# Reading roadmap: Weakly Mixing Polygonal Billiards

The proposed proof uses growth estimates for cohomology sections under the Kontsevich–Zorich cocycle, compact recurrence and the Veech eigenvalue criterion to exclude each fixed nonzero frequency in almost every direction. Countability of the spectrum and Fubini then give almost-everywhere ergodicity of products of directional flows. A rational-polygon approximation and Baire-category argument yields generic weak mixing for polygonal billiards.

Source SHA-256: `fa3e57d5d676bc1578099b6e5afad949068e7e40ea46baeea76ff1185bf2efd2`. Indexed 126,037 characters.

Edges point from prerequisite to consumer. Solid proof edges are attributed inferences, not formal certificates; references remain candidates.

## Internal claim index

### Thm 1.1 — `thm:billiard`

Source lines 100–102.

There exists a weakly mixing billiard flow in a polygon.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Let $n$ be given and, for all $k\in \mathbb N$, let $U_k$ be the set of polygons $P$ with $n\geq 3$ sides so that there exists $T>0$ such that, for all $f \in \LipP$, $$\int_{X(P)} |\frac{1}{T}\int_0^Tf(F^t(x,\theta),F^t(y,\psi))dt-\int_{X(P)}f|<\frac 1 k.$$ By Lemma \ref{lem:open}, $U_k$ is open for all $k\in \mathbb N$ as union (over $T>0$) of a family of open sets. By Corollary \ref{cor:dense} it is dense for all $k\in \mathbb N$. So by the Baire Category theorem there exists $P\in \cap_{k}U_k$. Because 1-Lipschitz functions have dense span in the set of continuous functions, the product of the billiard flow with itself on $X(P)$ is ergodic. In fact, if $P \in \cap_{k}U_k$ we have that, f

- `cor:dense` (proof_dependency; curated_source_check): By  Corollary \ref{cor:dense} it is dense for all $k\in \mathbb N$.
- `lem:open` (proof_dependency; curated_source_check): By Lemma \ref{lem:open}, $U_k$ is open

Candidate references: `lem:open` (candidate_proof_dependency); `cor:dense` (candidate_proof_dependency).

### Thm 1.2 — `thm:erg prod`

Source lines 122–124.

For a.e. $(\theta, \phi) \in S^1\times S^1$ the flow $F_{r_\theta \omega}^t \times F_{r_\phi \omega}^t $ is $\lambda^2 \times \lambda^2$ ergodic.

*Extractive statement; proof gist not independently reconstructed.*

- `thm:main` (proof_dependency; curated_source_check): is well known to imply Theorem~\ref{thm:erg prod}

### Thm 1.3 — `thm:main`

Source lines 126–128.

For every $\alpha\neq 0 $ and every translation surface $\omega$ we have $|\{\theta \in S^1:F_{r_\theta \omega}^t \text{ has eigenvalue } \alpha \}|=0$\,.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: We proceed with a proof by contradiction. Let $B \subset [-1,1]$ denote the set of points for which the conclusion of the theorem fails, and assume that the measure of $B$ is $\xi>0$. Let $\mathcal{K}$ be a compact set so that Lemma \ref{lem:in compact} and Corollary \ref{cor:better} hold with $c=.999$ and $\mu(Int(\mathcal{K}))>.999$. Let $a=\max\{t_0,\ell_{\mathcal{K}}\}$ where $t_0$ is as in Lemma \ref{lem:in compact} and $\ell_{\mathcal{K}}$ is as in Lemma \ref{lem:v crit}. We choose $\gamma_0 >0$ small enough so that Corollary \ref{cor:better} holds with the compact set $\mathcal{K}$, $\frac 1 2 |\log(\gamma_0 )|>T_0$, where $T_0$ is as in \S \ref{sec:key prop}, $\gamma_0\leq \gamma''_{

- `cor:better` (proof_dependency; curated_source_check): We now iteratively apply Corollary~\ref{cor:better}.
- `lem:transv` (proof_dependency; curated_source_check): By Lemma~\ref{lem:transv}, the section $\psi$ is transverse to any integer translate
- `prop:central` (proof_dependency; curated_source_check): we are assuming all but the first assumption of Proposition \ref{prop:central}

Candidate references: `lem:in compact` (candidate_proof_dependency); `cor:better` (candidate_proof_dependency); `lem:v crit` (candidate_proof_dependency); `sec:key prop` (candidate_proof_dependency); `prop:central` (candidate_proof_dependency); `eq:beta` (candidate_proof_dependency); `cond:last` (candidate_proof_dependency); `eq:weak stable density` (candidate_proof_dependency); `eq:density` (candidate_proof_dependency); `lem:transv` (candidate_proof_dependency); `eq:intwice` (candidate_proof_dependency); `thm:big quote` (candidate_proof_dependency); `lem:stay_proj_lip` (candidate_proof_dependency); `conc:growth 2` (candidate_proof_dependency); `eq:growth different form` (candidate_proof_dependency); `lem:stay lip` (candidate_proof_dependency); `cond:lip transport 2` (candidate_proof_dependency); `eq:meas lost 2` (candidate_proof_dependency); 2 more in the JSON/interactive graph.

### Lem 2.1 — `lem:KZbound`

Source lines 227–232.

Let $\text{dist}$ denote the hyperbolic distance on $SL(2,\mathbb{R})/SO(2)$. For any Abelian differential $\omega\in \mathcal{H}(\alpha)$ and for all $g\in SL(2,\mathbb{R})$, we have $\|KZ(g ,\omega)\|\leq e^{\text{dist}(g, Id)}$ and moreover $|KZ(g,\omega)v|\geq e^{-\text{dist}(g, Id)}|v|$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: For $g=g_t$ a diagonal element of $SL(2, \mathbb R)$, the upper bound holds for the Hodge norm by the first variation formulas~\cite[\S 2]{Fo} or \cite[\S 3.5]{FM}. The lower bounds also follows since $KZ(g_t,\omega)$ is invertible with inverse $K(g_{-t}, g_t\omega)$, hence for $t\geq 0$ $$ |v| = | KZ(g_{-t}, g_t\omega) KZ(g_t,\omega)v | \leq e^t | KZ(g_t,\omega)v | \,. $$ For a general $g \in SL(2, \mathbb R)$, the result follows from the $KAK$ decomposition of $SL(2, \mathbb R)$, since the action on the Hodge bundle of the group $K= SO(2) \subset SL(2, \mathbb R)$ of rotations is isometric.


Candidate references: `external:Fo` (external_reference); `external:FM` (external_reference).

### Lem 2.2 — `lem:in compact`

Source lines 240–243.

For every $c>0$ there is a compact set $\mathcal{K}$ and $t_0>0$ so that if $\omega \in \mathcal{K}$ then $|\{-1\leq s \leq 1:g_th_s\omega\in \mathcal{K} \}|>2-c$ for all $t>t_0$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: By \cite[Section 2.3]{ath thesis}, there exists $V$ a continuous, proper function on $\mathcal{H}$, and constants $c,\gamma, a, b, C$ and $\tau$ so that, for all $t\geq \tau$, $$\int_0^{2\pi} V(g_tr_\theta\omega)d\theta\leq ce^{-\gamma t}V(\omega)+b$$ and moreover, $a^{-1}V(\omega)<V(g\omega)<aV(\omega)$ for all $g\in SL(2,\mathbb{R})$ with $|g|<C$. By choosing $N$ large enough and using that $r_\theta=\hat h_{\tan(\theta)}g_{\log(\cos(\theta))}h_{-\tan(\theta)}$ we have the lemma for $\mathcal{K}=V^{-1}[0,N]$.


Candidate references: `external:ath thesis` (external_reference).

### Prop 3.1 — `prop:growth mechanism`

Source lines 286–297.

(Chaika-Eskin, \cite{CE}) If $\mathcal{F}\subset \FF$ is an equivariant subbundle, where the Kontsevich-Zorich cocycle acts strongly irreducibly and with a positive exponent, then there exists $\lambda:= \lambda_\mathcal F>0$ such that, for all $\delta,\epsilon >0$, and for all $L$ sufficiently large, there exist $\hat\epsilon>0$ and an open set $\hat{U}:= \hat U(\delta, L)$ so that $\nu(\hat{U})>1-\delta$, and for all $\omega\in \hat{U}$ and for all parallel horocycle sections $v$ at $\omega$, that is, for the all the maps $v : [-1,1] \to \mat

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `external:CE` (external_reference).

