Temporal Distributional Limit Theorems for Dynamical Systems

Abstract

Suppose $\{T^t\}$ is a Borel flow on a complete separable metric space X, $f:X\rightarrow \mathbb R$ is Borel, and $x\in X$. A temporal distributional limit theorem is a scaling limit for the distributions of the random variables $X_T:=\int _0^t f(T^s x)ds$, where t is chosen randomly uniformly from [0, T], x is fixed, and $T\rightarrow \infty$. We discuss such laws for irrational rotations, Anosov flows, and horocycle flows.

Appendices

Appendix 1: Veech Group of the Cylinder Suspension

Here we prove Proposition 4.2. We do this in two steps. First we construct many automorphisms $\psi :M\rightarrow M$, and then we show using the methods of [41] that they lift to automorphisms $\widetilde{\psi }:\widetilde{M}\rightarrow \widetilde{M}$.

1.1 The Automorphisms of M

Lemma 8.1

Let $\Gamma (q):=\{A\in \mathrm {SL}(2,\mathbb Z):A=I\mod q\}$. For every $A\in \Gamma (q)$ there exists a linear automorphism $\psi :M_0\rightarrow M_0$ which fixes $\bullet , \circ$, and has derivative A. Necessarily, $\psi |_M$ is an automorphism of M.

Proof

Every matrix $A\in \mathrm {SL}(2,\mathbb Z)$ determines an automorphism $\psi :\mathbb R^2/\mathbb Z^2\rightarrow \mathbb R^2/\mathbb Z^2$ which fixes $\bullet$ and has derivative A: Take $\psi ({x\atopwithdelims ()y}+\mathbb Z^2)=A{x\atopwithdelims ()y}+\mathbb Z^2.$ This automorphism fixes $\circ$ iff

$$\begin{aligned} A{p/q\atopwithdelims ()0}\in {p/q\atopwithdelims ()0}+\mathbb Z^2\ , \ A^{-1}{p/q\atopwithdelims ()0}\in {p/q\atopwithdelims ()0}+\mathbb Z^2. \end{aligned}$$

(8.1)

Write $A=\bigl ({\tiny \begin{array}{cc} a &{} b \\ c &{} d \end{array}}\bigr )$, $A^{-1}=\bigl ({\tiny \begin{array}{rr} d &{} -b \\ -c &{} a \end{array}}\bigr )$, then (8.1) holds iff $\frac{ap}{q}\in \frac{p}{q}+\mathbb Z$, $\frac{cp}{q}\in \mathbb Z$, $\frac{dp}{q}\in \frac{p}{q}+\mathbb Z$, $\frac{-cp}{q}\in \mathbb Z$. This is equivalent to

$$\begin{aligned} a=1(\mod q),\quad d=1(\mod q),\quad c=0(\mod q). \end{aligned}$$

In particular, if $A\in \Gamma (q)$, then $\psi :M_0\rightarrow M_0$ fixes $\bullet ,\circ$ and has derivative A. $\square$

1.2 Lifting the Automorphisms of M to $\widetilde{M}$

Theorem 8.2

(Hooper-Weiss) Let $M,\widetilde{M}$ be as above, then every automorphism of M fixing $\bullet ,\circ$ lifts to an automorphism of $\widetilde{M}$.

Theorem 8.2 can be easily deduced from general results in [41], but we decided to include a self-contained proof for completeness. The following proof, which uses the methods of [41], was explained to us by Pat Hooper and Barak Weiss.

First we give a general criterion for liftability of maps of M to maps of $\widetilde{M}$, and then we check this criterion for automorphisms.

