Local Limit Theorem for Randomly Deforming Billiards

Abstract

We study limit theorems in the context of random perturbations of dispersing billiards in finite and infinite measure. In the context of a planar periodic Lorentz gas with finite horizon, we consider random perturbations in the form of movements and deformations of scatterers. We prove a central limit theorem for the cell index of planar motion, as well as a mixing local limit theorem for piecewise Hölder continuous observables. In the context of the infinite measure random system, we prove limit theorems regarding visits to new obstacles and self-intersections, as well as decorrelation estimates. The main tool we use is the adaptation of anisotropic Banach spaces to the random setting.

Appendices

Appendix A. Proof of Lemma 4.5

Here we prove the Lemma 4.5, which was used in Sect. 4.2, especially used in the proof of Theorem 2.4.

Let us prove that (36) holds true. By density, it suffices to perform the estimate for $f \in \mathcal {C}^1({\bar{M}}_0)$. In the proof below, we use the fact that the invariant measure ${\bar{\mu }}_0$ is absolutely continuous with respect to the Lebesgure measure.

Choose $\ell \ge 1$ and fix ${\underline{\omega }}_\ell := (\omega _1, \ldots , \omega _\ell )$. Let g be as in the statement of the lemma. For brevity, denote by ${\bar{T}}_{\underline{\omega }_\ell }^\ell = \bar{T}_{\omega _\ell } \circ \cdots \circ {\bar{T}}_{\omega _1}$ the composition of random maps and by $\mathcal {L}_{\underline{\omega }_\ell }^\ell$ its associated transfer operator. Also, set $H_\ell ^p(g) = |g|_\infty + \sup _{C\in \mathcal C_{\omega _1,\ldots ,\omega _\ell }}C_{g_{|C}}^{(p)}$. We must estimate

$$\begin{aligned} {\mathbb {E}}_{{\bar{\mu }}_0} [f \, g ] = \int _{{\bar{M}}_0} f \, g \, d{\bar{\mu }}_0 = \int _{{\bar{M}}_0} \mathcal {L}_{{\underline{\omega }}_\ell }^\ell f \cdot g \circ ({\bar{T}}_{{\underline{\omega }}_\ell }^{\ell })^{-1} \, d{\bar{\mu }}_0 . \end{aligned}$$

To do this, we decompose ${\bar{M}}_0$ into a countable collection of local rectangles, each foliated by a smooth collection of stable curves on which we may apply our norms. This technique follows closely the decomposition used in [16, Lemma 3.4].

We partition each connected component of ${\bar{M}}_0 {\setminus } (\cup _{|k| \ge k_0} {\mathbb {H}}_k)$, into finitely many boxes $B_j$ whose boundary curves are elements of $\mathcal {W}^s$ and $\mathcal {W}^u$, as well as the horizontal boundaries of ${\mathbb {H}}_{\pm k_0}$. We construct the boxes $B_j$ so that each has diameter in $(\delta /2, \delta )$, for some $\delta >0$, and is foliated by a smooth foliation of stable curves $\{ W_\xi \}_{\xi \in \Xi _j}$, such that each curve $W_{\xi }$ is stretched completely between the two unstable boundaries of $B_j$. Indeed, due to the continuity of the cones $C^s(x)$ from (H1), we can choose $\delta$ sufficiently small that the family $\{ W_\xi \}_{\xi \in \Xi _j}$ is a family of parallel line segments.

We disintegrate the measure ${\bar{\mu }}_0$ on $B_j$ into a family of conditional probability measures $d\mu _{\xi } = c_\xi \cos \varphi \, dm_{W_\xi }$, $\xi \in \Xi _j$, where $c_\xi$ is a normalizing constant, and a factor measure $\lambda _j(\xi )$ on the index set $\Xi _j$. Since ${\bar{\mu }}_0$ is absolutely continuous with respect to Lebesgue measure on ${\bar{M}}_0$, we have $\lambda _j(\Xi _j) = \bar{\mu }_0(B_j) = \mathcal {O}(\delta ^2)$.

Similarly, on each homogeneity strip ${\mathbb {H}}_t$, $t \ge k_0$, we choose a smooth foliation of parallel line segments $\{ W_\xi \}_{\xi \in \Xi _t} \subset {\mathbb {H}}_t$ which completely cross ${\mathbb {H}}_t$. Due to the uniform transversality of the stable cone with $\partial {\mathbb {H}}_t$, we may choose a single index set $\Xi _t$ for each homogeneity strip. We again disintegrate ${\bar{\mu }}_0$ into a family of conditional probability measures $d\mu _\xi = c_\xi \cos \varphi \, dm_{W_\xi }$, $\xi \in \Xi _t$, and a transverse measure $\lambda _t(\xi )$ on the index set $\Xi _t$. This implies that $\lambda _t(\Xi _t) = {\bar{\mu }}_0({\mathbb {H}}_t) = \mathcal {O}(|t|^{-5})$ for each $|t| \ge k_0$.

Notice that on each homogeneity strip ${\mathbb {H}}_k$, the function $\cos \varphi$ satisfies,

$$\begin{aligned} |\log \cos \varphi (x)- \log \cos \varphi (y)| \le C d(x,y)^{1/3} \end{aligned}$$

(43)

for some uniform constant $C>0$ (uniform in k).

We are ready to estimate the required integral. Let $\mathcal {G}_\ell (W_\xi )$ denote the components of $(\bar{T}_{\underline{\omega }_\ell }^{\ell })^{-1} W_{\xi }$, with long pieces subdivided to have length between $\delta _0/2$ and $\delta _0$, as in the proof of Lemma 3.14.