### Prop 3.2 — `prop:easy restriction`

Source lines 308–316.

Let $\mathcal{F}\subset \FF$ be an equivariant subbundle of $\FF$. For any compact subset $\mathcal K$ there exists $\lambda:=\lambda_{\mathcal{F}, \mathcal K} \in [0,1)$ such that \item $\sup_{v\in {\mathcal{F}_\omega}}\frac{|KZ(g_{t},\omega)v|}{ |v| } \leq \exp \left( t - (1-\lambda )v_{\mathcal K} (\omega, t) \right)$; \item $\inf_{v\in {\mathcal{F}_\omega}} \frac{|KZ(g_{t},\omega)v|}{ |v| } \geq \exp \left( -t + (1-\lambda )v_{\mathcal K} (\omega, t) \right)$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Both estimates follow from the first variational formulas for the Hodge norm~\cite[\S 2]{Fo} or \cite[\S 3.5]{FM}.


Candidate references: `external:Fo` (external_reference); `external:FM` (external_reference).

### Lem 3.3 — `lem:Egood`

Source lines 340–342.

[Chaika-Eskin \cite{CE}, Lemma 2.11] For all $\epsilon>0$ we have $\underset{L \to \infty}{\lim}\nu(E_{good}(L,\epsilon))=1$.

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `external:CE` (external_reference).

### Lem 3.4 — `lemma:sublinear:tracking`

Source lines 347–367.

Sublinear Tracking

There exists $\lambda >0$ (depending only on $\mu$), and for $\mu^{\mathbb{N}}$-almost all $\bar{g} = (g_1, \dots, g_n, \dots) \in SL(2,\mathbb{R})^{\mathbb{N}}$ there exists $\bar{\theta} = \bar{\theta}(\bar{g}) \in \mathbb{R}$ such that \lim_{n \to \infty} \frac{1}{n} \log \| (g_{\lambda n} r_{\bar{\theta}}) (g_n \dots g_1)^{-1} \| = 0. Furthermore, the distribution of $\bar{\theta}$ is uniform, i.e \mu^{\mathbb{N}}\left( \{ \bar{g} \in SL(2,\mathbb{R})^{\mathbb{N}} : \bar{\theta}(\bar{g}) \in [\theta_1, \theta_2] \} \right) = \frac{|\theta_2

*Extractive statement; proof gist not independently reconstructed.*

### Lem 3.5 — `lem:sublin hor`

Source lines 375–390.

Sublinear Tracking

There exists $\lambda >0$ (depending only on $\mu$), and $\mu^{\mathbb{N}}$-almost all $\bar{g} = (g_1, \dots, g_n, \dots) \in SL(2,\mathbb{R})^{\mathbb{N}}$ there exists $\bar{s} = \bar{s}(\bar{g}) \in(-\infty, + \infty)$ such that \lim_{n \to \infty} \frac{1}{n} \log \| (g_{\lambda n} h_{\bar{s}}) (g_n \dots g_1)^{-1} \| = 0. Furthermore, the distribution of $\bar{s}$ is in the measure class of Lebesgue.

*Extractive statement; proof gist not independently reconstructed.*

### Lem 3.6 — `lem:good growth`

Source lines 401–403.

$\underset{L \to \infty}{\lim} \nu(\hat{E}_{good}(L,\tilde\epsilon))=1.$

*Extractive statement; proof gist not independently reconstructed.*

### Thm 3.7 — `thm:EMM 2.7`

Source lines 426–431.

(Eskin-Mirzakhani-Mohammadi) Let $U$ be an open set, $\nu$ be an $SL(2,\mathbb{R})$ invariant and ergodic measure. For any $\epsilon>0$ there exists a finite set of invariant manifolds, $\mathcal{Z}_1,\dots,\mathcal{Z}_n$ so that for any $\mathcal{C} \subset \supp(\nu) \setminus \cup_{i=1}^n\mathcal{Z}_i$, compact, there exists $T_1>0$ so that for all $\omega \in \mathcal{C}$ and for all $T\geq T_1$, $$\frac 1 {2T} \int_{-1}^1\int_0^T\chi_U(g_th_s\omega)dtds>\nu(U)-\epsilon.$$

*Extractive statement; proof gist not independently reconstructed.*

### Thm 3.8 — `thm:EMM 2.13`

Source lines 434–448.

(\cite[Proposition 2.13]{EskMirMo}) Let $\mathcal{M}\subset \mathcal{H}$ be an affine invariant submanifold. (In this proposition $\mathcal{M}=\emptyset$ is allowed.) Then there exists an $SO(2)$-invariant function $f_{\mathcal{M}}: \mathcal{H} \to [1,\infty]$ with the following properties: \item $f_{\mathcal{M}}(\omega)=\infty$ if and only if $\omega\in \mathcal{M}$, and $f_{\mathcal{M}}$ is bounded on compact subsets of $\mathcal{H}\setminus \mathcal{M}$. For any $\rho>0$, the set $\{\omega : f_{\mathcal{M}}(\omega)\leq \rho\}$ is a compact s

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `external:EskMirMo` (external_reference).

### Lem 3.9 — `lem:2.13 for hor`

Source lines 451–454.

\cite[Lemma 3.5]{AAEKMU} Let $f_{\mathcal{M}}$ be as in Theorem \ref{thm:EMM 2.13}. Then there exists a constant $b'>0$ so that for all $0<a<1$ there exists $\bar{t}_0= \bar{t}_0(a)$ such that for all $t> \bar{t}_0$ and for all $\omega \in \mathcal{H}\setminus \mathcal{M}$ we have $$\int_{-1}^1 f_{\mathcal{M}}(g_th_s \omega)ds < a f_{\mathcal{M}}(\omega)+ b'\,.$$

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `thm:EMM 2.13` (internal_reference); `external:AAEKMU` (external_reference).

### Prop 3.10 — `prop:cpct integer`

Source lines 461–466.

\cite[Proposition 3.7]{AAEKMU} Let $f_{\mathcal{M}}$ be as in Theorem \ref{thm:EMM 2.13} and let $b'>0$ and $\bar{t}_0=\bar{t}_0(a)$ be as in Lemma \ref{lem:2.13 for hor}. There exist $C_1> 1$ (independent of $\omega$ and $a$) such that for all $a\in (0,1)$, all $\rho > C_1 b'/a$, all $t\geq \bar{t}_0$ such that $e^t\in \mathbb N$ and all $N\in \mathbb N$, $$\int_{Z_{N-1}} f_{\mathcal{M}} (g_{Nt}h_s\omega)ds< (2a)^{N} f_{\mathcal{M}}(\omega)+(2a)^{N-1}b'\,.$$

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `thm:EMM 2.13` (internal_reference); `lem:2.13 for hor` (internal_reference); `external:AAEKMU` (external_reference).

### Cor 3.11 — `cor:ath`

Source lines 469–472.

(cf \cite[Theorem 1.1 (2)]{ath thesis}) There exists $\rho\in \mathbb{R}, \zeta<1$ so that for all $S$, $T$ large enough if $\omega \in \mathfrak{C}_\rho$ $$\text{\rm Leb}(\{s\in [-1,1] : \cup_{t\in [S,S+T]}\{g_t h_s \omega\} \cap \mathfrak{C}_\rho=\emptyset\})<\zeta^T.$$

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `external:ath thesis` (external_reference).

### Prop 3.12 — `prop:in good integer`

Source lines 473–480.

\cite[Proposition 3.9]{AAEKMU} (cf \cite[Theorem 1.5]{KKLM}) Let $\mathfrak{C}_\rho$ be as above for a function $f_{\mathcal M}$ as in Theorem \ref{thm:EMM 2.13}. Let us assume that $\omega \not \in \mathcal M$. For any $\delta,a \in (0,1)$ there exist $\rho_0>1$ and $t_0 >1$, depending only on $a$, such that for all $\rho \geq \rho_0$, all $t \geq t_0$ such that $e^t\in \mathbb N$, and all $N\in \mathbb{N}$, the set $$\{s\in [-1,1]: \frac 1 N\sum_{j=1}^{N}\chi_{\mathfrak{C}_\rho}(g_{jt}h_s\omega)<1- \delta\}$$ can be covered by at most $2^N C_

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `thm:EMM 2.13` (internal_reference); `external:AAEKMU` (external_reference); `external:KKLM` (external_reference).