Suppose $\psi :M\rightarrow M$ is a homeomorphism which fixes some point $x_0$, and fix some lift $\widetilde{x}_0\in \widetilde{M}$. For every $\widetilde{x}\in \widetilde{M}$:

• (a) Choose a smooth path $\widetilde{\gamma }_{\widetilde{x}}(t)$ from $\widetilde{x}_0$ to $\widetilde{x}$;

• (b) Form the curve $\psi \circ p\circ \widetilde{\gamma }_{\widetilde{x}}$. This is a smooth path from $x_0$ to $\psi (p(\widetilde{x}))$;

• (c) Let $\widetilde{\psi }(\widetilde{x}):=$endpoint of the lift of $\psi \circ p \circ \widetilde{\gamma }_{\widetilde{x}}$ to $\widetilde{M}$ at $\widetilde{x}_0$.

If we can show that $\widetilde{\psi }(\widetilde{x})$ is independent of the choice of $\widetilde{\gamma }_{\widetilde{x}}$, then it will be a simple matter to conclude that $\widetilde{\psi }:\widetilde{M}\rightarrow \widetilde{M}$ is a continuous map such that $p\circ \widetilde{\psi }=\psi \circ p$.

To see that this lift is invertible, we repeat the procedure for $\psi ^{-1}$, to obtain a continuous map $\widetilde{\psi ^{-1}}$ such that $p\circ \widetilde{\psi ^{-1}}=\psi ^{-1}\circ p$. So $p\circ \widetilde{\psi ^{-1}}\circ \widetilde{\psi }=p$, whence for every $\widetilde{x}\in \widetilde{M}$ there is a $k(\widetilde{x})\in \mathbb Z$ such that $(\widetilde{\psi ^{-1}}\circ \widetilde{\psi })(\widetilde{x})=D^{k(\widetilde{x})}(x)$, where D is a generator for the group of deck transformations. Since $\widetilde{\psi ^{-1}}\circ \widetilde{\psi }$ is continuous and $\widetilde{M}$ is connected, $k={{\mathrm{const}}}$. It follows that $D^{-k}\circ \widetilde{\psi ^{-1}}$ is the inverse of $\widetilde{\psi }$.

This general discussion reduces the problem of lifting $\psi :M\rightarrow M$ to a map on $\widetilde{M}$ to checking that the endpoint of the lift of $\psi \circ p \circ \widetilde{\gamma }_{\widetilde{x}}$ to $\widetilde{M}$ at $\widetilde{x}_0$ is independent of the choice of $\widetilde{\gamma }_{\widetilde{x}}$. Here is an obvious necessary and sufficient condition:

1.3 Liftability Criterion

Let $\gamma$ be a closed smooth loop in M which lifts to a closed loop in $\widetilde{M}$ , then $\psi \circ \gamma$, $\psi ^{-1}\circ \gamma$ lift to closed loops in $\widetilde{M}$.

Let $\alpha ,\beta$ denote the linear segments on M connecting $\bullet$ to $\circ$ with parameterizations $\alpha (t)={t\atopwithdelims ()1}$ $(0<t<p/q)$ , $\beta (t)={t\atopwithdelims ()1}$ $(p/q<t<1)$.

We call a smooth path $\gamma$ on M proper, if it intersects $\alpha ,\beta$ at most finitely many times, and all these intersections (if any) are transverse.

For such paths we can define the intersection numbers $i(\alpha ,\gamma ), i(\beta ,\gamma )$ which count the points in $\gamma \cap \alpha$ (resp. $\gamma \cap \beta$) with signs $+,-$ according to the orientation of the ordered pair $\varvec{\langle }\alpha ^\prime ,\gamma ^\prime \varvec{\rangle }$ (resp. $\varvec{\langle }\beta ^\prime ,\gamma ^\prime \varvec{\rangle }$).

Lemma 8.3

A closed proper path $\gamma$ on M lifts to a closed path on $\widetilde{M}$ iff $(q-p)\cdot i(\gamma ,\alpha )-p\cdot i(\gamma ,\beta )=0$

Proof

Let $\gamma$ be a closed proper path on M, and $\widetilde{\gamma }$ its lift to $\widetilde{M}$. Let $t_1<\cdots <t_N$ denote the times $\gamma$ intersects $\alpha \cup \beta$. Let $\sigma _i$ denote the sign of the intersection of $\gamma ,\alpha \cup \beta$ at time $t_i$.