$$\begin{aligned} \begin{aligned}&\int \mathcal {L}_{ \underline{\omega }_\ell }^\ell f \cdot g \circ (\bar{T}_{\underline{\omega }_\ell }^{\ell })^{-1} \, d{\bar{\mu }}_0 \\&\quad =\sum _j \int _{B_j} \mathcal {L}_{\underline{\omega }_\ell }^\ell f \cdot g \circ (\bar{T}_{\underline{\omega }_\ell }^{\ell })^{-1} d{\bar{\mu }}_0 + \sum _{|t| \ge k_0} \int _{{\mathbb {H}}_t} \mathcal {L}_{\underline{\omega }_\ell }^\ell f \cdot g \circ (\bar{T}_{\underline{\omega }_\ell }^{\ell })^{-1} d{\bar{\mu }}_0\\&\quad =\sum _j \!\! \int _{\Xi _j} \! \int _{W_\xi } \! \! \mathcal {L}_{\underline{\omega }_\ell }^\ell f \cdot g \circ (\bar{T}_{\underline{\omega }_\ell }^{\ell })^{-1} \, d\mu _\xi d\lambda _j(\xi )\\&\qquad + \sum _{|t| \ge k_0} \! \! \int _{\Xi _t} \! \int _{W_\xi } \! \! \mathcal {L}_{\underline{\omega }_\ell }^\ell f \cdot g \circ (\bar{T}_{\underline{\omega }_\ell }^{\ell })^{-1} \, d\mu _\xi d\lambda _t(\xi ) \\&\quad =\sum _j \int _{\Xi _j} \sum _{W_{\xi ,i} \in \mathcal {G}_\ell (W_\xi )} \int _{W_{\xi ,i}} f \, g \, c_\xi \cos \varphi \circ \bar{T}_{{\underline{\omega }}_\ell }^\ell \, J_{W_{\xi ,i}}{\bar{T}}_{{\underline{\omega }}_\ell }^\ell \, dm_{W_{\xi ,i}} d\lambda _j(\xi ) \\&\qquad + \sum _{|t| \ge k_0} \int _{\Xi _t} \sum _{W_{\xi ,i} \in \mathcal {G}_\ell (W_\xi )} \int _{W_{\xi ,i}} f \, g \, c_\xi \cos \varphi \circ {\bar{T}}_{{\underline{\omega }}_\ell }^\ell \, J_{W_{\xi ,i}}\bar{T}_{{\underline{\omega }}_\ell }^\ell \, dm_{W_{\xi ,i}} d\lambda _t(\xi ) \, . \end{aligned} \end{aligned}$$

Next we use the assumption that g is Hölder continuous on connected componts of ${\bar{M}}_0 {\setminus } (\cup _{k=1}^\ell \bar{T}_{\omega _1}^{-1} \circ \cdots \circ {\bar{T}}_{\omega _k}^{-1} (\mathcal {S}_{0,H}))$. Since elements of $\mathcal {G}_\ell (W_\xi )$ are also subdivided according to these singularity sets, we have that g is Hölder continuous on each $W_{\xi , i} \in \mathcal {G}_\ell (W_\xi )$. Thus,

$$\begin{aligned} \begin{aligned}&\int _{W_{\xi , i}} f \, g \, c_\xi \cos \varphi \circ \bar{T}_{{\underline{\omega }}_\ell }^\ell \, J_{W_{\xi ,i}}\bar{T}_{{\underline{\omega }}_\ell }^\ell \, dm_{W_{\xi ,i}}\\&\quad \le |f|_w |g|_{\mathcal {C}^p(W_{\xi ,i})} c_\xi |\cos \varphi \circ {\bar{T}}_{{\underline{\omega }}_\ell }^\ell | _{\mathcal {C}^p(W_{\xi ,i})} |J_{W_{\xi ,i}}{\bar{T}}_{{\underline{\omega }}_\ell }^\ell |_{\mathcal {C}^p(W_{\xi ,i})} \\&\quad \le |f|_w H_\ell ^p(g) |J_{W_{\xi ,i}}\bar{T}_{{\underline{\omega }}_\ell }^\ell |_{\mathcal {C}^0(W_{\xi ,i})} \frac{C}{|W_{\xi }|}, \end{aligned} \end{aligned}$$

where we used (43) in the last estimate, as well as the fact that the normalizing constant $c_\xi$ is proportional to $|W_\xi |^{-1}$. This implies that

$$\begin{aligned} {\mathbb {E}}_{{\bar{\mu }}_0} [ f \, g ]&\le C |f|_w H_\ell ^p(g) \Big ( \sum _j \int _{\Xi _j} \sum _{W_{\xi ,i} \in \mathcal {G}_\ell (W_\xi )} |J_{W_{\xi ,i}}\bar{T}_{{\underline{\omega }}_\ell }^\ell |_{\mathcal {C}^0(W_{\xi ,i})} |W_\xi |^{-1} \, d\lambda _j(\xi ) \\&\quad + \sum _{|t| \ge k_0} \int _{\Xi _t} \sum _{W_{\xi ,i} \in \mathcal {G}_\ell (W_\xi )} |J_{W_{\xi ,i}}{\bar{T}}_{{\underline{\omega }}_\ell }^\ell |_{\mathcal {C}^0(W_{\xi ,i})} |W_\xi |^{-1} \, d\lambda _t(\xi ) \Big ). \end{aligned}$$

Now $\sum _{W_{\xi ,i} \in \mathcal {G}_\ell (W_\xi )} |J_{W_{\xi ,i}}\bar{T}_{{\underline{\omega }}_\ell }^\ell |_{\mathcal {C}^0(W_{\xi ,i})}$ is bounded by a uniform constant independent of $\xi$ and ${\underline{\omega }}_\ell$ by [16, Lemma 5.5(b)]. Moreover, $\int _{\Xi _j} |W_{\xi }|^{-1} d \lambda _j(\xi ) \le C\delta _0$ for some constant $C>0$ since we chose our foliation to be comprised of long cone-stable curves. We conclude that the first term to the right hand side of the last inequality is uniformly bounded by $C_1|f|_w H_\ell ^p(g)$ since the sum over j is finite.

For the second term on the right hand side of the last inequality, we again use [16, Lemma 5.5(b)] as well as the fact that $|W_{\xi }|^{-1} = \mathcal {O}(t^3)$ for $\xi \in \Xi _t$, while $\lambda _t(\Xi _t) = \mathcal {O}(t^{-5})$. Thus

$$\begin{aligned} \sum _{|t| \ge k_0} \int _{\Xi _t}|W_{\xi }|^{-1}d \lambda _t(\xi )\le \sum _{|t| \ge k_0} C t^{-2} \le C k_0^{-1} . \end{aligned}$$

We conclude that

$$\begin{aligned} \left| {\mathbb {E}}_{{\bar{\mu }}_0} [f \, g] \right| \le K_1 |f|_w H^p_\ell (g) , \end{aligned}$$

for some uniform constant $K_1$ depending on $\bar{\mathcal {F}}_{\vartheta _0}$, but not on f, $\ell$ or $\underline{\omega }_\ell$. This completes the proof of (36).