### Cor 3.13 — `cor:all together`

Source lines 483–494.

Let $U$ be an open set whose boundary has measure zero. For all $\epsilon>0$ there exist numbers $\xi\in (0,1)$, $t_0>0$, $N_0\in \mathbb{N}$ and a set $V_\epsilon := V_\epsilon(U) \subset \supp (\nu)$, open in $\supp (\nu)$, and $T_2\in \mathbb{R}^+$ so that for any $T>T_2$: [label=(\roman*)] \item $\nu(V_\epsilon)>1-\epsilon;$ \item for $\nu$ almost every $\omega$ we have $\underset{S \to \infty}{\liminf}\, \frac{|\{ 0\leq t\leq S: g_t\omega\in V_\epsilon\}|}{S}>1-\epsilon;$ \item if $\omega \in V_\epsilon$ we have $|\frac 1 {2T} \int_{-1}^1\

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: First, (ii) follows free of charge by the fact that $V_\epsilon$ is open and Lemma \ref{lem:gen set}. Given an open set $U$ and $\epsilon>0$, we then construct an open set $V_\epsilon$ with properties (i), (iii) and (iv). For the given $U$ and $\epsilon>0$, we apply Theorem \ref{thm:EMM 2.7} to obtain a finite number of manifolds $\mathcal{Z}_1,\dots,\mathcal{Z}_n$. For each of these manifolds we build a function $f_i$ as in Theorem \ref{thm:EMM 2.13} and let $\mathfrak{C}^{(i)}_s=\{\omega:f_i(\omega) \leq s\}$. Applying Proposition \ref{prop:in good integer} to each $f_i$ with $\delta<\frac \epsilon n$ and $a<\frac 1 {2 (2C_1)^{\frac 1 \delta}}$ (where $C_1$ is as in the proposition), there


Candidate references: `lem:gen set` (candidate_proof_dependency); `thm:EMM 2.7` (candidate_proof_dependency); `thm:EMM 2.13` (candidate_proof_dependency); `prop:in good integer` (candidate_proof_dependency).

### Lem 3.15 — `lem:gen set`

Source lines 499–505.

Let $S$ be a set so that $\nu(\partial S)=0$. If $\omega$ is Birkhoff generic then $$\underset{T \to \infty}{\lim}\, \frac 1 T\int_0^T\chi_S(g_t\omega) dt=\nu(S).$$ If $U$ is an open set and $\omega$ is Birkhoff generic $$\underset{T \to \infty}{\liminf} \, \frac 1 T\int_0^T\chi_U(g_t\omega) dt\geq \nu(U).$$

*Extractive statement; proof gist not independently reconstructed.*

### Lem 4.2 — `lem:stay lip`

Source lines 549–553.

If $\phi:[a,b] \to H^1(M,\mathbb R)$ is a $K$-Lipschitz horocycle section at $\omega$, then $KZ(g_t)(\phi):[e^{2t}a,e^{2t}b] \to H^1(M, \mathbb R)$ given by $$KZ(g_t)(\phi)(s)=KZ(g_t,h_{se^{-2t}}\omega) \phi(se^{-2t})$$ is a $Ke^{-t}$-Lipschitz horocycle section at $g_t\omega$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: By definition $\phi:[a,b] \to H^1(M,\mathbb R)$ is a $K$-Lipschitz horocycle section at $\omega$ if the parallel transport $\phi_\omega: I \to H^1_\omega(M,\mathbb R)$ is a $K$-Lipschitz map with respect to the Hodge norm. We recall that the map $\phi_\omega$ is defined, for all $s\in I$, as a composition $\phi_\omega (s) = \pi_\omega(s) \circ \phi(s)$ with the parallel transport $\pi_\omega (s): H^1_{h_s\omega}(M,\mathbb R) \to H^1_\omega(M,\mathbb R)$. By the commutativity of parallel transport and the KZ cocycle, we have $$ \pi_{g_t\omega} (s) KZ(g_t,h_{se^{-2t}}\omega)\phi(se^{-2t}) = KZ(g_t,\omega)\phi_\omega (se^{-2t})\,, $$ hence by Lemma \ref{lem:KZbound} the map defined as $KZ(g_t,\


Candidate references: `lem:KZbound` (candidate_proof_dependency).

### Rem 4.4 — `rem:Lipschitz_1`

Source lines 574–576.

Using Lemma \ref{lem:KZbound} to bound the change of Hodge norm from above and below under parallel transport along a horocycle, we obtain that there exists $C_{a,b}$, depending only on $b-a$ so that for any section $\phi: [a,b] \to H^1(M,\mathbb{R})$ projectively $\kappa$-Lipschitz at $\omega$, and for any $s \in [a,b]$, the section $\phi$ is also projectively $C_{a,b} \kappa$-Lipschitz at $h_{s}\omega$.

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `lem:KZbound` (internal_reference).

### Rem 4.5 — `rem:Lipschitz_2`

Source lines 578–583.

For any $K>0$, every $K$-Lipschitz horocycle section $\phi: [a,b] \to H^1(M,\mathbb{R})$ at $\omega$, is projectively $\kappa$-Lipschitz at $\omega$ if $ \max_{s\in [a,b]} \Vert \phi_\omega (s) \Vert_\omega > K (b-a)$ with $$ \kappa := K / \big(\max_{s\in [a,b]} \Vert \phi_\omega (s) \Vert_\omega - K(b-a) \big)\,. $$

*Extractive statement; proof gist not independently reconstructed.*

### Lem 4.6 — `lem:stay_proj_lip`

Source lines 584–596.

If $\phi:[a,b] \to H^1(M,\mathbb R)$ is a projectively $\kappa$-Lipschitz horocycle section at $\omega$, then $KZ(g_t)(\phi):[e^{2t}a,e^{2t}b] \to H^1(M, \mathbb R)$ given by $KZ(g_t)(\phi)(s)=KZ(g_t,h_{se^{-2t}}\omega) \phi(se^{-2t})$ is a projectively $\kappa$-Lipschitz horocycle section at $g_t\omega$. If in addition $\phi:[a,b] \to \mathcal F \subset \FF$, then for any compact set $\mathcal K$ there exists $\lambda_{\mathcal F, \mathcal K}\in [0,1)$ such that the following holds. For any $\omega$ and for any $t>0$, let $$ v_{\mathcal K} (\o

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: By Lemma~\ref{lem:stay lip}, if the horocycle section $\phi$ is $K$-Lipschitz at $\omega$, then $KZ(g_t)(\phi)_{g_t\omega}$ is $Ke^{-t}$-Lipschitz, and by Lemma~\ref{lem:KZbound} it follows that $\Vert KZ(g_t)(\phi)_{g_t\omega} (s) \Vert \geq e^{-t} \Vert \pi_\omega(\phi) (s) \Vert$, for all $s\in [a,b]$. It follows immediately that if $\Vert \pi_\omega(\phi) (s) \Vert_\omega \geq K/\kappa$, then $$ \Vert KZ(g_t)(\phi)_{g_t\omega} (s) \Vert \geq e^{-t} \Vert \phi_{\omega} (s) \Vert \geq K e^{-t}/\kappa\,, $$ hence $KZ(g_t)(\phi)$ is projectively $\kappa$-Lipschitz at $g_t\omega$ establishing the first claim. \noindent If $\phi:[a,b] \to \FF$, it follows from Proposition \ref{prop:easy restri


Candidate references: `lem:stay lip` (candidate_proof_dependency); `lem:KZbound` (candidate_proof_dependency); `prop:easy restriction` (candidate_proof_dependency).

### Lem 4.7 — `lem:growth from E`

Source lines 626–633.