Every intersection with $\alpha$ increases the index of the square containing $\widetilde{\gamma }(t_i)$ by $\sigma _i(q-p)$. Every intersection of $\gamma$ with $\beta$ decreases the index of the square containing $\widetilde{\gamma }(t_i)$ by $\sigma _i p$. The lifted loop closes iff the total change is zero. $\square$

Lemma 8.4

Let $H_1(M_0,P,\mathbb Z)$ be the relative homology group, where $P=\{\bullet ,\circ \}$. Let $\omega$ denote a (non-closed!) smooth path connecting $\bullet$ and $\circ$ (e.g. $\alpha ,\beta$).

• (a) For all proper loops $\gamma$ in M, $i(\omega ,\gamma )$ only depends on the homology classes $[\![\omega [\!],[\![\gamma [\!]\in H_1(M_0,P,\mathbb Z)$.

• (b) $i(\cdot ,\cdot )$ is bilinear on $H_1(M_0,P,\mathbb Z)\times H_1(M_0,P,\mathbb Z)$.

• (c) If $\psi :M_0\rightarrow M_0$ is a diffeomorphism which fixes $\bullet ,\circ$ and $\gamma , \psi ^{\pm 1}\circ \gamma$ are proper, then $i(\psi ^{\pm 1}\circ \omega ,\psi ^{\pm 1}\circ \gamma )={\pm \sigma \cdot i(\omega ,\gamma )}$ where $\sigma =1$ when $\psi$ preserves orientation, and $\sigma =-1$ if it doesn’t.

Proof

We think of $M_0$ as of a simplicial complex. Form the space $M^*:=M_0\uplus CP$ where CP is a cone over P. In our case this means that we attach to M a path from $\bullet$ to $\circ$ which does not intersect M. Then $\alpha$ is a part of a loop $\alpha ^*$ in $M^*$ and for every proper path $\gamma \subset M$, $i(\alpha ,\gamma )=i(\alpha ^*,\gamma )$.

Denote the homology classes in $H_1(M^*,\mathbb Z)$ by $[\![\cdot ]\!]$. By [56], p. 43, $i(\alpha ^*,\gamma )$ only depends on the (absolute) homology classes $[\alpha ^*],[\gamma ]\in H_1(M^*,\mathbb Z)$. By [56], p. 13, these classes only depend on the relative homology classes $[\![\alpha ]\!],[\![\gamma ]\!]\in H_1(M_0,P,\mathbb Z)$. This proves part (a). Parts (b) and (c) are immediate. $\square$

Lemma 8.5

Suppose $\psi :M_0\rightarrow M_0$ is an automorphism which fixes $\bullet ,\circ$, and let $\psi _*:H_1(M_0,P,\mathbb Z)\rightarrow H_1(M_0,P,\mathbb Z)$ denote the homomorphism it induces. Let

$$\begin{aligned} \,[\![\omega ]\!]:=q(q-p)[\![\alpha ]\!]-pq[\![\beta ]\!]\in H_1(M_0,P,\mathbb Z), \end{aligned}$$

then there are $0\ne m,n\in \mathbb Z$ such that $m\cdot \psi _*[\![\omega ]\!]=n\cdot [\![\omega ]\!]$.

Proof

The holonomy of a smooth path $\gamma$ in M is defined to be the vector $\text {hol}(\gamma )={\text {hol}_x(\gamma )\atopwithdelims ()\text {hol}_y(\gamma )}\in \mathbb R^2$ given by

$$\begin{aligned} \text {hol}(\gamma ):=\text {endpoint}(\widetilde{\gamma })-\text {beginning}(\widetilde{\gamma }) \end{aligned}$$

for some (any) lift $\widetilde{\gamma }$ of $\gamma$ to $\mathbb R^2$.

Two homotopic paths have the same holonomy. Therefore, $\text {hol}$ defines a homomorphism $\pi _1(M,x_0)\rightarrow \mathbb Z$. Since $\mathbb Z$ is abelian, $\text {hol}(\gamma )$ defines a homomorphism $\text {hol}:H_1(M,\mathbb Z)\rightarrow \mathbb Z$.

