## There is no positively expansive homeomorphism

Let $f$ be a homeomorphism on a compact metric space $(X,d)$. Then $f$ is said to be $\mathbb{Z}$-expansive, if there exists $\delta>0$ such that for any two points $x,y\in X$, if $d(f^nx,f^ny)<\delta$ for all $n\in\mathbb{Z}$, then $x=y$. The constant $\delta$ is called the expansive constant of $f$.

Similarly one can define $\mathbb{N}$-expansiveness if $f$ is not invertible. An interesting phenomenon observed by Schwartzman states that

Theorem. A homeomorphism $f$ cannot be $\mathbb{N}$-expansive (unless $X$ is finite).

This result was reported in Gottschalk–Hedlund’s book Topological Dynamics (1955), and a proof was given in King’s paper A map with topological minimal self-joinings in the sense of del Junco (1990). Below we copied the proof from King’s paper.

Proof. Suppose on the contrary that there is a homeo $f$ on $(X,d)$ that is $\mathbb{N}$-expansive. Let $\delta>0$ be the $\mathbb{N}$-expansive constant of $f$, and $d_n(x,y)=\max\{d(f^k x, f^k y): 1\le k\le n\}$.

It follows from the $\mathbb{N}$-expansiveness that $N:=\sup\{n\ge 1: d_n(x,y)\le\delta \text{ for some } d(x,y)\ge\delta\}$ is a finite number. Pick $\epsilon\in(0,\delta)$ such that $d_N(x,y)<\delta$ whenever $d(x,y)<\epsilon$.

Claim. If $d(x,y)<\epsilon$, then $d(f^{-n} x, f^{-n}y)<\delta$ for any $n\ge 1$.

Proof of Claim. If not, we can prolong the $N$-string since $f^{k}=f^{k+n}\circ f^{-n}$.

Recall that a pair $(x,y)$ is said to be $\epsilon$-proximal, if $d(f^{n_i}x, f^{n_i}y)<\epsilon$ for some $n_i\to\infty$. The upshot for the above claim is that any $\epsilon$-proximal pair is $\delta$-indistinguishable: $d(f^{n}x, f^{n}y)<\delta$ for all $n$.

Cover $X$ by open sets of radius $< \epsilon$, and pick a finite subcover, say $\{B_i:1\le i\le I\}$. Let $E=\{x_j:1\le j\le I+1\}$ be a subset consisting of $I+1$ distinct points. Then for each $n\ge 0$, there are two points in $f^n E$ share the room $B_{i(n)}$, say $f^nx_{a(n)}$, and $f^nx_{b(n)}$. Pick a subsequence $n_i$ such that $a(n_i)\equiv a$ and $b(n_i)\equiv b$. Clearly $x_a\neq x_b$, and $d(f^{n_i}x_a,f^{n_i}x_b)<\epsilon$. Hence the pair $(x_a,x_b)$ is $\epsilon$-proximal and $\delta$-indistinguishable. This contradicts the $\mathbb{N}$-expansiveness assumption on $f$. QED.

## Area under holomorphic maps

Let $f$ be a map from $(x,y)\in \mathbb{R}^2$ to $(a,b)\in \mathbb{R}^2$. The area form $dA=dx\wedge dy$ gives the Jacobian $dA=da\wedge db= J(x,y)dx\wedge dy$, where $J(x,y)=a_xb_y- a_yb_x$.

Now consider the complex setting, where $\displaystyle dA=\frac{i}{2} dz\wedge d\bar z$. Let $f$ be a map from $z\in \mathbb{C}$ to $w\in \mathbb{C}$. Then $\displaystyle dA=\frac{i}{2} dw\wedge d\bar w= \frac{i}{2}f'(z)\overline{f'(z)} dz\wedge d\bar z$. So this time the Jacobian $J(z)$ becomes $f'(z)\overline{f'(z)}$.