Let $\mathcal{F} \subset \hat{\mathcal{F}}$ be a subbundle where $KZ$ acts strongly irreducibly and with a positive exponent. There exists $\lambda >0$ such that the following holds. For any $\delta, \epsilon>0$ and for all sufficiently large $L>0$, there exist $\kappa_0:= \kappa_0 (a,b,\delta, L)$ and an open set $\hat U:=\hat U(\delta, L)$ (as in Proposition \ref{prop:growth mechanism}) with $\nu(\hat U)>1-\delta$, such that for all $\kappa \in (0, \kappa_0)$, for all $t\geq 0$, for all $s_0 \in [a+e^{-2t},b-e^{-2t}]$ and for all horocycle se

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: The lemma follows by Proposition~\ref{prop:growth mechanism} applied to the horocycle $\{ h_s g_t (h_{s_0} \omega) \vert s \in [-1, 1]\}$ at $g_t h_{s_0}\omega$. Let us consider the image $KZ(g_t) (\phi)$ of the section $\phi$ for $s\in [s_0-e^{-2t}, s_0 + e^{-2t}]$. By definition this section is the restriction of a horocycle section at $g_t\omega$ and after reparametrization can be regarded as a horocycle section at $g_t h_{s_0} \omega= h_{e^{2t}s_0} g_t\omega$, defined as $KZ(g_t, h_{s_0+e^{-2t}s}\omega) \phi(s_0+ e^{-2t}s)$ for $s\in [-1, 1]$. By Lemma~\ref{lem:stay_proj_lip} since $\phi$ is projectively $\kappa$-Lipschitz at $h_{s_0} \omega$, then $KZ(g_t) (\phi)$, and its reparametriza


Candidate references: `prop:growth mechanism` (internal_reference); `prop:growth mechanism` (candidate_proof_dependency); `lem:stay_proj_lip` (candidate_proof_dependency).

### Lem 4.8 — `lem:Zgrowth from V`

Source lines 688–705.

Let $\mathcal{F}\subset \hat{\mathcal{F}}$ be a subbundle where $KZ$ acts strongly irreducibly with a positive exponent. For any bounded interval $[a,b] \subset \mathbb R$, for all $\delta>0$, for all $L>0$ sufficiently large, there exists $\kappa_0:= \kappa_0(a,b,\delta, L)$ such that the following holds. For all $\kappa \in (0, \kappa_0)$, there exists $T_3>0$ and $t_0>0$ such that, for all $s_0 \in [a+e^{-2t},b-e^{-2t}]$, whenever \item $g_th_{s_0}\omega \in \hat V_\epsilon:= V_\epsilon (\hat U(\delta, L))$, \item $ \phi: [a,b] \to \mathcal{

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Let $\delta>0$ and $L>0$ sufficiently large be fixed. By Corollary \ref{cor:all together} (iii) (and the fact that $\nu(\hat{U})>1-\delta$), since $g_th_{s_0} \omega \in \hat V_\epsilon$, we have that, for all $T>T_2$, \int_0^T\int_{-1}^1 \chi_{\hat{U}}(g_\ell h_s g_th_{s_0}\omega)dsd\ell \\ = { e^{2t}} \int_0^T\int_{-e^{-2t}}^{e^{-2t}} \chi_{\hat{U}}(g_\ell g_th_{s_0+s}\omega)dsd\ell>2T(1-\delta -\epsilon). That is, $$ \sum_{j=0}^{\lfloor \frac {T } {L}\rfloor-1}\int_0^L e^{2t} \int_{-e^{-2t}}^{e^{-2t}} \chi_{\hat{U}}(g_\ell g_{jL} g_th_{s_0+s}\omega)dsd\ell >2T-2T(\epsilon+\delta)-L.$$ From this we have that there exists $a \in [0,L]$ so that $$\sum_{j=0}^{\lfloor \frac {T} {L}\rfloor-1} e


Candidate references: `cor:all together` (candidate_proof_dependency); `rem:Lipschitz_1` (candidate_proof_dependency); `lem:stay_proj_lip` (candidate_proof_dependency); `lem:growth from E` (candidate_proof_dependency); `lem:KZbound` (candidate_proof_dependency).

### Lem 4.9 — `lem:comparison`

Source lines 767–773.

Let $(\Omega,\mu)$ be a probability space and $F_i:(\Omega,\mu) \to \{0,1\}$ be a sequence of random variables such that there exists $0<\rho<1$ so that for any $j$, the conditional probability that $F_j$ is $1$ given $F_1,\dots,F_{j-1}$ is at least $\rho$. Let $G_i:(\Omega,\mu) \to \{0,1\}$ be independent and so that $\mu(G_i^{-1}(1))=\rho$. Then for all $\ell$ and $r$, $$\mu(\{\omega:\sum_{i=1}^\ell F_i(\omega)\leq r\})\leq \mu(\{\omega:\sum_{i=1}^\ell G_i(\omega)\leq r\}).$$

*Extractive statement; proof gist not independently reconstructed.*

### Cor 4.10 — `cor:standard large deviations`

Source lines 776–780.

Let $(\Omega,\mu)$ be a probability space and $F_j:(\Omega,\mu) \to \{0,1\}$ be a sequence of random variables such that there exists $0<\rho<1$ so that for any $j$, the conditional probability that $F_j$ is $1$ given $F_1,\dots,F_{j-1}$ is at least $\rho$. For all $\epsilon>0$ there exists $C_1,C_2>0$ so that $$\mu(\{\omega:\sum_{j=1}^\ell F_j(\omega)\leq (\rho-\epsilon)\ell\})\leq C_1e^{-C_2 \ell}.$$

*Extractive statement; proof gist not independently reconstructed.*

### Cor 4.11 — `cor:large deviations`

Source lines 781–786.

Let $(\Omega,\mu)$ be a probability space, $k \in \mathbb{N}$ and $F_j:(\Omega,\mu) \to \{0,1\}$ be a sequence of random variables such that there exists $0<\rho<1$ so that for any $j$, the conditional probability that $F_j$ is $1$ given $F_1,...,F_{j-k}$ is at least $\rho$. For all $\epsilon>0$ there exists $C_3,C_4>0$ so that $$\mu(\{\omega:\sum_{j=1}^\ell F_j(\omega)\leq (\rho-\epsilon)\ell\})\leq C_3e^{-C_4 \ell}.$$

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Let us consider the $k$ sequences of random variables: $(F_{1+jk})_{j\in \mathbb N}$, $(F_{2+jk})_{j\in \mathbb N}, \dots, (F_{k+jk})_{j \in \mathbb N}$. For each $i\in \{1, \dots, k\}$ the sequence $(F_{i+jk})_{j\in \mathbb N}$ satisfies the hypothesis of Corollary~\ref{cor:standard large deviations}. In fact, by assumption the conditional probabability that $F_{i +jk}$ is $1$ given $F_1,\dots, F_{i+jk-k}$ is at least $\rho$. Since $\{F_{i},\dots, F_{i +(j-1)k}\} \subset \{F_1,\dots, F_{i+jk-k}\}$, it follows that the conditional probability that $F_{i +jk}$ is $1$ given $F_{i+1},\dots, F_{i +(j-1)k}$ is also at least $\rho$. By Corollary~\ref{cor:standard large deviations} we therefore hav


Candidate references: `cor:standard large deviations` (candidate_proof_dependency).

### Prop 5.1 — `prop:better`

Source lines 813–839.