We now work in simplicial homology. If $[\![\gamma ]\!]\in H_1(M_0,P,\mathbb Z)$ equals zero, then $\gamma =\partial c+c_p$ where c is a finite linear combination of 2-cells in M and $c_p$ is zero or a finite linear combination of 1-cells in P. Since $P=\{\bullet ,\circ \}$, there are no 1-cells in P, so $c_p=0$ and $\gamma$ is homologous to zero. So $\text {hol}(\gamma )=\vec {0}$. We see that $[\![\gamma ]\!]=0$ implies that $\text {hol}(\gamma )=\vec {0}$.

It follows that $\text {hol}$ determines a homomorphism $\text {hol}:H_1(M_0,P,\mathbb Z)\rightarrow \mathbb Q$. The range of values is $\mathbb Q$, because absolute cycles in $M_0$ have integral holonomies, and paths connecting $\bullet$ to $\circ$ have rational holonomies.

Calculating, we find that $\text {hol}([\![\omega ]\!])=(q-p)\cdot \frac{p}{q}\vec {e}_1-p\cdot \frac{q-p}{q}\vec {e}_1=\vec {0}$, and $\text {hol}(\psi _*[\![\omega ]\!])=d\psi \bigl (\text {hol}([\![\omega ]\!])\bigr )=\vec {0}$ (here we use the fact that $\psi$ has constant derivative). Thus

$$\begin{aligned} \,[\![\omega ]\!], \psi _*[\![\omega ]\!]\in W:=\{[\![\gamma ]\!]\in H_1(M_0,P,\mathbb Z): \text {hol}([\![\gamma ]\!])=\vec {0}\}. \end{aligned}$$

The plan now is to show that W spans a one-dimensional linear vector space over $\mathbb Q$. This implies that $\exists m,n\in \mathbb Z\setminus \{0\}$ s.t. $m\cdot [\![\omega ]\!]=n\cdot \psi _*[\![\omega ]\!]$.

Step 1 $\exists [\![\gamma _1]\!],[\![\gamma _2]\!],[\![\gamma _3]\!]$ s.t. $H_1(M_0,P,\mathbb Z)={{\mathrm{span}}}_\mathbb Z\{[\![\gamma _1]\!],[\![\gamma _2]\!],[\![\gamma _3]\!]\}$.

Proof The long exact sequence for relative homology states the existence of homomorphisms a, b, c, d such that the following sequence is exact ([56], p. 13):

$$\begin{aligned} \cdots \rightarrow H_1(P)\xrightarrow []{} H_1(M_0)\xrightarrow []{a} H_1(M_0,P)\xrightarrow []{b} H_0(P)\xrightarrow []{c} H_0(M_0)\xrightarrow []{d} H_0(M_0,P) \end{aligned}$$

In our case $H_1(P)=0$, $H_1(M_0)=H_1(\mathbb T^2)=\mathbb Z^{2}$, $H_0(P)=\mathbb Z^2$, $H_0(M_0)=\mathbb Z$, and $H_0(M_0,P)=0$ (see e.g. [56], pp. 3, 4, 12, 39), so

$$\begin{aligned} 0\rightarrow \mathbb Z^2\xrightarrow []{a} H_1(M_0,P)\xrightarrow []{b} \mathbb Z^2\xrightarrow []{c}\mathbb Z\xrightarrow []{d} 0\text { is exact}. \end{aligned}$$

We now chase arrows. Since $\ker (d)=\mathbb Z$, $c:\mathbb Z^2\rightarrow \mathbb Z$ is onto, so $\ker (c)\cong \mathbb Z$. So $\text {Im}(b)\cong \mathbb Z$. Choose $[\![\gamma _1]\!]\in H_1(M_0,P)$ such that $b[\![\gamma _1]\!]$ generates $\text {Im}(b)$. Next by exactness, $a:\mathbb Z^2\rightarrow H_1(M_0,P)$ is one-to-one so there are $[\![\gamma _2]\!]$, $[\![\gamma _3]\!]\in H_1(M_0,P)$ which generate $\text {Im}(a)=\text {ker}(b)$.