Suppose $\displaystyle f(z)=\sum_{n\ge 0} a_n z^n$ is a holomorphic map on the unit disk $D$. Then
$\displaystyle J(z)=\sum_{n,m\ge 0}nm a_n \bar a_m z^{n-1} \bar z^{m-1}$, the area of $f(D)$ is $\displaystyle \int_D J_f(z) dA$.

Using polar coordinate, we have $dA= rdr\, d\theta$, $\displaystyle z^{n-1} \bar z^{m-1}=r^{n+m-2}e^{i\theta(n-m)}$,
and $\displaystyle \int_D r^{n+m-2}e^{i\theta(n-m)} rdr\, d\theta= 0$ if $n\neq m$, and $=\frac{\pi}{n}$ if $n=m$.

So $\displaystyle |f(D)|=\sum_{n\ge 0} n^2 |a_n|^2\cdot \frac{\pi}{n}=\pi \sum_{n\ge 0} n |a_n|^2$.

## An interesting lemma about the Birkhoff sum

A few days ago I attended a lecture given by Amie Wilkinson. She presented a proof of Furstenberg’s theorem on the Lyapunov exponents of random products of matrices in $SL(2,\mathbb{R})$.

Let $\lambda$ be a probability measure on $SL(2,\mathbb{R})$, $\mu=\lambda^{\mathbb{N}}$ be the product measure on $\Omega=SL(2,\mathbb{R})^{\mathbb{N}}$. Let $\sigma$ be the shift map on $\Omega$, and $A:\omega\in\Omega\mapsto \omega_0\in SL(2,\mathbb{R})$ be the projection. We consider the induced skew product $(f,A)$ on $\Omega\times \mathbb{R}^2$. The (largest) Lyapunov exponent of $(f,A)$ is defined to be the value $\chi$ such that $\displaystyle \lim_{n\to\infty}\frac{1}{n}\log\|A_n(\omega)\|=\chi$ for $\mu$-a.e. $\omega\in \Omega$.

To apply the ergodic theory, we first assume $\int\log\|A\| d\lambda < \infty$. Then $\chi(\lambda)$ is well defined. There are cases when $\chi(\lambda)=0$:

(1) the generated group $\langle\text{supp}\lambda\rangle$ is compact;

(2) there exists a finite set $\mathcal{L}=\{L_1,\dots, L_k\}$ of lines that is invariant for all $A\in \langle\text{supp}\lambda\rangle$.

Furstenberg proved that the above cover all cases with zero exponent:
$\chi(\lambda) > 0$ for all other $\lambda$.

## 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.

## The real decay rate

Let $f$ be a $C^2$ uniform expanding map on the 1-torus $\mathbb{T}$, $\mu$ be the unique ACIP of $f,$ which is exponentially mixing. That is, there exists $\lambda\in(0,1)$ such that $|C(\phi,\psi,f^n)|\le C\lambda^n$ for any two Lipschitz functions $\phi,\psi$ on $\mathbb{T}$, where
$C(\phi,\psi,f^n,\mu)=\int \phi\circ f^n\cdot \psi d\mu -\mu(\phi)\cdot\mu(\psi)$ be the correlation function.

Let $h$ be a $C^2$ diffeomorphism on $\mathbb{T}$, $g=h^{-1}fh$ be the induced map, and $h_\ast \nu=\mu.$ The new correlation function

$C(\phi,\psi,g^n,\nu)=\int \phi\circ g^n\cdot \psi d\nu-\nu(\phi)\cdot\nu(\psi)$
$=\int \hat\phi\circ f^n(hx)\cdot \hat\psi(hx) d\nu-\nu(\phi)\cdot\nu(\psi)$
$=\int \hat\phi\circ f^n \cdot \hat\psi d\mu-\mu(\hat\phi)\cdot\mu(\hat\psi)$,

where $\hat\phi=\phi\circ h^{-1}$. Therefore, the two smoothly conjugate systems $(g,\nu)$ and $(f,\mu)$ have the same mixing rate.