Let $\kappa_0$ and $\mathcal{F}\subset \hat{\mathcal{F}}$ be as in Lemma \ref{lem:Zgrowth from V}. There exist constants $\sigma>0,\tau>0$ so that for all large enough compact sets $\mathcal{K}$, there exists $\gamma_{\mathcal K}>0$ such that all horocycle sections $\phi:[-1, 1] \to \mathcal F \subset \FF$ at $\omega$, under the conditions that, for some $s_0 \in [-1,1]$ we have [label=(\alph*)] \item $e^t $ large enough and $\gamma \in (0, \gamma_{\mathcal K})$ ; \item $g_t h_{s_0} \omega \in \mathcal{K}$, \item the section $\phi_{t,s_0} \colo

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: For all $c, \gamma >0$, let $\mathcal B:= \mathcal B(c, \gamma)$ denote the set of $s \in [-1,1]$ such that for all $\ell \in [\frac 1 2 |\log \gamma|,(\frac 1 2 +c)|\log \gamma|]$ we have $g_{\ell} h_sg_{t}h_{s_0} \omega \not\in \mathcal{K}$. We first prove that for any $c>0$, and for $\gamma>0$ sufficiently small, the Lebesgue measure of $\mathcal B$ is polynomially small in $\gamma>0$. To prove an upper bound on the measure of $\mathcal B$ we apply Corollary~\ref{cor:ath} with $S=\frac 1 2 |\log \gamma|$ and $T:=c|\log \gamma|$ sufficiently large to derive that there exists $\zeta\in (0,1)$ such that $$\text{Leb}(\mathcal B) \leq \zeta^{c |\log \gamma|} = \gamma^{c |\log \zeta|} .$$ Let $

- `lem:key step` (proof_dependency; curated_source_check): Lemma \ref{lem:key step} and the previous paragraph establishes  the measure lower bound
- `lem:dist_integers` (proof_dependency; curated_source_check): By applying the above argument to the curve $\phi_{t,s_0}-z_{t,s_0}$

Candidate references: `lem:Zgrowth from V` (internal_reference); `cond:lip transport` (internal_reference); `lem:key step` (candidate_proof_dependency); `cor:ath` (candidate_proof_dependency); `eq:phi_t_s_0` (candidate_proof_dependency); `eq:better lem eq` (candidate_proof_dependency); `eq:meas lost` (candidate_proof_dependency); `conc:time` (candidate_proof_dependency); `conc:in cpct` (candidate_proof_dependency); `conc:growth` (candidate_proof_dependency); `lem:KZbound` (candidate_proof_dependency); `cond:lip transport` (candidate_proof_dependency); `cond:max size` (candidate_proof_dependency); `lem:dist_integers` (candidate_proof_dependency); `eq:max=gamma` (candidate_proof_dependency).

### Cor 5.2 — `cor:better`

Source lines 841–866.

Let $\kappa_0$ be as in Lemma \ref{lem:Zgrowth from V}. Let $\tilde{\mathcal{F}}$ be a direct sum of continuously varying strongly irreducible subbundles, each with a positive exponent. There exist constants $\sigma>0,\tau>0$ so that for all large enough compact sets $\mathcal{K}$, there exists $\gamma_{\mathcal K}>0$ such that all horocycle sections $\phi:[-1, 1] \to \FF$ at $\omega$, under the conditions that, for some $s_0 \in [-1,1]$ we have [label=(\alph*)] \item $e^t $ large enough and $\gamma \in (0, \gamma_{\mathcal K})$ ; \item $g_t h_

*Extractive statement; proof gist not independently reconstructed.*

- `prop:better` (proof_dependency; curated_source_check): The proposition formally implies the following corollary, with different choices of constants.

Candidate references: `lem:Zgrowth from V` (internal_reference).

### Lem 5.3 — `lem:dist_integers`

Source lines 876–884.

For any compact subset $\mathcal K$ of the moduli space there exists a constant $\gamma'_{\mathcal K}>0$ such that for $0<\gamma <\gamma'_{\mathcal K}$ the following holds. Let $\phi: [-1, 1] \to H^1(M, \mathbb R)$ be a $\gamma$-Lipschitz horocycle section at $\omega$. If $g_{\ell} \omega \in \mathcal K$ and $\max_{s\in [-1,1]} \Vert \phi(s)\Vert_{\mathbb Z }\leq \gamma$, then there exists a parallel section $z(s) \in H^1_{h_s \omega} (M,\mathbb Z)$ such that, for all $\ell \leq 3\vert \log \gamma\vert /4$ and for all $s \in [-1, 1]$ we have $$

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: We remark that over any compact set $\mathcal K$ the Hodge length of the shortest vector of the integer lattice $H^1(M, \mathbb Z)$ has a positive minimum $\delta_{\mathcal K}>0$. Since, by hypothesis, $\max \Vert \phi(s)\Vert_{\mathbb Z } \leq \gamma$ it follows that there exists $z \in H^1_{\omega}(M, \mathbb Z)$ such that $$ \vert \phi(0) -z \vert = \max_{s\in [-1,1]} \Vert \phi(s)\Vert_{\mathbb Z } \leq \gamma\,. $$ Let $z(s) \in H^1_{h_{s}\omega}(M, \mathbb Z)$ denote the section given by the parallel transport of $z \in H^1_{h_{s_0}\omega}(M, \mathbb Z)$, that is, the section such that $\pi_\omega(s) z(s)=z$ for all $s\in [-1,1]$. Similarly, let $\bar \phi(s) \in H^1_{h_{s}\omega}(M, \

### Lem 5.4 — `lem:key step`

Source lines 930–945.

Let $T_0>0$ be sufficiently large and so that $e^{T_0} \in \mathbb{N}$ and let $T > T_0$. Let $\kappa \in (0,\kappa_0)$ and let $\psi: [-1,1] \to H^1(M, \mathbb R)$ be a horocycle section at $\omega \in \mathcal K$ such that $\psi$ is projectively $\kappa $-Lipschitz. For all $j \in \{0, \dots, \lfloor \frac T {T_0} \rfloor-1\}$, for $s\in [-1,1]$, let $$ \Phi_{jT_0}(s) := KZ(g_{jT_0}, h_s\omega)\psi (s)\,. $$ If $\epsilon, \delta>0$ are small enough there exists $\sigma'>0$ such that \text{\rm Leb}\Big(\{s\in [-1,1]: \# \{j \in \{0,\dots, \lfl

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Let us recall that by definition, for $s\in [-1,1]$, $$ \Phi_{jT_0}(s) := KZ(g_{jT_0}, h_s\omega)\phi(s)\,. $$ We say that the pair $(j,s)$ is good if $g_{jT_0}h_s\omega\in V_\epsilon. $ Observe that if $(j,s)$ is good then, since by hypothesis the horocycle section $\phi$ is projectively $\kappa$-Lipschitz, by Lemma \ref{lem:Zgrowth from V} for sufficiently large $T_0>0$, we have \text{\rm Leb} \Big(\Big\{s'\in [s-e^{-2jT_0 },s +e^{-2jT_0)}]: |\Phi_{(j+1)T_0, s_0}(s') | \\ < e^{\mu T_0} |\Phi_{jT_0, s_0}(s') )| \Big\}\Big)< \eta e^{-2jT_0}. We now wish to use this estimate and Corollary \ref{cor:large deviations} to complete the proof of the lemma. To satisfy the assumptions of Corollary \r

- `lem:Zgrowth from V` (proof_dependency; curated_source_check): by Lemma \ref{lem:Zgrowth from V} for sufficiently large $T_0>0$, we have

Candidate references: `lem:Zgrowth from V` (candidate_proof_dependency); `cor:large deviations` (candidate_proof_dependency); `lem:KZbound` (candidate_proof_dependency); `eq:same size` (candidate_proof_dependency); `eq:useful growth` (candidate_proof_dependency); `eq:cond good` (candidate_proof_dependency); `cor:all together` (candidate_proof_dependency); `c:large dev` (candidate_proof_dependency); `eq:plenty in` (candidate_proof_dependency); `eq:in at integers` (candidate_proof_dependency).

### Rem 5.5 — `rem:line:946`

Source lines 946–951.

Let $\mathcal{F}\subset \FF$ be an equivariant subbundle, where the Kontsevich-Zorich cocycle acts strongly irreducibly with a positive exponent, and let $\lambda$ be the largest such exponent. Let $v:[-1,1]\to \mathcal{F}$ be a parallel horocycle section at $\omega$. By the above lemma and Lemma \ref{lem:KZbound} one has that for any $\epsilon'>0$ there exists $ c>0$ and $T_1\in \mathbb{R}$ so that for all $T>T_1$ we have $$\lambda(\{s\in [-1,1]:|KZ(g_{ T },h_s\omega)v(s)|<e^{(\lambda-\epsilon') T}\})<e^{-cT}.$$ More generally, one can prove s

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `lem:KZbound` (internal_reference); `prop:in good integer` (internal_reference); `lem:stay_proj_lip` (internal_reference).

### Lem 6.1 — `lem:v crit`

Source lines 1123–1128.