To prove (37), we follow the proof of Lemma 3.14. Note that for $f \in \mathcal {C}^1({\bar{M}}_0)$, $W \in \mathcal {W}^s$, and a test function $\psi$, we have

$$\begin{aligned} \int _W \mathcal {L}_{u, \omega _\ell } \ldots \mathcal {L}_{u, \omega _1}(fg) \, \psi \, dm_W = \sum _{W_i} \int _{W_i} fg\, e^{iu \cdot S_\ell } \, \psi \circ {\bar{T}}^\ell _{{\underline{\omega }}_\ell } \, J_{W_i}\bar{T}^\ell _{{\underline{\omega }}_\ell } \, dm_{W_i} \, , \end{aligned}$$

where the sum is taken over $W_i \in \mathcal {G}_\ell (W)$, the components of $({\bar{T}}^\ell _{{\underline{\omega }}_\ell })^{-1}W$, subdivided as before. This is the same type of expression as in [16, eq. (5.24)] or [16, eq. (4.4)], but now the test function is

$$\begin{aligned} g\, e^{iu \cdot S_\ell } \, \psi \circ {\bar{T}}^\ell _{\underline{\omega }_\ell } \, J_{W_i}{\bar{T}}^\ell _{{\underline{\omega }}_\ell } \end{aligned}$$

rather than simply $\psi \circ {\bar{T}}^\ell _{{\underline{\omega }}_\ell } \, J_{W_i}{\bar{T}}^\ell _{{\underline{\omega }}_\ell }$. Since $S_\ell$ is constant on each $W_i \in \mathcal {G}_\ell (W)$, and we have assumed that g is (uniformly in $\ell$) Hölder continuous on each $W_i \in \mathcal {G}_\ell (W)$, the proof of the Lasota–Yorke inequalities follows as in the proof of [16, Proposition 5.6]. The bound (37) then follows as in the proof of Lemma 3.14.

Remark A.1

As a consequence of this lemma, if $g:{\bar{M}} \rightarrow {\mathbb {R}}$ is a bounded measurable function such that, for every $\underline{\omega }=(\omega _k)_{k\ge 0}\in E^{{\mathbb {N}}}$, there exists positive integer $\ell _{\underline{\omega }}$ such that $g(\cdot ,\underline{\omega })$ is p-Hölder on every connected component (uniformly on $\underline{\omega }$) of ${\bar{M}}_0{\setminus }\left( \cup _{k=0}^{\ell _{\underline{\omega }}-1} \bar{T}_{\omega _0}^{-1} \circ \cdots \circ \bar{T}_{\omega _{\ell (\underline{\omega })-1}}^{-1} (\mathcal {S}_{0,H})\right)$. Then, for every $f\in \widetilde{{\mathcal {B}}}_w$, we have

$$\begin{aligned} \left| {\mathbb {E}}_{{\bar{\mu }}}[gf]\right|= & {} \left| \int _{E}{\mathbb {E}}_{{\bar{\mu }}_0}[g(\cdot ,\underline{\omega }) f(x,\underline{\omega })]\, d\eta (\underline{\omega })\right| \\= & {} K_1\Vert f\Vert _{{\widetilde{\mathcal {B}}}_w} \left( \Vert g\Vert _\infty +\sup _{\underline{\omega }\in E^{{\mathbb {N}}}}\sup _{C\in \mathcal C_{\omega _1,\ldots ,\omega _\ell (\omega )}}C_{(g(\cdot ,\underline{\omega }))_{|C}}^{(p)}\right) \, , \end{aligned}$$

with the same notations as in the previous lemma. Therefore, ${\mathbb {E}}_{{\bar{\mu }}} [ g \cdot ]$ is in ${\widetilde{\mathcal {B}}}_w'$.

Appendix B. Proof of Lemma 4.10

Note that ${\mathcal {V}}_n=n+2\sum _{1\le k<\ell \le n}\mathbf 1_{\{S_\ell =S_k,{\mathcal {I}}_\ell ={\mathcal {I}}_k\}}$. Hence

$$\begin{aligned} Var_{{\bar{\mu }}}({\mathcal {V}}_n)=4\sum _{1\le k_1<\ell _1\le n}\sum _{1\le k_2<\ell _2\le n} D_{k_1,\ell _1,k_2,\ell _2}, \end{aligned}$$

with $D_{k_1,\ell _1,k_2,\ell _2}:={\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2})-{\bar{\mu }}(E_{k_1,\ell _1}){\bar{\mu }}(E_{k_2,\ell _2})$. It follows that

$$\begin{aligned} \left| Var_{{\bar{\mu }}}({\mathcal {V}}_n)-8(A_2+A_3)\right| \le 8(A_1+A_4), \end{aligned}$$

(44)

with

$$\begin{aligned} A_1:= & {} \sum _{1\le k_1<\ell _1\le k_2<\ell _2\le n} \left| D_{k_1,\ell _1,k_2,\ell _2}\right| ,\quad A_2:=\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n} D_{k_1,\ell _1,k_2,\ell _2}\, ,\\ A_3:= & {} \sum _{1< k_1< k_2<\ell _2<\ell _1\le n} D_{k_1,\ell _1,k_2,\ell _2},\quad A_4:=\sum _{(k_1,k_2,\ell _1,\ell _2)\in E_n\cup F_n} \left| D_{k_1,\ell _1,k_2,\ell _2}\right| \, , \end{aligned}$$

with

$$\begin{aligned} E_n:= & {} \{(k_1,k_2,\ell _1,\ell _2)\in \{1,\ldots ,n\}\ :\ k_1=k_2<\min (\ell _1,\ell _2)\},\\ F_n:= & {} \{(k_1,k_2,\ell _1,\ell _2)\in \{1,\ldots ,n\}\ :\ \max (k_1, k_2)<\ell _1=\ell _2\}. \end{aligned}$$

We will start with the two easiest estimates: the estimates of the error terms $A_1$ and $A_4$. The method we will use to estimate the main terms $A_2$ and $A_3$ differs from [31].

