of $\alpha$ and $r$, the above equation has a $L^2$-solution, but no continuous solution. In particular the corresponding system is minimal but not ergodic.

## Uniform Ergodic Theorem

Today I saw a paper on arxiv entitled with “semiuniform ergodic theorem”. This is the first time I saw such a theorem and I want to take a note about it.

Let’s start with $f:X\to X$, a homeomorphism on a compact metric space. Let $\mathcal{M}(f)$ be the collection of $f$-invariant probability measures on $X$. Let $\phi:X\to \mathbb{R}$ be a continuous function. Birkhoff ergodic theorem states that the time-average $\displaystyle \frac{1}{n}\sum_{k=0}^{n-1}\phi(f^kx)\to\phi^\ast(x)$ for almost every $x\in X$, where $\phi^\ast$ is a almost every defined, measurable function.

A special case is that $\mathcal{M}(f)$ is a singleton. Such a map is called uniquely erogdic. In this case the time-average converges uniformly to a constant $\phi_f$ for every $x\in X$.

The uniform ergodic theorem concerns an intermediate case, that $\mathcal{M}(f,\phi)=\{\mu(\phi):\mu\in \mathcal{M}(f)\}$ is a singleton (say also $\phi_f$), and states that the time-average of $\phi$ converges uniformly to a constant $\phi_f$ for every $x\in X$.

For the general case, $\mathcal{M}(f,\phi)$ is a compact interval, say $[\underline{\phi},\overline{\phi}]$. Let’s show that the time average will fall close to this interval uniformly on $X$.

Proof. We will derive a contradiction by assuming the contrary that, there exists $\delta>0$, $x_k\in X$ and $n_k\to+\infty$ such that $\displaystyle \frac{1}{n}\sum_{i=0}^{n_k-1}\phi(f^ix_k)\notin[\underline{\phi}-\delta,\overline{\phi}+\delta]$ for every $k\ge1$. Then consider the sequence $\displaystyle \frac{1}{n}\sum_{i=0}^{n_k-1}\delta{f^ix_k}$. Passing to a subsequence if necessary, we can assume that it converges to some $\mu\in\mathcal{M}(f)$, which will force $\displaystyle \frac{1}{n}\sum_{i=0}^{n_k-1}\phi(f^ix_k)\to a\in[\underline{\phi},\overline{\phi}]$ and contradict the choice of $(x_k,n_k)$. Q.E.D.

Then let’s consider a subadditive sequence $\Phi=\{\phi_n:n\ge1\}$. Denote $\mu(\Phi)=\inf_{n\ge1} \frac{\mu(\phi_n)}{n}$ and $\mathcal{M}(f,\Phi)=\{\mu(\Phi):\mu\in\mathcal{M}(f)\}$. Then Semiuniform Erogdic Theorem concerns one-side estimate similar to UET. It states that if $\mathcal{M}(f,\Phi)\subset(-\infty,a)$, then there exist $\delta>0$ and $N\ge 1$ such that $\displaystyle\frac{\phi_n(x)}{n}\le a-\delta$ for every $n\ge N$, and every $x\in X$.

Proof. We will derive a contradiction by assuming the contrary that, for each $k\ge1$, there exist $x_k\in X$ and $n_k\to+\infty$ such that $\displaystyle \frac{1}{n}\Phi_{n_k}(x_k)\ge a-1/k$ for every $k\ge1$. Then consider the sequence $\displaystyle \frac{1}{n}\sum_{i=0}^{n_k-1}\delta{f^ix_k}$. Passing to a subsequence if necessary, we can assume that it converges to some $\mu\in\mathcal{M}(f)$. Then we use a common trick to show $\mu(\Phi)\ge a$ and hence contradicts the choice of $a$. Note that it suffices to show $\mu(\phi_N)/N \ge a$ for each $N\ge1$. From now on let’s fix $N\ge1$.

Let $0\le i\le N-1$ and decompose $n_k=i+b_iN+a_i$ for some $0\le a_i\le N-1$. So $\displaystyle \phi_{n_k}(x_k)\le \phi_i(x_k)+\sum_{j=0}^{b_i-1}\phi_N(f^{jN+i}x_k)+\phi_{a_i}(f^\ast x_k)$. Summing over $i$ and divide both sides by $b_iN^2$, we get
$displaystyle \frac{\phi_{n_k}(x_k)}{b_iN}\le\frac{A_N}{b_i}+\frac{1}{b_iN^2}\sum_{j=0}^{b_iN-1}\phi_N(f^j x_k)+\frac{A_N}{b_i}$. Now passing $k\to\infty$ we get $b_i\to\infty$ and hence $\displaystyle a\le \liminf_{k\to\infty} \frac{\phi_{n_k}(x_k)}{b_iN}\le\frac{\mu(\phi_N)}{N}$. Q.E.D.

## Basic set of a smooth flow: Bowen’s trichotomy

This is a note taken from Bowen, Periodic Orbits for Hyperbolic Flows, American Journal of Mathematics (1972), 1–30.

Let $f:M\to M$ be a transitive Anosov flow. Then

1. either it is mixing: then strong stable and strong unstable manifold everywhere dense on $M$
2. or it is a suspension: choose the closure of a non-dense stable manifold as a cross-section and the induced roof-function is constant.

Anosov proved this in volume-preserving case and Plante proved it for the general case (Anosov flows, 1972).

Let $M$ be a closed manifold and $\phi_t:M\to M$ be a $C^1$ flow. Let $\Omega$ be a closed, invariant subset without fixed points, that is, the vector field of $\phi$ does not admits zeros on $\Omega$. Then $\Omega$ is said to be a basic set of $\phi$ if
$(\Omega,\phi_t)$ is (topologically) transitive and hyperbolic,
– close orbits are dense in $\Omega$
$(\Omega,\phi_t)$ is isolated: $\Omega=\bigcap_{\mathbb{R}}\phi_t(U)$ for some open neighborhood $U$ of $\Omega$.
The last one is equivalent to the local product structure.

Theorem 3.2 in [B]. There are three mutually exclusive types:
1. $\Omega$ consists of a single closed orbit of $\phi$;
2. the strong stable manifold $W^s(p)$ is dense in $\Omega$ for for each $p\in\Omega$;
3. $(\Omega,\phi_t)$ is the constant suspension of a Axiom A homeomorphism.

As remarked by Bowen, this is first proved by Anosov for volume-preserving Anosov flow, by Plante for general Anosov flow. We should view the following as a proof for Anosov flow case for first reading, and then take the induced topology on $\Omega$ for basic sets.