(Veech) Let $\alpha$ be a nontrivial eigenvalue of $F^t_{\omega}$. For every compact set $\mathcal{K}$ there exists $\ell_{\mathcal{K}}$ so that, if there exist $\ell \geq \ell_{\mathcal{K}}$ and a diverging sequence $(t_i) \subset \mathbb{R}^+$ with the property that $g_{t_i}\omega \in \mathcal{K}$, $g_{t_i-\ell}\omega \in \mathcal{K}$ and $g_{t_i+\ell}\omega \in \mathcal{K}$, then $$\underset{i \to \infty}{\lim} \| KZ(g_{t_i},\omega)(\alpha\Im(\omega))\|_{\mathbb{Z}}=0. $$

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: For each compact set $\mathcal{K}$ there exist constants $a_{\mathcal{K}}, \, b_{\mathcal{K}}$ and $d_{\mathcal{K}}>0$ so that if $\omega' \in \mathcal{K}$ then we can write $\omega'$ as a zippered rectangles over an interval $I$ of length at most $a_{\mathcal{K}}$, in the horizontal direction and one endpoint a singularity (that is, as a suspension with piece-wise constant roof function). Moreover the heights of the rectangles are at most $b_{\mathcal{K}}$. Lastly $d_{\mathcal{K}}$ is the length of the shortest saddle connection on all $\omega'\in \mathcal K$. Because there are singularities on each endpoint of the rectangles, if $u$ is the width of a rectangle, there is a saddle connection

### Lem 6.2 — `lem:transv`

Source lines 1164–1167.

The section $\psi$ has range in $SL(2,\mathbb{R})$ subspace at $\omega$ and it is transverse to any integer translate of the stable subbundle of the $SL(2,\mathbb{R})$ subbundle.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: The stable stable subspace of the $SL(2,\mathbb{R})$ subspace at $h_s\omega$ is generated by the cohomology class $[\Re (h_s\omega)]\in H^1_{h_s\omega}(M, \mathbb R)$. For any $z \in H^1_{\omega} (M, \mathbb Z)$ we consider the equation (with $a \in \mathbb R$) $$ \psi(s) - z = a \Re(h_s\omega) = a (\Re (\omega) + s \Im(\omega)) \,. $$ By definition of $\psi(s)$ we derive $$ z \wedge \Im(\omega) &=- a \Re(\omega) \wedge \Im(\omega) \, \\ z \wedge \Re(\omega) &= (as -\alpha \sec (\arctan s) ) \Re(\omega) \wedge \Im(\omega)\,. $$ Thus the coefficient $a \in \mathbb R$ is uniquely determined (given $z \in H^1_{\omega} (M, \mathbb Z)$) by the first equation, hence the second equation has a uniqu

### Prop 6.3 — `prop:central`

Source lines 1185–1195.

For every compact set $\mathcal K$ there exists a constant $\gamma''_{\mathcal K}>0$ such that the following holds. If there exist $t_0>0$ and an integer class $z \in H^1_{g_{t_0}\omega} (M, \mathbb Z)$ such that \item $g_{t_0} \omega \in \mathcal K$ and $\{t>0\vert g_t\omega \in \mathcal K\}$ has upper density at least $\frac 3 4 $; \item $\vert KZ(g_{t_0}, \omega) (\alpha [\Im(\omega)] ) -z \vert \in (0, \gamma''_{\mathcal K})$,\footnote{Note that this condition cannot hold if $\alpha=0$.} \item $KZ(g_{t_0}, \omega) (\alpha [\Im(\omega)] ) -z

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Let $\gamma''_{\mathcal K}$ be so small that the minimum Hodge distance between points of the lattice $H^1_\omega (M, \mathbb Z)$ for $\omega \in \mathcal K$ is at least $10\gamma''_{\mathcal K}$. By the isometry property of the Kontsevich--Zorich cocycle on $E_0$, we have that for all $t\geq t_0$, \Vert KZ(g_{t_0} , \omega) (\alpha [\Im(\omega)] ) \Vert_{\mathbb Z} = \vert KZ(g_{t_0}, \omega) (\alpha [\Im(\omega)] ) - z\vert \\ = \vert KZ(g_t, \omega) (\alpha [\Im(\omega)] ) - KZ(g_{t-t_0}, g_{t_0} \omega)(z)\vert \leq \gamma''_{\mathcal K}\,, hence, whenever $g_{t} \omega \in \mathcal K$, since $KZ(g_{t-t_0}, g_{t_0} \omega)(z) \in H^1_{g_t\omega} (M, \mathbb Z)$, by the choice of the cons


Candidate references: `lem:v crit` (candidate_proof_dependency).

### Thm 6.4 — `thm:big quote`

Source lines 1214–1215.

(c.f. \cite{EskMir}, \cite{FMZ}, \cite{Fi17}) The Hodge bundle (hence the symplectic orthogonal of the $SL(2,\mathbb R)$ subbundle) splits as a sum of an $SL(2,\mathbb R)$-invariant continuous (in fact, algebraic) subbundle on which the Kontsevich--Zorich cocycle acts isometrically and a finite number of $SL(2,\mathbb R)$-invariant continuous (algebraic) irreducible subbundles on each of which the cocycle has a strictly positive exponent. After passing to a finite cover, the irreducible components of the decomposition can be taken to be strongl

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: By \cite[Theorem A.6]{EskMir} the Kontsevich--Zorich cocycle on Hodge bundle is semi-simple, in the sense that it can be split as a direct sum of $SL(2,\mathbb R)$-invariant measurable subbundles, and each irreducible component is either isotropic or symplectic. By \cite[Theorem A.5]{EskMir} if the subbundle is isotropic then all of the exponents are zero. By \cite[Theorem A.4]{EskMir}, if all the exponents are zero then the cocycle acts isometrically. It follows that the Hodge bundle can be split as a sum of an isometric component, equal to the sum of all isotropic irreducible components, and a finite sum of symplectic irreducible components with at least one non-zero exponent. Each symplec


Candidate references: `external:EskMir` (external_reference); `external:FMZ` (external_reference); `external:Fi17` (external_reference).

### Cor 6.5 — `cor:big quote`

Source lines 1223–1230.

Let $\phi$ be a horocycle section. For almost every $s\in \mathbb R$, one of the following two (mutually exclusive) possibilities holds \item $\phi(s)$ {belongs to a continuous} $SL(2,\mathbb{R})$-equivariant subbundle where the Kontsevich-Zorich cocycle acts isometrically. \item $\phi(s)$ has a non-trivial projection onto a continuous $SL(2,\mathbb{R})$-equivariant subbundle where the Kontsevich-Zorich cocycle has a positive exponent and acts strongly irreducibly.

*Extractive statement; proof gist not independently reconstructed.*

### Lem 7.1 — `lem:open`

Source lines 1368–1376.

For every $\epsilon>0$ and $T>0$ we have \{P: \int_{X(P)} \Big\vert \frac{1}{T}\int_0^Tf(F^t(x,\theta),&F^t(y,\psi))dt-\int_{X(P)}f \Big\vert <\epsilon \\ &\text{ for all }f\in \LipP\} is open.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Let $\epsilon<\frac 1 4$. It suffices to show that for any $P$ which belongs to the set in formula~\eqref{eq:open} there exists $\delta_P>0$ so that any $\delta_P$-perturbation also belongs to the set~\eqref{eq:open}. Because $P$ belongs to the set~\eqref{eq:open}, it satisfies the inequality in ~\eqref{eq:open} for $\epsilon-\epsilon'''$ for some $\epsilon'''>0$. We choose $\delta>0$ so that Lemma \ref{lem:lip in lip} implies that, if $Q$ is a $\delta$-perturbation of $P$, then $Lip_1(X(Q))\subset Lip_{1+\frac {\epsilon'''}9}(X(P))$, formula \eqref{eq:close wm averages} is satisfied with $\epsilon=\frac {\epsilon'''}9$ and $$ \Big\vert \int_{X(Q)} f\circ \pi_{P,Q}^{-1}-\int_{X(P)}f \Big\ver


Candidate references: `eq:open` (candidate_proof_dependency); `lem:lip in lip` (candidate_proof_dependency); `eq:close wm averages` (candidate_proof_dependency).

### Cor 7.2 — `cor:dense`

Source lines 1378–1381.