Due to Lemma 4.9,

$$\begin{aligned} A_1\le \, I^2\sum _{1\le k_1<\ell _1\le k_2<\ell _2\le n}\frac{C_1\alpha ^{k_2-\ell _1}}{(\ell _1-k_1)(\ell _2-k_2)} =O(n(\log n)^2)=o(n^2). \end{aligned}$$

Let us now prove that $A_4=o(n^2)$ by writing

$$\begin{aligned}&\sum _{(k_1,k_2,\ell _1,\ell _2)\in E_n} \left| D_{k_1,\ell _1,k_2,\ell _2}\right| \\&\quad \le 2\sum _{1\le k<\ell _1\le \ell _2\le n} \left( {\bar{\mu }}(E_{k,\ell _1}\cap E_{k,\ell _2})+{\bar{\mu }}(E_{k,\ell _1}){\bar{\mu }}(E_{k,\ell _2})\right) \\&\quad \le 2\sum _{1\le k<\ell _1\le \ell _2\le n} \left( {\bar{\mu }}(S_{\ell _1}=S_{\ell _2}=S_k) +{\bar{\mu }}(S_{\ell _1}=S_k){\bar{\mu }}(S_{\ell _2}=S_k)\right) \\&\quad \le 2\sum _{1\le k<\ell _1\le \ell _2\le n} \left( {\mathbb {E}}_{{\bar{\mu }}}\left[ {\mathcal {H}}_{0,\ell _2-\ell _1}{\mathcal {H}}_{0, \ell _1-k}({\mathbf {1}})\right] + {\mathbb {E}}_{{\bar{\mu }}} \left[ \mathcal H_{0, \ell _1-k}({\mathbf {1}}) \right] {\mathbb {E}}_{{\bar{\mu }}} \left[ {\mathcal {H}}_{0, \ell _2-k}({\mathbf {1}}) \right] \right) \\&\quad \le K'_0\sum _{1\le k<\ell _1\le \ell _2\le n} \left( \frac{1}{(\ell _1-k)(\ell _2-\ell _1+1)} +\frac{1}{(\ell _1-k)(\ell _2-k)}\right) \end{aligned}$$

for some $K'_0>0$ due to Theorem 4.2, since ${\mathbb {E}}_{{\bar{\mu }}}[\cdot ]$ is a continuous linear operator on ${\widetilde{\mathcal {B}}}_1$ and since ${\mathbf {1}}\in {\widetilde{\mathcal {B}}}_1$. This leads to $\sum _{(k_1,k_2,\ell _1,\ell _2)\in E_n} \left| D_{k_1,\ell _1,k_2,\ell _2}\right| =O(n(\log n)^2)$. Analogously, we obtain $\sum _{(k_1,k_2,\ell _1,\ell _2)\in F_n} \left| D_{k_1,\ell _1,k_2,\ell _2}\right| =O(n(\log n)^2)$. Hence $A_4=o(n^2)$.

For $A_2$, we study separately the terms ${\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2})$ and the terms ${\bar{\mu }}(E_{k_1,\ell _1}){\bar{\mu }}(E_{k_2,\ell _2})$. First by Lemma 4.8,

$$\begin{aligned}&\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n} {\bar{\mu }}(E_{k_1,\ell _1}){\bar{\mu }}( E_{k_2,\ell _2})\nonumber \\&\quad =c_1^2\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n}\left( (\ell _1-k_1)^{-1}+O((\ell _1-k_1)^{-3/2})\right) \nonumber \\&\qquad \left( (\ell _2-k_2)^{-1}+O((\ell _2-k_2)^{-3/2})\right) \nonumber \\&\quad =o(n^2)+c_1^2\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n}\frac{1}{(\ell _1-k_1)(\ell _2-k_2)}\, , \end{aligned}$$

(45)

where we used the fact that

$$\begin{aligned}&\sum _{1\le k_1<k_2<\ell _1<\ell _2\le n}\frac{1}{\ell _1-k_1} \frac{1}{(\ell _2-k_2)^{\frac{3}{2}}}\\&\quad \le \sum _{m_1,m_2,m_3,m_4=1}^{n}\frac{1}{m_2+m_3}\frac{1}{(m_3+m_4)^{\frac{3}{2}}}\\&\quad \le n\sum _{m_3=1}^n\sum _{m_2=1}^n\frac{1}{m_2+m_3}\sum _{m_4=1}^{n}\frac{1}{(m_3+m_4)^{\frac{3}{2}}}\\&\quad = O\left( n\sum _{m_3=1}^n\log n \, m_3^{-\frac{1}{2}}\right) \\&\quad = O(n^{\frac{3}{2}}\log n)=o(n^2)\, . \end{aligned}$$

Therefore, due to the Lebesgue dominated convergence theorem, we obtain

$$\begin{aligned}&\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n} {\bar{\mu }}(E_{k_1,\ell _1}){\bar{\mu }}( E_{k_2,\ell _2}) \nonumber \\&\quad = o(n^2)+c_1^2 n^2\int _{\frac{1}{n}\le \frac{\lfloor nx\rfloor }{n}<\frac{\lfloor ny\rfloor }{n}<\frac{\lceil nz\rceil }{n}<\frac{\lceil nt\rceil }{n}\le 1}\frac{dxdydzdt}{\left( \frac{\lceil nz\rceil }{n}-\frac{\lfloor nx\rfloor }{n}\right) \left( \frac{\lceil nt\rceil }{n}-\frac{\lfloor ny\rfloor }{n}\right) } \nonumber \\&\quad \sim c_1^2 n^2\int _{0<x<y<z<t<1}\frac{dxdydzdt}{(z-x)(t-y)} \nonumber \\&\quad = c_1^2\frac{\pi ^2}{12}n^2 = \frac{n^2}{48\det \Sigma ^2}\left( \sum _{a=1}^I {\bar{\mu }}({\mathcal {I}}_0=a)^2\right) ^2. \end{aligned}$$

(46)

The rest of the estimate of $A_2$ is new (it is different from [31]). Fix for the moment $1\le k_1< k_2< \ell _1<\ell _2\le n$. Note that

$$\begin{aligned}&{\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2})\\&\quad =\sum _{a,b=1}^I{\bar{\mu }}\left( {\bar{T}}^{-k_1}{\bar{O}}_a\cap \bar{T}^{-k_2}{\bar{O}}_b\cap {\bar{T}}^{-\ell _1}({\bar{O}}_a)\right. \\&\left. \qquad \cap {\bar{T}}^{-\ell _2}\bar{O}_b\cap \{S_{k_2}-S_{k_1}=-(S_{\ell _1}-S_{k_2})=S_{\ell _2}-S_{\ell _1}\}\right) \, . \end{aligned}$$