Assuming $h$ is close to identity, we see that $g=h^{-1}fh$ is also $C^2$ uniformly expanding, and one may derive the mixing rate of $(g,\nu)$ independently. However, this new rate may be different (better or worse) from $\lambda$. For example, $f$ could be the linear expanding ones and archive the best possible rate among its conjugate class. Could one detect this rate from $(g,\nu)$ itself?

In the general case, two expanding maps on $\mathbb{T}$ are only topologically conjugate (via full shifts). So it is possible that the decay rate varies in the topologically conjugate classes.

## Dynamics of the Weil-Petersson flow: basic geometry of the Weil-Petersson metric II

In the first post of this series, we planned to discuss in the third and fourth posts the proof of the following ergodicity criterion for geodesic flows in incomplete negatively curved manifolds of Burns-Masur-Wilkinson:

Theorem 1 (Burns-Masur-Wilkinson) Let $latex {N}&fg=000000$ be the quotient $latex {N=M/Gamma}&fg=000000$ of a contractible, negatively curved, possibly incomplete, Riemannian manifold $latex {M}&fg=000000$ by a subgroup $latex {Gamma}&fg=000000$ of isometries of $latex {M}&fg=000000$ acting freely and properly discontinuously. Denote by $latex {overline{N}}&fg=000000$ the metric completion of $latex {N}&fg=000000$ and $latex {partial N:=overline{N}-N}&fg=000000$ the boundary of $latex {N}&fg=000000$.Suppose that:

• (I) the universal cover $latex {M}&fg=000000$ of $latex {N}&fg=000000$ is geodesically convex, i.e., for every $latex {p,qin M}&fg=000000$, there exists an unique geodesic segment in $latex {M}&fg=000000$ connecting $latex {p}&fg=000000$ and $latex {q}&fg=000000$.
• (II) the metric completion $latex {overline{N}}&fg=000000$ of $latex {(N,d)}&fg=000000$ is compact.
• (III) the boundary $latex {partial N}&fg=000000$ is volumetrically cusplike, i.e., for…

View original post 7,622 more words

## Dynamics of the Weil-Petersson flow: basic geometry of the Weil-Petersson metric I

Today we will define the Weil-Petersson (WP) metric on the cotangent bundle of the moduli spaces of curves and, after that, we will see that the WP metric satisfies the first three items of the ergodicity criterion of Burns-Masur-Wilkinson (stated as Theorem 3 in the previous post).

In particular, this will “reduce” the proof of the Burns-Masur-Wilkinson theorem (of ergodicity of WP flow) to the verification of the last three items of Burns-Masur-Wilkinson ergodicity criterion for the WP metric and the proof of the Burns-Masur-Wilkinson ergodicity criterion itself.

We organize this post as follows. In next section we will quickly review some basic features of the moduli spaces of curves. Then, in the subsequent section, we will start by recalling the relationship between quadratic differentials on Riemann surfaces and the cotangent bundle of the moduli spaces of curves. After that, we will introduce the Weil-Petersson and the Teichmüller metrics…

View original post 7,184 more words

## Dynamics of the Weil-Petersson flow: Introduction

Boris Hasselblatt and Françoise Dal’bo are organizing the event “Young mathematicians in Dynamical Systems” at CIRM (Luminy/Marseille, France) from November 25 to 29, 2013.

This event is part of the activities around the chaire Jean-Morlet of Boris Hasselblatt. Among the topics scheduled in this event, there is a mini-course by Keith Burns and myself around the dynamics of the Weil-Petersson (WP) geodesic flow.

In our mini-course, Keith and I plan to cover some aspects of Burns-Masur-Wilkinson theorem on the ergodicity of WP flow and, maybe, some points of our joint work with Masur and Wilkinson on the rates of mixing of WP flow.

In order to help me prepare my talks, I thought it could be a good idea to make my notes available on this blog.

So, this post starts a series of 6 posts (vaguely corresponding the 6 lectures of the mini-course) on the dynamics of…

View original post 4,677 more words

## Martingale and its application to dynamical systems

In the last week of May I attended two lectures given by Professor Matthew Nicol.