For every $\epsilon>0$ there exists $N_\epsilon$ such that the following holds. If $P$ is a rational polygon with the property that the group of reflections about its sides (translated to the origin) contains a rotation by $\frac{2\pi} {M}$ with $M\geq N_\epsilon$, then there exists $T_0$ such that for all $T\geq T_0$ and $f \in \LipP$ we have $$ \int_{X(P)} \vert \frac{1}{T}\int_0^Tf(F^t(x,\theta),F^t(y,\psi))dt-\int_{X(P)}f|<\epsilon.$$

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Consider the flat surface $M_P$ obtained by unfolding $P$. By Theorem~\ref{thm:erg prod} we have that for almost every pair of directions the product flow is ergodic on $M_P\times M_P$. Considering these flows on $T_1(P)$, by our assumption on the group of reflections, they equidistribute in the product of the table cross $M\geq N_\epsilon$ evenly spaced copies of a discrete set in $S^1$. The corollary follows.

- `thm:erg prod` (proof_dependency; curated_source_check): By Theorem~\ref{thm:erg prod} we have that for almost every pair of directions the product flow is ergodic

Candidate references: `thm:erg prod` (candidate_proof_dependency).

### Lem 7.4 — `lem:lip in lip`

Source lines 1408–1410.

For every table $P$ and for every $\epsilon>0$, there exists~$\delta>0$ such that if $Q$ is a $\delta$-perturbation of $P$, then $$Lip_{1-\epsilon}(X(Q))\subset \phi_{P,Q}^{-1}(\LipP)\,.$$

*Extractive statement; proof gist not independently reconstructed.*

### Lem 7.5 — `lem:lip tolerance`

Source lines 1412–1419.

If $P$ is in the set described by \eqref{eq:open} then there exists $\epsilon'>0$ so that $$ \int_{X(P)} \Big\vert \frac{1}{T}\int_0^Tf(F^t(x,\theta),F^t(y,\psi))dt-&\int_{X(P)}f \Big\vert<\epsilon \\ & \text{ for all }f\in Lip_{1+\epsilon'}(X(P)). $$

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: The table $P$ is compact and $\LipP$ is bounded and equicontinuous, so by the Arzel\`a-Ascoli Theorem, $\LipP$ is compact in $\|\cdot\|_{\sup}$. Thus, there exists $0<\epsilon''<\frac 1 9$ so that \int_{X(P)} \Big\vert\frac{1}{T} \int_0^Tf(F^t(x,\theta),F^t(y,\psi))dt-&\int_{X(P)}f \Big\vert<\epsilon-\epsilon'' \,, \\ &\text{ for all }f\in \LipP. Indeed, for each $f \in \LipP$ there exists $\epsilon_f$ so that $$ \int_{X(P)} \Big\vert\frac{1}{T} \int_0^Tf(F^t(x,\theta),F^t(y,\psi))dt-&\int_{X(P)}f \Big\vert<\epsilon-\epsilon_f. $$ Thus, there is an open neighborhood $U_f$ of $f$ that, for all $h \in U_f$, $$ \int_{X(P)} \Big\vert\frac{1}{T} \int_0^Th(F^t(x,\theta),F^t(y,\psi))dt-&\int_{X(P)}


Candidate references: `eq:open` (internal_reference); `eq:eps2` (candidate_proof_dependency).

### Prop 7.6 — `prop:line:1447`

Source lines 1447–1455.

(Vorobets \cite[Proposition 2.3]{vorobets}) Let $Q$ be a $\delta$-perturbation of $P$. For each $t>0$ there exists $B \subset P \times S^1$ dependent on the polygons $P,Q$, the map $\phi:= \phi_{P,Q}$ and $t$ and of measure at most $C_3(C_1t+C_2)^3 \delta$ such that for each $(x,\theta) \in P \times S^1 \setminus B$ and for each $\tau$, $0\leq \tau \leq t$, at least one of the following two possibilities hold: \item The distance between $\pi_P^{(1)}F_p^t(x,v)$ and $\phi^{-1}(\pi_Q^{(1)}F_Q^t(\phi(x),v)$ is at most $C_4(C_1t+C_2)^2 \delta$ and t

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `external:vorobets` (external_reference).

### Cor 7.7 — `eq:close wm averages`

Source lines 1456–1461.

For any $f \in Lip_c(X(P))$, $T$ and $\epsilon>0$ there exists $\delta>0$ so that for any $Q$, a $\delta$-perturbation of $P$ we have \int_{X(Q)} \Big\vert \int_0^T f\circ \phi_{P,Q}^{-1}(F^t(x,\theta),F^t(y,\psi))dt \\ -\int_0^Tf(F^t(x,\theta),F^t(y,\psi))dt \Big\vert <\epsilon.

*Extractive statement; proof gist not independently reconstructed.*

## External inputs and references

### [AAEKMU] AAEKMU — `external:AAEKMU`

Source lines 1474–1476.

Al-Saqban, H; Apisa, P; Erchenko, A; Khalil,~O; Mirzadeh,~S; Uyanik,~C. Exceptional Directions for the Teichm\"uller Geodesic Flow and Hausdorff Dimension, J. Eur. Math. Soc. 23 (2021), no. 5, 1423–-1476. DOI 10.4171/JEMS/1037


Candidate references: `sec:background` (external_reference); `lem:2.13 for hor` (external_reference); `prop:cpct integer` (external_reference); `prop:in good integer` (external_reference).

### [Ath] ath thesis — `external:ath thesis`

Source lines 1476–1477.

Athreya, J. Quantitative recurrence and large deviations for Teichmuller geodesic flow. Geom. Dedicata 119 (2006), 121--140.


Candidate references: `lem:in compact` (external_reference); `cor:ath` (external_reference).

### [AuAvDel] AAD — `external:AAD`

Source lines 1477–1478.

Aulicino, D; Avila, A; Delecroix, V. Announcement


Candidate references: `section:1` (external_reference); `sec:background` (external_reference).

### [AvDel] AD — `external:AD`

Source lines 1478–1479.

Avila, A; Delecroix, V. Weak mixing directions in non-arithmetic Veech surfaces J. Amer. Math. Soc. 29, no. 4 (2016), 1167--1208.


Candidate references: `section:1` (external_reference); `sec:background` (external_reference).

### [AvFo] AF — `external:AF`

Source lines 1479–1482.

Avila, A; Forni, G. Weak mixing for interval exchange transformations and translation flows Ann. of Math. 165 (2007), no. 2, 637--664. \bibitem[BoTr] {BoTr} Bobok,~J; Troubetzkoy, S. Topologically weakly mixing polygonal billiards, Proceedings AMS. DOI: https://doi.org/10.1090/proc/14135


Candidate references: `section:1` (external_reference); `sec:background` (external_reference).

### [BKM] BKM — `external:BKM`

Source lines 1482–1483.

Boldrighini, C; Keane, M; Marchetti, F, Billiards in polygons. Ann. Probab.~6 (1978), no. 4, 532--540.


Candidate references: `section:1` (external_reference).

### [CasPros] CP — `external:CP`

Source lines 1483–1485.

Casati, G; Prosen, T, Mixing property of triangular billiards. Phys. Rev. Lett. 83 (1999), 4729--4732.


Candidate references: `section:1` (external_reference).

### [ChEs] CE — `external:CE`

Source lines 1485–1486.

Chaika, J; Eskin, A. Every flat surface is Birkhoff and Oseledets generic in almost every direction. J. Mod. Dyn. 9 (2015), 1--23.


Candidate references: `section:1` (external_reference); `sec:background` (external_reference); `prop:growth mechanism` (external_reference); `lem:Egood` (external_reference); `sec:grow` (external_reference); `thm:main` (external_reference).

### [ChHu] CH — `external:CH`

Source lines 1486–1487.

Chaika, J.; Hubert, P. Circle averages and disjointness in typical flat surfaces on every Teichm\"uller disc, Bull. London Math. Soc. 49, no. 5 (2017), 755--769.


Candidate references: `section:1` (external_reference).

### [CheMar] CM — `external:CM`

Source lines 1487–1489.

Chernov, N; Markarian, R. Chaotic billiards. Mathematical Surveys and Monographs 127. American Mathematical Society, Providence, RI, 2006. xii+316 pp.


Candidate references: `section:1` (external_reference).

### [EskMir] EskMir — `external:EskMir`

Source lines 1489–1490.

Eskin, A; Mirzakhani, M. \textit{Invariant and stationary measures for the $SL(2,\mathbb{R})$ action on moduli space.} Publ. Math. Inst. Hautes \'Etudes Sci. 127 (2018), 95--324.