Using now (23) as for (24), we observe that $\mathbf 1_{\{S_{k_2}-S_{k_1}=-(S_{\ell _1}-S_{k_2})=S_{\ell _2}-S_{\ell _1}\}}$ is equal to the following quantity

$$\begin{aligned} \frac{1}{(2\pi )^4}\int _{([-\pi ,\pi ]^2)^2}e^{i u\cdot ((S_{k_2}-S_{k_1})+(S_{\ell _1}-S_{k_2}))} e^{i v\cdot ((S_{\ell _2}-S_{\ell _1})+(S_{\ell _1}-S_{k_2}))} \, du\, dv \, , \end{aligned}$$

which is also equal to

$$\begin{aligned} \begin{aligned}&\frac{1}{(2\pi )^4}\int _{([-\pi ,\pi ]^2)^2}e^{i u\cdot (S_{k_2}-S_{k_1})}e^{i(u+v)\cdot (S_{\ell _1}-S_{k_2})} e^{i v\cdot (S_{\ell _2}-S_{\ell _1})} \, du\, dv \\&\quad = \frac{1}{(2\pi )^4}\int _{([-\pi ,\pi ]^2)^2} e^{i u\cdot S_{k_2 - k_1} \circ {\bar{T}}^{k_1}} e^{i(u+v)\cdot S_{\ell _1 - k_2} \circ \bar{T}^{k_2}} e^{i v\cdot S_{\ell _2 - \ell _1} \circ {\bar{T}}^{\ell _1}} \, du\, dv \, . \end{aligned} \end{aligned}$$

Now using the P-invariance and ${\bar{T}}$-invariance of ${\bar{\mu }}$ and several times the formula $P^m(f.g\circ {\bar{T}}^m)=gP^m(f)$, we obtain

$$\begin{aligned}&{\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2})\\&\quad = \sum _{a,b=1}^I\frac{1}{(2\pi )^4}\int _{([-\pi ,\pi ]^2)^2}{\mathbb {E}}_{{\bar{\mu }}} \left[ \mathbf 1_{{\bar{O}}_b} P_v^{\ell _2-\ell _1}\left( {\mathbf {1}}_{{\bar{O}}_a} P_{u+v}^{\ell _1-k_2}\left( {\mathbf {1}}_{{\bar{O}}_b}P_u^{k_2-k_1}(\mathbf 1_{{\bar{O}}_a})\right) \right) \right] \, du\, dv\, . \end{aligned}$$

Due to our spectral assumptions, we observe that

$$\begin{aligned} P_u^n= \lambda _u^n\Pi _u + O(\alpha ^n)\, , \end{aligned}$$

up to defining $\lambda _u=e^{-\frac{1}{2} \Sigma ^2 u\cdot u}$ for u outside $[-\beta ,\beta ]^2$ and so, proceding as in the proof of Theorem 4.2, we obtain that, for every $n\ge 2$ and every $u,v\in [-\pi ,\pi ]^2$,

$$\begin{aligned} P_u^n= & {} e^{-\frac{n}{2} \Sigma ^2u\cdot u}{\mathbb {E}}_{{\bar{\mu }}}[\cdot ]{\mathbf {1}} + O(\alpha ^n)+O(e^{-2na|u|^2}(|u|+n|u|^3))\\= & {} e^{-\frac{n}{2} \Sigma ^2u\cdot u}{\mathbb {E}}_{{\bar{\mu }}}[\cdot ]\mathbf 1 + O(e^{-n a|u|^2}|u|)\, , \end{aligned}$$

and $|\lambda _u^n|\le e^{- 2a|u|^2}$ for some $a>0$ (such that $e^{-2a|\pi |^2}>\alpha ^n$, $\max (\lambda _u^{n-1},e^{-\frac{n-1}{2}\Sigma ^2u\cdot u})\le e^{- 2an|u|^2}$) since $n|u|^2e^{-2n a|u|^2}= O(e^{-n a|u|^2})$. Therefore, we obtain

$$\begin{aligned}&{\mathbb {E}}_{{\bar{\mu }}} \left[ {\mathbf {1}}_{{\bar{O}}_b} P_v^{\ell _2-\ell _1}\left( {\mathbf {1}}_{\bar{O}_a} P_{u+v}^{\ell _1-k_2}\left( {\mathbf {1}}_{\bar{O}_b}P_u^{k_2-k_1}({\mathbf {1}}_{{\bar{O}}_a})\right) \right) \right] \nonumber \\&\quad =({\bar{\mu }}({\bar{O}}_a){\bar{\mu }}({\bar{O}}_b))^2 e^{-\frac{1}{2}Q(\Sigma u,\Sigma v)} +O\left( (|u|+|v|)e^{-naQ(u,v)}\right) \, , \end{aligned}$$

(47)

where we have set

$$\begin{aligned} Q(u,v):= & {} (\ell _2-\ell _1)|v|^2+(\ell _1-k_2)|u+v|^2+(k_2-k_1) |u|^2\\= & {} (\ell _2-k_2)|v|^2+2(\ell _1-k_2)u\cdot v+(\ell _1-k_1) |u|^2\\= & {} (A_Q (u,v))\cdot (A_Q(u,v))=|A_Q(u,v)|^2\, , \end{aligned}$$

with $A^2_Q:=\left( \begin{array}{cccc} \ell _1-k_1&{}0&{}\ell _1-k_2&{}0\\ 0&{}\ell _1-k_1&{}0&{}\ell _1-k_2\\ \ell _1-k_2&{}0&{}\ell _2-k_2&{}0\\ 0&{}\ell _1-k_2&{}0&{}\ell _2-k_2\end{array}\right)$ which is symmetric with determinant

$$\begin{aligned} \det A^2_Q= & {} (\ell _1-k_1)^2(\ell _2-k_2)^2+(\ell _1-k_2)^4-2(\ell _1-k_2)^2(\ell _1-k_1)(\ell _2-k_2)\nonumber \\= & {} ((k_2-k_1)(\ell _1-k_2)+(k_2-k_1)(\ell _2-\ell _1)+(\ell _1-k_2)(\ell _2-\ell _1))^2.\nonumber \\ \end{aligned}$$

(48)