Let $(\Omega,\mu)$ be a prob space with a $\sigma$-algebra $\mathcal{B}$. Let $\mathcal{F}\prec \mathcal{B}$ be a sub $\sigma$-algebra.

Example. $f(x)=2x (\text{mod} 1)$ on $\mathbb{T}$, and $\mathcal{B}$ be the Borel $\sigma$-algebra. Let $\mathcal{F}=f^{-1}\mathcal{B}$. Note that $(0.2,0.3)\notin\mathcal{F}$.

Let $Y$ be a $\mathcal{B}$-measurable r.v. and $Y\in L^1(\mu)$. The conditional expectation $E(Y|\mathcal{F})$ is the unique $\mathcal{F}$-measurable r.v. $Z$ satisfying $Z^{-1}(a,b)\in \mathcal{F}$ for all $(a,b)$, and $\int_F Z d\mu=\int _A Y d\mu$ for all $A\in \mathcal{F}$.

Note that $E(Y|\mathcal{F})=Y$ if and only if $Y$ is $\mathcal{F}$-measurable; and $E(Y|\mathcal{F})=E(Y)$ if $Y$ is independent of $\mathcal{F}$.

Let $(X_n)_{n\ge 0}$ be a stationary ergodic process with stationary initial distribution $\mu$. A basic problem is to find sufficient conditions on $(X_n)_{n\ge 0}$ and on functions $\phi\in L^2_0(\mu)$ such that $\displaystyle S_n(\phi)=\sum_{k=1}^n \phi(X_k)$ satisfies the central limit theorem (CLT) $\displaystyle \frac{1}{\sqrt{n}}S_n(\phi) \to N(0,\sigma^2)$, where the limit variance is given by $\displaystyle \sigma^2(\phi)=\lim_{n\to\infty}\frac{1}{n}E(S^2_n(\phi))$.

Let $f$ be a conservative diffeomorphism on $(M,m)$. There are two operators: $\phi\mapsto U\phi=\phi\circ f$, and $\phi\mapsto P\phi$ via $\int P\phi\cdot \psi=\int \phi\cdot \psi\circ f$ for all test function $\psi$.

Property. $PU(\phi)=\phi$ (vol-preserving) and $UP(\phi)=E(\phi|f^{-1}\mathcal{B})$.

Let $\mathcal{F}_n$ be an increasing sequence of $\sigma$-algebras. Then a sequence of r.v. $S_n$ is called a martingale w.r.t. $\mathcal{F}_n$, if $S_n$ is $\mathcal{F}_n$-measurable, $E(S_{n+1}|\mathcal{F}_n)=S_n$.

Let $\mathcal{F}_n$ be a decreasing sequence of $\sigma$-algebras. Then a sequence of r.v. $S_n$ is called a reverse martingale w.r.t. $\mathcal{F}_n$, if $S_n$ is $\mathcal{F}_n$-measurable, $E(S_{n}|\mathcal{F}_m)=S_m$ for any $n\le m$.

Theorem. Let $\{X_n:n\ge 1\}$ be a stationary ergodic sequence of (reverse) martingale differences w.r.t. $\{\mathcal{F}_n\}$. Suppose $E(X_n)=0$, and $\sigma^2=\text{Var}(X_i)>0$. Then $\displaystyle \frac{1}{\sigma\sqrt{n}}\sum_{i=1}^n X_i \to N(0,1)$ in distribution.

Gordin: Suppose $(f,m)$ is ergodic. Consider the Birkhoff sum $\displaystyle \sum_{i=1}^n \phi\circ f^i$ for some $\phi$ with $\int \phi=0$. The time series $\phi\circ f^i$ can be approximated by martingale differences provided the correlations decay quickly enough.

Suppose there exists $p(n)$ with $\sum p(n) < \infty$, such that $\|P^n\phi\|\le C\cdot p(n)\|\phi\|$. Then define $\displaystyle g=\sum_{n\ge 1}P^n\phi$, and let $X=\phi+g-g\circ f$.