Candidate references: `section:1` (external_reference); `sec:grow` (external_reference); `thm:big quote` (external_reference).

### [EskMirMo] EskMirMo — `external:EskMirMo`

Source lines 1490–1491.

Eskin, A; Mirzakhani, M; Mohammadi, A. \textit{Isolation, equidistribution, and orbit closures for the $SL(2,\mathbb{R})$ action on moduli space.} Ann. of Math. 182 (2015), no. 2, 673--721.


Candidate references: `section:1` (external_reference); `sec:grow` (external_reference); `thm:EMM 2.13` (external_reference).

### [FH19] FH19 — `external:FH19`

Source lines 1491–1493.

Ferenczi, S; Hubert, P. Rigidity of square-tiled interval exchange transformations, J. Mod. Dynam. 14 (2019), 153-177.


Candidate references: `section:1` (external_reference).

### [Fi16] Fi16 — `external:Fi16`

Source lines 1493–1494.

Filip, S. Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, { Invent. math.} 205 (2016), 617--670. https://doi.org/10.1007/s00222-015-0643-3

### [Fi17] Fi17 — `external:Fi17`

Source lines 1494–1496.

Filip, S. Zero Lyapunov exponents and monodromy of the Kontsevich--Zorich cocycle, { Duke Math. J.} 166 (2017) (4), 657--706.


Candidate references: `thm:big quote` (external_reference).

### [Fo02] Fo — `external:Fo`

Source lines 1496–1497.

Forni, G. Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. 155, no. 1, (2002), 1--103.


Candidate references: `lem:KZbound` (external_reference); `prop:easy restriction` (external_reference).

### [Fo24] FoSurv — `external:FoSurv`

Source lines 1497–1498.

Forni, G. Effective unique ergodicity and weak mixing of translation flows, Modern Aspects of Dynamical Systems (C. Ulcigrai C. Bonanno, C. Sorrentino, ed.), Lecture Notes in Mathematics (LNM)-CIME Foundation Subseries, vol. 2347, Springer, 2024, pp. 161--221.


Candidate references: `sec:main proof` (external_reference).

### [FoMa] FM — `external:FM`

Source lines 1498–1500.

Forni, G; Matheus, C. Introduction to Teichm\"uller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, in Lectures from the Bedlewo Summer School 2011, edited by F. Rodriguez Hertz. J. Mod. Dynam. 8 (3-4), 2014, 271--436.


Candidate references: `sec:background` (external_reference); `lem:KZbound` (external_reference); `prop:easy restriction` (external_reference).

### [FoMaZo] FMZ — `external:FMZ`

Source lines 1500–1501.

Forni, G; Matheus, C; Zorich, A. Lyapunov spectrum of invariant subbundles of the Hodge bundle Ergod. th. \& Dynam. Sys. 34 (2014) 353-408.


Candidate references: `sec:background` (external_reference); `thm:big quote` (external_reference).

### [Gut] Gut — `external:Gut`

Source lines 1501–1502.

Gutkin, E. Billiard dynamics: an updated survey with the emphasis on open problems. Chaos 22, No. 2, 026116 (2012).


Candidate references: `section:1` (external_reference).

### [GutKa] GK — `external:GK`

Source lines 1502–1504.

Gutkin, E; Katok, A. Weakly mixing billiards, Springer Lecture Notes in Math. 1345 (1988), 163--176.


Candidate references: `section:1` (external_reference).

### [KKLM] KKLM — `external:KKLM`

Source lines 1504–1505.

Kadyrov, S; Kleinbock, D; Lindenstrauss, E; Margulis, G. Singular systems of linear forms and non-escape of mass in the space of lattices. Journal d'Analyse Math\'ematique 133, (1) (2017) 253--277.


Candidate references: `prop:in good integer` (external_reference).

### [Kat87] Ka87 — `external:Ka87`

Source lines 1505–1506.

Katok, A. B. The growth rate for the number of singular and periodic orbits for a polygonal billiard, Commun. Math. Phys. 111 (1987), 151--160.


Candidate references: `section:1` (external_reference).

### [Kat05] KaBil — `external:KaBil`

Source lines 1506–1509.

Katok, A. B. Billiard table as a playground for a mathematician. Surveys in modern mathematics, 216--242, London Math. Soc. Lecture Note Ser. 321, Cambridge Univ. Press, Cambridge, 2005.


Candidate references: `section:1` (external_reference).

### [KT] KT — `external:KT`

Source lines 1509–1510.

Katok, A. B; Thouvenot, J.-P. {\em Spectral properties and combinatorial constructions in ergodic theory}, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, 649--743.


Candidate references: `section:1` (external_reference).

### [KMS] KMS — `external:KMS`

Source lines 1510–1511.

Kerckhoff, S; Masur, H; Smillie, J. Ergodicity of Billiard Flows and Quadratic Differentials, Ann. of Math. 124, no. 2, (1986), 293--311.


Candidate references: `section:1` (external_reference).

### [MaTr] MaTr — `external:MaTr`

Source lines 1511–1512.

M\'alaga Sabogal, A; Troubetzkoy, S. \text{it} {\it Weakly mixing polygonal billiards.} Bull. Lond. Math. Soc. 49 (2017), no. 1, 141--147.


Candidate references: `section:1` (external_reference).

### [Mas82] Ms82 — `external:Ms82`

Source lines 1512–1514.

Masur, H. Interval Exchange Transformations and Measured Foliations, Ann. of Math. 115, No. 1 (1982), 169--200.

### [Mas90] Ms90 — `external:Ms90`

Source lines 1514–1515.

Masur, H. The growth rate of trajectories of a quadratic differential, Ergod. Th. \& Dynam. Sys. 10 (1990), 151--176.


Candidate references: `section:1` (external_reference).

### [MT] MT — `external:MT`

Source lines 1515–1516.

Masur, H; Tabachnikov, S. Rational billiards and flat structures. Handbook in Dynamical Systems Vol. 1A, Elsevier, 2002, 1015--1089.


Candidate references: `section:1` (external_reference).

### [Sch13] Sch13 — `external:Sch13`

Source lines 1516–1517.

Scheglov, D. Growth of periodic orbits and generalized diagonals for typical triangle billiards, J. Mod. Dynam. 7 (2013), 31--44.


Candidate references: `section:1` (external_reference).

### [Sch20] Sch — `external:Sch`

Source lines 1517–1518.

Scheglov, D. Complexity growth of a typical triangular billiard is weakly exponential, JAMA 142, 105--124 (2020). https://doi.org/10.1007/s11854-020-0134-3


Candidate references: `section:1` (external_reference).

### [Tab] Tab — `external:Tab`

Source lines 1518–1519.

Tabachnikov, S. Geometry and billiards. Student Mathematical Library 30. American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005. xii+176 pp.


Candidate references: `section:1` (external_reference).

### [Ve84] veech metric 1 — `external:veech metric 1`

Source lines 1519–1520.

Veech, W. The metric theory of interval exchange transformations. I. Generic spectral properties. Amer. J. Math. 106 (1984), no. 6, 1331--1359.


Candidate references: `sec:background` (external_reference); `sec:main proof` (external_reference).

### [Vo97] vorobets — `external:vorobets`

Source lines 1520–1521.

Vorobets, Y. Ergodicity of billiards in polygons. Mat. Sb. 188, no. 3 (1997), 65--112; Sb. Math. 188, no. 3 (1997), 389--434


Candidate references: `sec:background` (external_reference); `defin:line:1396` (external_reference); `prop:line:1447` (external_reference).

## Coverage and limits

{"mode": "native_latex_index", "claims": 51, "sections": 7, "external_references": 35, "source_bibliography_entries": 35, "unresolved_bibliography": 0, "unresolved": 1}

- Reference edges are not certified proof dependencies. Edge direction is prerequisite → consumer.
- Numbers reflect simple source counters; labels and line anchors are authoritative.
- Extractive statements and proof openings are marked; they are not reconstructed proof gists.
- Macros, included files and custom counters require an expanded source or annotated fallback.
- Eleven proof-use edges were checked in the native source. This is a partial dependency audit; the remaining references are candidates. Decision colors belong only to the panel view.