Due to the form of $A^2_Q$, we observe that $A^2_Q$ has eigenvectors of the forms $(*,0,*,0)$ and $(0,*,0,*)$, that it has two double eigenvalues of sum (without multiplicity) $\ell _1-k_1+\ell _2-k_2$ and of product (without multiplicity) $\sqrt{\det A_Q^2}$. Therefore its dominating eigenvalue is smaller than the sum and so is less than $4\max (k_2-k_1,\ell _1-k_2,\ell _2-\ell _1)$ and so (using the fact that the product of the two eigenvalues is larger than the maximum times the median of these three values) the smallest eigenvalue of $A^2_Q$ cannot be smaller than a quarter of the median of $k_2-k_1,\ell _1-k_2,\ell _2-\ell _1$, that we denote by $med (k_2-k_1,\ell _1-k_2,\ell _2-\ell _1)$. So

$$\begin{aligned}&\int _{([-\pi ,\pi ]^2)^2} e^{-nQ(\Sigma u,\Sigma v)}\, dudv\, =(\det \Sigma )^{-2}\int _{(\Sigma [-\pi ,\pi ]^2)^2} e^{-nQ( u,v)}\, dudv\\&\quad =(\det A_Q)^{-1}(\det \Sigma )^{-2}\int _{A_Q(\Sigma ([-\pi ,\pi ]^2)^2)} e^{-|(x,y)|^2}\, dxdy\\&\quad =(\det A_Q)^{-1}(\det \Sigma )^{-2}\left( \int _{({\mathbb {R}}^2)^2} e^{-|(x,y)|^2}\, dxdy + O(e^{-a_1 {med (k_2-k_1,\ell _1-k_2,\ell _2-\ell _1)^2}})\right) \\&\quad =(2\pi )^2(\det A_Q)^{-1}(\det \Sigma )^{-2}\left( 1 + O(e^{-a_1 {med (k_2-k_1,\ell _1-k_2,\ell _2-\ell _1)^2}})\right) \, , \end{aligned}$$

for some $a_1>0$. Moreover

$$\begin{aligned}&\int _{({\mathbb {R}}^2)^2} |(u,v)|e^{-naQ(u,v)}\, dudv =(\det A_Q)^{-1}\int _{({\mathbb {R}}^2)^2} |A_Q^{-1}(u,v)| e^{-a|(x,y)|^2}\, dxdy\\&\quad =O\left( (\det A_Q)^{-1}\, med(k_2-k_1,\ell _1-k_2,\ell _2-\ell _1)^{-\frac{1}{2}} \right) \, . \end{aligned}$$

Therefore

$$\begin{aligned}&{\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2})\nonumber \\&=\frac{\left( \sum _{a=1}^I{\bar{\mu }}({\bar{O}}_a)^2\right) ^2}{(2\pi )^{2}\, \det A_Q\, \det \Sigma ^{2}}\left( 1+ O\left( med(k_2-k_1,\ell _1-k_2,\ell _2-\ell _1)^{-\frac{1}{2}} \right) \right) \, . \end{aligned}$$

(49)

But using (48),

$$\begin{aligned}&\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n} (\det A_Q)^{-1}\\&\quad =\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n}\frac{1}{(k_2-k_1)(\ell _1-k_2)+(k_2-k_1)(\ell _2-\ell _1)+(\ell _1-k_2)(\ell _2-\ell _1)}\\&\quad =\sum _{m_1,m_2,m_3,m_4\ge 1\ :\ m_1+m_2+m_3+m_4\le n}\frac{1}{m_2m_3+m_2m_4+m_3m_4}\\&\quad =n^2\int _{(0,+\infty )^4} \frac{{\mathbf {1}}_{\left\{ \frac{\lceil ny_1\rceil }{n}+\frac{\lceil ny_2\rceil }{n}+\frac{\lceil ny_3\rceil }{n}+\frac{\lceil ny_4\rceil }{n}\le 1\right\} }}{\frac{\lceil ny_2\rceil }{n} \frac{\lceil ny_3\rceil }{n} +\frac{\lceil ny_2\rceil }{n} \frac{\lceil ny_4\rceil }{n}+\frac{\lceil ny_3\rceil }{n}\frac{\lceil ny_4\rceil }{n}}\, dy_1\, dy_2\, dy_3\, dy_4\\&\quad \sim n^2\int _{(0,+\infty )^4} \frac{\mathbf 1_{\left\{ y_1+y_2+y_3+y_4\le 1\right\} }}{y_2y_3 +y_2y_4+y_3y_4}\, dy_1\, dy_2\, dy_3\, dy_4\, , \end{aligned}$$

due to the dominated convergence theorem. Therefore

$$\begin{aligned}&\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n} (\det A_Q)^{-1}\sim n^2\nonumber \\&\quad \int _{(0,+\infty )^3} \frac{(1-y_2-y_3-y_4)\mathbf 1_{\left\{ y_2+y_3+y_4\le 1\right\} }}{y_2y_3 +y_2y_4+y_3y_4}\, dy_2\, dy_3\, dy_4=n^2J\, . \end{aligned}$$

(50)

Analogously

$$\begin{aligned} \begin{aligned}&\sum _{1\le k_1< k_2< \ell _1<\ell _2\le n} (\det A_Q)^{-1}\, (med(k_2-k_1,\ell _1-k_2,\ell _2-\ell _1))^{-\frac{1}{2}} \\&\quad =\sum _{m_1,m_2,m_3,m_4\ge 1\ :\ m_1+m_2+m_3+m_4\le n}\frac{1}{(m_2m_3+m_2m_4+m_3m_4)\, med(m_2,m_3,m_4)^{\frac{1}{2}}}\\&\quad \le n\sum _{1\le m_2\le m_3\le m_4\le n}\frac{1}{(m_2m_3+m_2m_4+m_3m_4)\, m_3^{\frac{1}{2}}}\\&\quad \le n\sum _{1\le m_2\le m_3\le m_4\le n}\frac{1}{ m_3^{\frac{3}{2}}m_4} \; \le \; n \log n \sum _{m_2=1}^n\sum _{m_3=m_2}^n m_3^{-\frac{3}{2}}\\&\quad \le n \log n \sum _{m_2=1}^n O( m_2^{-\frac{1}{2}})=O(n^{\frac{3}{2}}\log n)=o(n^2)\, . \end{aligned} \end{aligned}$$

(51)

