Mathematics.
Dynamics.

### Some special actions

Consider the conjugate action $\rho$ of $GL(2,R)$ on $M(2,R)$: $\rho_A(M) = A MA^{-1}$.

1. This action $\rho$ factors through an action of $PGL(2,R)$.

2. There exists a 3D invariant subspace $E=\{M\in M(2,R): tr(M)=0\}$.

3. The determinant $\det M$ is an invariant quadratic form on $E$, and the signature of this form is $(-, - ,+)$.

Let $Q=x_1^2 + x_2^2 - x_3^2$ be a quadratic form on $R^3$, whose isometry group is $O(2,1)=\{A\in M(3,R): A^TgA=g\}$, where $g=\mbox{diag}\{1, 1, -1\}$.

This induces an injection $PGL(2,R) \subset O(2,1)$, and an identification between $PSL(2,R)$ and the connected component of $O(2,1)$.

The action $O(2,1)$ on $R^3$ passes on to the projective space $P^2$. The cone $C=Q^{-1}(0)$ is invariant, and separates $P^2$ into two domains: one of them is homeomorphic to a disk, which the other is a Mobius band. This induces an action of $PSL(2,R)$ on the disk.

### Equilibrim states

Let $S=\{1,\dots, l\}$ be the space of symbols, $A=(a_{ij})$ be an $l\times l$ matrix with $a_{ij}\in\{0,1\}$, $\Sigma_A$ be the set of sequences $x=(x_n)$ that is $A$-admissible. Consider the dynamical system $(\Sigma_A, \sigma)$ We assume this system is mixing.

Let $f:\Sigma_A \to \mathbb{R}$ be a Holder potential, which induces a transfer operator $L_f$ on the space of continuous functions: $\phi(x) \mapsto L_f\phi(x):=\sum_{\sigma y =x} e^{f(x)}\phi(x)$.

Let $\lambda$ be the spectral radius of $L_f$. Then $\lambda$ is also an eigenvalue of $L_f$, which is called the principle eigenvalue. Moreover, there exists a positive eigenfunction $h$ such that $L_f h =\lambda h$. Replacing $f$ by $f-\log\lambda$, we will assume $\lambda =1$.

Consider the conjugate action $L_f^{\ast}$ on the space of functional (or sign measures). There is a positive eigenmeasure $\nu$ such that $L_f^{\ast} \nu =\nu$.

We normalize the pair $(h,\nu)$ such that $\int h d\nu =1$. Then the measure $\mu:= h \nu$ is a $\sigma$-invariant probability measure. It is called the equilibrium state of $(\Sigma_A, \sigma, f)$.

Two continuous functions $f, g:\Sigma_A \to \mathbb{R}$ is called cohomologous if there exists a continuous function $\phi:\Sigma_A \to \mathbb{R}$ such that
$f(x)-g(x) =\phi(\sigma x) -\phi(x)$.

Let $f, g:\Sigma_A \to \mathbb{R}$ be cohomologous. Then the two operators $L_f$ and $L_g$ are different, but $\lambda(f) =\lambda(g)=1$.
Their eigenfunctions and eigenmeasures are different, but the associated equilibrium states are the same.

To find a natural representative in the class $[f]$ of functions that are cohomologous to $f$, we set $g(x)=f(x)+ \log h(x) -\log h(\sigma x)$. Then we have

1). $\displaystyle L_g1(x)=\sum_{\sigma y =x} e^{g(y)}\cdot 1= \sum_{\sigma y =x} e^{f(y)}h(y)/h(x)=\frac{L_fh(x)}{h(x)}=1$. So $1$ is the eigenfunction of $L_g$.

2). $\displaystyle \int \phi dL_g^{\ast} \mu=\int L_g\phi d\mu =\int L_f(\phi h)d\nu =\int \phi\cdot h dL_f^{\ast}\nu =\int \phi h d\nu =\phi d\mu$.
So $\mu$ is the eigenmeasure of $L_g$.

From this point of view, we might pick $g(x)=f(x)+ \log h(x) -\log h(\sigma x)$ as the representative of $[f]$.