For every $[\![\xi ]\!]\in H_1(M_0,P,\mathbb Z)$ there is $k\in \mathbb Z$ s.t. $b[\![\xi ]\!]=k\cdot b[\![\gamma _1]\!]$. So $[\![\xi ]\!]-k\cdot [\![\gamma _1]\!]\in \ker (b)={{\mathrm{span}}}_\mathbb Z\{[\![\gamma _2]\!],[\![\gamma _3]\!]\}$, whence $[\![\xi ]\!]\in {{\mathrm{span}}}_\mathbb Z\{[\![\gamma _1]\!],[\![\gamma _2]\!],[\![\gamma _3]\!]\}$. Since $[\![\xi ]\!]$ was arbitrary, the step is proved.

Step 2 $\dim {{\mathrm{span}}}_{\mathbb Q}\{\text {hol}([\![\gamma _1]\!]), \text {hol}([\![\gamma _2]\!]), \text {hol}([\![\gamma _3]\!])\}=2$.

Proof By step 1, ${{\mathrm{span}}}_\mathbb Z\{\text {hol}([\![\gamma _1]\!]), \text {hol}([\![\gamma _2]\!]), \text {hol}([\![\gamma _3]\!])\}=\text {hol}(H_1(M_0,P,\mathbb Z))$. The last set can be easily seen to equal $\mathbb Z^2$, so it contains two vectors which are linearly independent over $\mathbb Z$, whence also over $\mathbb Q$.

Step 3 Completion of the proof.

By step 1, every $[\![\gamma ]\!]\in W$ equals $\sum _{i=1}^3 a_i[\![\gamma _i]\!]$ for some $a_i\in \mathbb Z$ which solve

$$\begin{aligned} a_1\text {hol}_x([\![\gamma _1]\!])+a_2\text {hol}_x([\![\gamma _2]\!])+a_3\text {hol}_x([\![\gamma _3]\!])&=0\\ a_1\text {hol}_y([\![\gamma _1]\!])+a_2\text {hol}_y([\![\gamma _2]\!])+a_3\text {hol}_y([\![\gamma _3]\!])&=0. \end{aligned}$$

This is a system of linear equations with rational coefficients. By step 2, the rank is two. So the space of solutions over $\mathbb Q$ is one-dimensional over $\mathbb Q$. In particular, $[\![\omega ]\!]=\sum _{i=1}^3 a_i[\![\gamma _i]\!]$ and $\psi _*[\![\omega ]\!]=\sum _{i=1}^3 b_i[\![\gamma _i]\!]$ where $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$ are linearly dependent over $\mathbb Q$, and $\exists m,n\in \mathbb Z\setminus \{0\}$ s.t. $m\cdot [\![\omega ]\!]=n\cdot \psi _*[\![\omega ]\!]$. $\square$

Proof of Theorem 8.2 We check the liftability criterion: Let $\gamma$ be a smooth loop in M, and suppose $\gamma$ lifts to a closed loop in $\widetilde{M}$. We show that $\psi ^{\pm 1}\circ \gamma$ lift to closed loops in $\widetilde{M}$. Obviously this property only depends on the homotopy class of $\gamma$, so there is no loss of generality in assuming that $\gamma ,\psi \circ \gamma$, and $\psi ^{-1}\circ \gamma$ are proper.