Equations (49), (50) and (51) lead to

$$\begin{aligned} \sum _{1\le k_1< k_2< \ell _1<\ell _2\le n}{\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2})=\frac{\left( \sum _{a=1}^I{\bar{\mu }}(\bar{O}_a)^2\right) ^2}{((2\pi )^2\det \Sigma ^2)}J+o(n^2)\, . \end{aligned}$$

Combining this with (46), we conclude that

$$\begin{aligned} A_2\sim \frac{n^2}{\det \Sigma ^2}\left( \sum _{a=1}^I{\bar{\mu }}(\mathcal I_0=a)^2\right) ^2\left( \frac{-1}{48}+\frac{J}{4\pi ^2} \right) \, . \end{aligned}$$

(52)

The study of $A_3$ is the most delicate. We can observe that both sums $\sum _{1\le k_1< k_2<\ell _2<\ell _1\le n} {\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2})$ and $\sum _{1\le k_1< k_2<\ell _2<\ell _1\le n}{\bar{\mu }}(E_{k_1,\ell _1}){\bar{\mu }}(E_{k_2,\ell _2})$ are in $O(n^2\log n)$. However, we will see that their difference is in $n^2$. Once again our proof differs from the one in [31] and is based on the same idea as the one used to prove $A_2$. We set $E_{k,\ell }(b):=E_{k,\ell }\cap \{{\mathcal {I}}_k=b\}$. Due to the first part of Lemma 4.8,

$$\begin{aligned} A_3&=\sum _{1\le k_1< k_2< \ell _2< \ell _1\le n} {\bar{\mu }}(E_{k_1,\ell _1}\cap E_{k_2,\ell _2}) -{\bar{\mu }}(E_{k_1,\ell _1}){\bar{\mu }}(E_{k_2,\ell _2})\nonumber \\&=o(n^{2})+\sum _{1\le k_1< k_2< \ell _2< \ell _1\le n} \sum _{a,b=1}^I\left( -I_{k_1,k_1,l_1,l_2}+ {\bar{\mu }}(O_{k_1,k_2,l_1,l_2}\cap S_{k_1,k_2,l_1,l_2})\right) \nonumber \\&=o(n^{2})+\sum _{1\le k_1< k_2< \ell _2 < \ell _1\le n} \sum _{a,b=1}^I\left( -I_{k_1,k_1,l_1,l_2}\right) \end{aligned}$$

(53)

$$\begin{aligned}&\quad +\sum _{1\le k_1< k_2< \ell _2 < \ell _1\le n} \sum _{a,b=1}^I\left( \frac{1}{(2\pi )^2}\int _{[-\pi ,\pi ]^2}{\mathbb {E}}_{{\bar{\mu }}}\left[ {\mathbf {1}}_{\bar{O}_a}P_u^{\ell _1-\ell _2}\left( {\mathbf {1}}_{{\bar{O}}_b}\mathcal H_{0,\ell _2-k_2}\right. \right. \right. \nonumber \\&\left. \left. \left. \quad \left( {\mathbf {1}}_{{\bar{O}}_b}P_u^{k_2-k_1}\left( {\mathbf {1}}_{\bar{O}_a}\right) \right) \right) \right] \, du\right) \, , \end{aligned}$$

(54)

where

$$\begin{aligned} I_1(k_1,k_1,l_1,l_2)= & {} \frac{({\bar{\mu }}(\bar{O}_a))^2{\bar{\mu }}(E_{k_2,\ell _2}(b))}{2\pi \sqrt{\det \Sigma ^2}(\ell _1-k_1)},\\ O_{k_1,k_2,l_1,l_2}= & {} {\bar{O}}_a\cap {\bar{T}}^{-(k_2-k_1)}{\bar{O}}_b\cap \bar{T}^{-(\ell _2-k_1)}{\bar{O}}_b \cap {\bar{T}}^{-(\ell _1-k_1)}\bar{O}_a,\\ S_{k_1,k_2,l_1,l_2}= & {} \{S_{\ell _2-k_2}\circ {\bar{T}}^{k_2-k_1}=0\} \cap \{S_{\ell _1-\ell _2}\circ {\bar{T}}^{\ell _2-k_1}=-S_{k_2-k_1}\}. \end{aligned}$$

Now, as we did for (47) (and using Theorem 4.2), we get that

$$\begin{aligned}&{\mathbb {E}}_{{\bar{\mu }}}\left[ {\mathbf {1}}_{{\bar{O}}_a}P_u^{\ell _1-\ell _2}\left( {\mathbf {1}}_{\bar{O}_b}{\mathcal {H}}_{0,\ell _2-k_2}\left( {\mathbf {1}}_{\bar{O}_b}P_u^{k_2-k_1}\left( {\mathbf {1}}_{\bar{O}_a}\right) \right) \right) \right] \\&\quad =({\bar{\mu }}({\bar{O}}_a))^2 e^{-\frac{(\ell _1-\ell _2)+(k_2-k_1)}{2}|\Sigma u|^2}{\mathbb {E}}_{{\bar{\mu }}}\left[ {\mathbf {1}}_{{\bar{O}}_b}\mathcal H_{0,\ell _2-k_2}{\mathbf {1}}_{{\bar{O}}_b}\right] +O\left( \frac{|u|}{\ell _2-k_2}e^{-na|u|^2}\right) \, . \end{aligned}$$

Therefore

$$\begin{aligned}&\frac{1}{(2\pi )^2}\int _{(-\pi ,\pi )^2}{\mathbb {E}}_{{\bar{\mu }}}\left[ {\mathbf {1}}_{{\bar{O}}_a}P_u^{\ell _1-\ell _2}\left( {\mathbf {1}}_{\bar{O}_b}{\mathcal {H}}_{0,\ell _2-k_2}\left( {\mathbf {1}}_{\bar{O}_b}P_u^{k_2-k_1}\left( {\mathbf {1}}_{\bar{O}_a}\right) \right) \right) \right] \, du\nonumber \\&\quad = \frac{({\bar{\mu }}({\bar{O}}_a))^2{{\bar{\mu }}}\left( E_{k_2,\ell _2}(b)\right) }{2\pi (\ell _1-\ell _2+k_2-k_1)\sqrt{\det \Sigma ^2}} +O\left( \frac{1}{(\ell _2-k_2)(\ell _1-\ell _2+k_2-k_1)^{\frac{3}{2}}}\right) \, . \end{aligned}$$

(55)