Property. Let $f:M\to M$ be such that $f^{-n}\mathcal{B}$ is decreasing. $\displaystyle S_n=\sum_{i=1}^n X\circ f^i$ is a reverse martingale with respect to $f^{-n}\mathcal{B}$.

Proof. Note that $PX=P\phi+Pg-PUg=0$. Then $E(X|f^{-1}\mathcal{B})=UP(X)=U0=0$.
Let $k < n$. It remains to show $E(X\circ f^k|f^{-n}\mathcal{B})=0$. To this end, we pick an element $A\in f^{-n}\mathcal{B}$ and write it as $A=f^{-k-1}C$ for some $C\in f^{k+1-n}\mathcal{B}$. Then $\displaystyle \int_A X\circ f^k dm=\int_{f^{-1}C}X dm =\int_{f^{-1}C} E(X|f^{-1}\mathcal{B}) dm=\int_{f^{-1}C}0 dm=0$. This completes the proof.

Three theorems of Gordin. Let $(\Omega,\mu,T)$ be an invertible $\mu$-preserving ergodic system, $X\in L^1(\mu)$ and $X_k(x)=X(T^kx)$ be a strictly stationary ergodic sequence.

(*) $\displaystyle \limsup_{n\to\infty}\frac{1}{\sqrt{n}}E|S_n| < \infty$

Theorem 1. Suppose there exists $\displaystyle \mathcal{F}_k\subset T^{-1}\mathcal{F}_k=\mathcal{F}_{k+1}$ such that $\displaystyle \sum_{k\ge 0} E|E(X_0|\mathcal{F}_{-k})|<\infty$, $\displaystyle \sum_{k\ge 0} E|X_0-E(X_0|\mathcal{F}_{k})| < \infty$. Then (*) implies $\displaystyle \lambda:=\lim_{n\to\infty}\frac{1}{\sqrt{n}}E|S_n|$ exists, and $\displaystyle \frac{1}{\sqrt{n}}S_n\to N(0,\lambda^2\pi/2)$ in distribution (degenerate if $\lambda=0$).

–Mixing condition. Let $\displaystyle \alpha(n):=\sup\{P(A\cap B)-P(A)P(B):A\in\mathcal{F}^0_{-\infty}, B\in\mathcal{F}^{\infty}_n\}$.

Theorem 2. Suppose for some $1/p+1/q=1$, $X\in L^p(\mu)$ and $\displaystyle \sum_{n\ge 1}\alpha(n)^{1/q} < \infty$. Then (*) implies the conclusion of Theorem 1.

–uniform mixing condition. Let $\displaystyle \phi(n):=\sup\{P(B|A)-P(B):A\in\mathcal{F}^0_{-\infty}, \mu(A) > 0, B\in\mathcal{F}^{\infty}_n\}$.

Theorem 3. Suppose $X\in L^1(\mu)$ and $\displaystyle \sum_{n\ge 1}\phi(n) < \infty$. Then (*) implies the conclusion of Theorem 1.

Cuny–Merlevede: not only the CLT, but also the ASIP holds under the above conditions.

Note that we started with an invariant measure $m$. The operator $U$ and $P$ can be defined for all non-conservative maps. To emphasize the difference, we use $\hat P$. Suppose $\hat P h=h$ for some $h\in L^1(m)$. Then $\mu=hm$ is an absolutely continuous invariant prob. measure:

$\displaystyle \int \phi\circ f d\mu=\int \phi\circ f h dm=\int \phi\cdot \hat P h dm=\int \phi hdm=\int\phi d\mu$.

Then we can rewrite $\displaystyle P\phi=\frac{1}{h}\hat P(h\phi)$, in the sense that $\displaystyle \int P(\phi)\cdot \psi d\mu=\int \phi\cdot \psi\circ f d\mu =\int \phi h\cdot \psi\circ f dm$
$\displaystyle =\int\hat P(\phi h)\cdot \psi dm \int \frac{1}{h}\hat P(\phi h)\cdot \psi d\mu$.