Since $\gamma$ lifts to a closed loop in $\widetilde{M}$, $i([\![\omega ]\!], [\![\gamma ]\!])=0$ (Lemma 8.3). Thus by Lemma 8.4(c),

$$\begin{aligned} i(\psi _*[\![\omega ]\!],[\![\psi \circ \gamma ]\!])=i([\![\psi \circ \omega ]\!],[\![\psi \circ \gamma ]\!])=\pm i([\![\omega ]\!], {[\![\gamma ]\!]})=0. \end{aligned}$$

By Lemma 8.5, there are $m,n\in \mathbb Z\setminus \{0\}$ s.t. $m\psi _*[\![\omega ]\!]=n[\![\omega ]\!]$, so

$$\begin{aligned} 0&= m\cdot i(\psi _*[\![\omega ]\!],[\![\psi \circ \gamma ]\!])=i(m\cdot \psi _*[\![\omega ]\!],[\![\psi \circ \gamma ]\!])=i(n\cdot [\![\omega ]\!],[\![\psi \circ \gamma ]\!])\\&= n\cdot i([\![\omega ]\!],[\![\psi \circ \gamma ]\!]), \text { whence }i([\![\omega ]\!],[\![\psi \circ \gamma ]\!])=0. \end{aligned}$$

Since $i([\![\omega ]\!],[\![\psi \circ \gamma ]\!])=0$, $\psi \circ \gamma$ lifts to a closed loop in $\widetilde{M}$ (Lemma 8.3). A similar argument shows that $\psi ^{-1}\circ \gamma$ lifts to a closed loop in $\widetilde{M}$ as well.

We see that $\psi$ satisfies the liftability criterion. By the discussion at the beginning of the section, $\psi$ has an invertible continuous lift to $\widetilde{M}$. $\square$

Appendix 2: Maximal Growth

To prove Theorem 5.9 we need the following fact.

Proposition 9.1

• (a) If $0<a<||\omega ||_s$ then there is $T_0$ such that for $T\ge T_0$ there exists $\vec v$ such that ${\mathcal {W}}_g(\omega , \vec v, T)>aT.$

• (b) If $||\omega ||_s< a$ then there is $T_0$ such that for $T\ge T_0$ for all $\vec v$ we have ${\mathcal {W}}_g(\omega , \vec v, T)<aT.$

• (c) If $0<a<||\omega ||_s$ then there is $T_0$ such that for $T\ge T_0$ there exists $\vec v$ such that ${\mathcal {W}}_g(\omega , \vec v, T)<-aT.$

• (d) If $||\omega ||_s< a$ then there is $T_0$ such that for $T\ge T_0$ for all $\vec v$ we have ${\mathcal {W}}_g(\omega , \vec v, T)>-aT.$

Proof

This proposition is well known but we sketch the proof to make the paper self contained.

• (a) Fix ${\varepsilon }>0.$ By the definition of the stable norm there is a finite set of closed curves $\gamma _1,$ $\gamma _2,\dots , \gamma _m$ such that

  $$\begin{aligned} \sum _{j=1}^m r_j \text {length}(\gamma _j)=1\quad \text {and}\quad \sum _{j=1}^m r_j \omega (\gamma _j)\ge ||\omega ||_s-{\varepsilon }. \end{aligned}$$

  Since geodesics minimize length in its homotopy class, we may assume, increasing $r_j$ if necessary, that $\gamma _j$ are geodesics. Using the specification property of geodesic flow we see that there are numbers $n_0$ and L such that for each T there are numbers $t_j$ and geodesic $\Gamma$ of length T such that denoting $n_j=[T r_j]$ we have

  $$\begin{aligned} d(\Gamma (t_j+t), \gamma _j(t))\le 1 \text { for } t\in [0, n_j-n_0], \text { and } \left| t_j-\sum _{i=1}^{j-1} n_i\right| \le L. \end{aligned}$$

  By convexity

  $$\begin{aligned} \int _\Gamma \omega =\sum _j n_j \omega (\gamma _j)+O(1). \end{aligned}$$

  Since

  $$\begin{aligned} \sum _j n_j \omega (\gamma _j)=T \sum _j r_j \omega (\gamma _j) +O(1) \end{aligned}$$

  part (a) follows.