### Notes. Some basic terms

1. Let $R$ be a commutative ring, $S$ be a multiplicatively closed subset in the sense that $a,b\in S \Rightarrow ab \in S$. Then we consider the localization $S^{-1}R$ as the quotient $S\times R/\sim$, where $(r,a)\sim (s,b)$ if $(br-as)t=0$ for some $t\in S$.

Let $f\in R$. We can construct a m.c.subset $S=\{f^n: n\ge 0\}$, and denote the corresponding local ring by $R_f=S^{-1}R$.

Let $p\triangleleft R$ be a prime ideal of $R$. Then $S=R\backslash p$ is m.c. We denote the corresponding local ring by $R_p=S^{-1}R$.

Let $\text{Spec}R$ be the set of all prime ideals of $R$. For each ideal $I\triangleleft R$, let $V_I=\{p\in \text{Spec}R: p\supset I\}$. The Zariski topology on $\text{Spec}R$ is defined that the closed subsets are exactly $\{V_I: I\triangleleft R\}$.

A basis for the Zariski topology on $\text{Spec}R$ can be constructed as follows. For each $f\in R$, let $D_f\subset \text{Spec}R$ to be the set of prime ideals not containing $f$. Then each $D_f= \text{Spec}R\backslash V_{(f)}$ is open.

The points corresponding to maximal ideals $m \triangleleft R$ are closed points in the sense that the singleton $\{m\}=V_m$.

In the case $R=C[x_1, \dots, x_n]$, we see that each maximal ideal $m=\langle x_1-a_1,\dots, x_n-a_n \rangle$ corresponds to a point $(a_1,\dots, a_n)\in C^n$. So one can interprat this as $C^n \subset X= \text{Spec} R$. A non-max prime ideal $p$ (a non-closed point) corresponds an affine variety $P$, which is a closed subset in $C^n$. Then $p$ is called the generic point of the varity $P$.

2. Let $(M,\omega)$ be a symplectic manifold, $G$ be a Lie group acting on $M$ via symplectic diffeomorphisms. Let $\mathfrak{g}$ be the Lie algebra of $G$. Each $\xi \in \mathfrak{g}$ induces a vector field $\rho(\xi):x\in M \mapsto \frac{d}{dt}\Big|_{t=0}\Big(\exp(t\xi)\cdot x\Big)$. Note that $\rho(g^{-1}\xi g)=g_\ast \rho(\xi)$, and $\rho([\xi,\eta])=[\rho(\xi),\rho(\eta)]$.

Consider the 1-form induced by the contraction $\iota_{\rho(\xi)}\omega$. Clearly this 1-form is closed: $d\iota_{\rho(\xi)}\omega=L_{\rho(\xi)}\omega=0$ since $G$ preserves the form $\omega$.

Then the action is called weakly Hamiltonian, if for every $\xi\in \mathfrak{g}$, the one-form $\iota_{\rho(\xi)} \omega$ is exact: $\iota_{\rho(\xi)} \omega=dH_\xi$ for some smooth function $H_{\xi}$ on $M$. Although $H_\xi$ is only determined up to a constant $C_\xi$, the constant $\xi \mapsto C_\xi$ can be chosen such that the map $\xi\mapsto H_\xi$ becomes linear.

The action is called Hamiltonian, if the map $\mathfrak{g} \to C^\infty(M)$, $\xi\mapsto H_\xi$ is a Lie algebra homomorphism with respect to Poisson structure. Then $\rho(\xi)=X_{H_\xi}$ and $H_{g^{-1}\xi g}(x)=H_\xi(gx)$.

A moment map for a Hamiltonian $G$-action on $(M,\omega)$ is a map $\mu: M\to \mathfrak{g}^\ast$ such that $H_\xi(x)=\mu(x)\cdot \xi$ for all $\xi\in \mathfrak{g}$. In other words, for each fixed point $x\in M$, the map $\xi \mapsto H_\xi(x)$ from $\mathfrak{g}$ to $\mathbb{R}$ is a linear functional on $\mathfrak{g}$ and is denoted by $\mu(x)$. Also note that $\mu(gx)\cdot \xi=H_\xi(gx)=H_{g^{-1}\xi g}(x)$. So $\mu(gx)=g\mu(x)g^{-1}$.

### 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…

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