• (b) Assume by contradiction that for every $T_0$ there are $T>T_0$ and $\vec {v}$ s.t. ${\mathcal {W}}_g(\omega ,\vec {v},T)\ge aT$. Let $\tilde{\Gamma }:=\{g^t(\vec {v}\}_{0<t<T}$, then $\int _{\tilde{\Gamma }} \omega \ge aT$. By Anosov’s Closing Lemma there is a closed geodesic $\Gamma$ with $|\int _\Gamma \omega -\int _{\tilde{\Gamma }} \omega |$ bounded by a constant independent of T and $\vec {v}$. Thus if T is sufficiently large then $\omega ([\Gamma ]/\text {length}(\Gamma ))\ge a> ||\omega ||_s$ giving a contradiction.

Parts (c),(d) follow from parts (a),(b) by substituting $-\vec {v}$ for $\vec {v}$. $\square$

Proof of Theorem 5.9

Given $T>0$, $0<t<T$, let $\vec {w}:=-h^{t/T} g^{\ln T} \vec {v}$, then we have by (5.4)

$$\begin{aligned} {\mathcal {W}}_h(\omega , \vec v, t)={\mathcal {W}}_g(\omega , \vec v, \ln T)+ {\mathcal {W}}_g(\omega , \vec w, \ln T)+O(1). \end{aligned}$$

(9.1)

Let us assume to fix our notation that ${\mathcal {W}}_g(\omega , \vec v, \ln T)\ge 0.$ By Proposition 9.1(c) for each $\varepsilon ,$ if T is sufficiently large then we can find $\vec u\in T^1 M$ such that

$$\begin{aligned} {\mathcal {W}}_g(\omega , \vec u, \ln T) \le -(||\omega ||_s-{\varepsilon }) \ln T. \end{aligned}$$

(9.2)

The vector $-\vec u$ does not need to belong to $\text {Hor}(g^{\ln T}\vec v),$ but since our surface is compact, there exists L such that

$$\begin{aligned} \exists t\in [0,1], \tilde{t}\in [-L, L], \text { and }r\in [-L, L]\text { s.t.}\quad \vec u=\tilde{h}^{\tilde{t}} g^r \vec w, \end{aligned}$$

(9.3)

where $\vec w=h^t (-g^{\ln T}\vec v)$ and $\tilde{h}$ denotes the stable horocycle flow.

To show (9.3) it suffices to find $\tilde{L}$ such that every pair $\vec u, \vec v$ can be joined by a three leg path consisting of orbits of g,  h and $\tilde{h}$ respectively so that each leg is shorter that $\tilde{L}.$ To see this represent $T^1 M$ by $T^1 F$ where F is a compact subset of $\mathrm {PSL}(2,\mathbb R)$ and use the $NAN^{-}$ decomposition. Now apply the geodesic flow to shorten the stable leg.

(9.3) shows that

$$\begin{aligned} {\mathcal {W}}_g(\omega , \vec w, \ln T)={\mathcal {W}}_g(\omega , \vec u, \ln T)+O(1). \end{aligned}$$

Thus (9.2) tells us that the $\lim \inf$ in (5.9) is greater than $||\omega ||_s-\varepsilon .$ Since ${\varepsilon }$ is arbitrary, (5.9) follows.

To prove (5.10) it remains to bound

$$\begin{aligned} \limsup _{T\rightarrow \infty } \max _{t\le T} \frac{\left| {\mathcal {W}}_h(\omega , \vec v, t)\right| }{\ln T} . \end{aligned}$$

By Ergodic Theorem for almost all $\vec v$, $\displaystyle \lim _{T\rightarrow \infty } \frac{{\mathcal {W}}_g(\omega , \vec v, \ln T)}{\ln T}=0.$ Thus by (9.1)

$$\begin{aligned} \limsup _{T\rightarrow \infty }\max _{t\le T} \frac{\left| {\mathcal {W}}_h(\omega , \vec v, t)\right| }{\ln T}= \lim \sup \frac{\left| {\mathcal {W}}_g(\omega , \vec w, \ln T)\right| }{\ln T}, \end{aligned}$$

which is less than $||\omega ||_s$ by parts (a) and (c) of Proposition 9.1. $\square$