We will now prove that the term in O in this last formula is negligable. Indeed its sum over $\{1\le k_1\le k_2\le \ell _2\le \ell _1\le n\}$ is in O of the following quantity:

$$\begin{aligned} \sum _{m_1+m_2+m_3+m_4\le n}\left( \frac{1}{m_3(m_4+m_2)^{\frac{3}{2}}}\right)\le & {} n\log n\sum _{m_2=1}^n\sum _{m_4=1}^n(m_4+m_2)^{-\frac{3}{2}}\\\le & {} O\left( n\log n\sum _{m_2=1}^nm_2^{-\frac{1}{2}}\right) =O(n^{\frac{3}{2}}\log n)=o(n^2)\, . \end{aligned}$$

This combined with (54) and (55) leads to

$$\begin{aligned} A_3= & {} o(n^{2})+\sum _{1\le k_1< k_2< \ell _2\le \ell _1\le n} \sum _{a,b=1}^I\frac{({\bar{\mu }}({\bar{O}}_a))^2{\bar{\mu }}\left( E_{k_2,\ell _2}(b)\right) }{2\pi \sqrt{\det \Sigma ^2}}\\&\left( \frac{1}{\ell _1-\ell _2+k_2-k_1}-\frac{1}{\ell _1-k_1}\right) \, , \end{aligned}$$

i.e.

$$\begin{aligned} A_3= & {} o(n^{2})+\frac{\sum _{a}^I({\bar{\mu }}(\mathcal I_0=a))^2}{2\pi \sqrt{\det \Sigma ^2}}\sum _{m_1+m_2+m_3+m_4\le n}\left( \frac{c_1}{m_3}+O(m_3^{-\frac{3}{2}})\right) \\&\frac{m_3}{(m_2+m_4)(m_2+m_3+m_4)}\\= & {} o(n^{2})+c_1^2\sum _{m_1+m_2+m_3+m_4\le n}\frac{1}{(m_2+m_4)(m_2+m_3+m_4)}\, , \end{aligned}$$

since

$$\begin{aligned}&\sum _{m_1+m_2+m_3+m_4\le n}\frac{1}{m_3^{\frac{1}{2}}(m_2+m_4)(m_2+m_3+m_4)}\\&\quad =O\left( n\sum _{m_2,m_3,m_4=1}^nm_3^{-\frac{1}{2}}(m_2m_4)^{-1}\right) =o(n^2)\, . \end{aligned}$$

Therefore, due to the Lebesgue dominated convergence theorem,

$$\begin{aligned} A_3 \sim n^{2}c_1^2\int _{y_1,y_2,y_3,y_4>0:y_1+y_2+y_3+y_4<1} \frac{1}{(y_2+y_4)(y_2+y_3+y_4)}\, dy_1dy_2dy_3dy_4 \sim \frac{c_1^2}{2} n^{2} . \end{aligned}$$

To conclude the proof of the lemma, we use the estimate for $A_3$ together with (44) and (52) to obtain,

$$\begin{aligned} 8A_2 + 8A_3&= 4c_1^2 n^2 + \frac{8n^2}{\det \Sigma ^2} \left( \sum _{a=1}^I {\bar{\mu }}({\bar{O}}_a)^2 \right) ^2 \left( \frac{-1}{48} + \frac{J}{4\pi ^2} \right) \\&= \frac{n^2}{\det \Sigma ^2} \left( \sum _{a=1}^I {\bar{\mu }}(\bar{O}_a)^2 \right) ^2 \left[ \frac{2J+1}{\pi ^2} - \frac{1}{6} \right] . \end{aligned}$$

This finished the proof.

Appendix C. Spectrum of $\mathcal {P}_u$

In this appendix, we are interested in the spectrum of the family of operators $\mathcal {P}_u$. We start by stating a result for the unperturbed operators ${\mathcal {L}}_{u,0}$.

Lemma C.1

Let $u\in {\mathbb {R}}^2$, $h\in \mathcal {B}$ and $\lambda \in {\mathbb {C}}$ be such that ${\mathcal {L}}_{u,0}h=\lambda h$ in ${{\mathcal {B}}}$ and $|\lambda | \ge 1$. Then either $h\equiv 0$ or $u\in 2\pi {\mathbb {Z}}^2$, $\lambda =1$ and h is ${\bar{\mu }}_0$-almost surely constant.

Proof

Recall that for $\psi \in \mathcal {C}^p({\bar{M}}_0)$, we have $\psi \circ \bar{T}_0^n \in \mathcal {C}^p({\bar{T}}^{-n}\mathcal {W}^s)$. Note that

$$\begin{aligned} {\mathcal {L}}_{u,0} h(\psi )=h(e^{iu \cdot \Phi _0}\psi \circ {\bar{T}}_0). \end{aligned}$$

Thus for $n\ge 1$,

$$\begin{aligned} {\mathcal {L}}_{u,0}^n h(\psi )= h(e^{iu \cdot S_n \Phi _0 }\psi \circ {\bar{T}}_0^n), \end{aligned}$$

where $S_n \Phi _0 =\Phi _0+\Phi _0\circ {\bar{T}}_0+\cdots +\Phi _0\circ {\bar{T}}_0^{n-1}$ denotes the partial sum. By [16, Lemma 3.4], using the invariance of h,

$$\begin{aligned}&|h(\psi )| = |\lambda |^{-n} |h(e^{iu \cdot S_n\Phi _0}\psi \circ \bar{T}_0^n) | \nonumber \\&\quad \le C |\lambda |^{-n} | h |_w \big (|e^{i u \cdot S_n\Phi _0} \psi \circ {\bar{T}}_0^n|_\infty + C^{(p)}_{{\bar{T}}_0^{-n} \mathcal {W}^s}(e^{iu\cdot S_n\Phi _0}\cdot \psi \circ {\bar{T}}_0^n) \big ) , \end{aligned}$$

(56)

where $C^{(p)}_{{\bar{T}}_0^{-n}\mathcal {W}^s}(\cdot )$ denotes the Hölder constant of exponent p measured along elements of ${\bar{T}}_0^{-n} \mathcal {W}^s$. Since $|e^{i u \cdot S_n \Phi _0}| = 1$ and $S_n \Phi _0$ is constant on each element of ${\bar{T}}_0^{-n} \mathcal {W}^s$, we have

$$\begin{aligned} \begin{aligned}&C^{(p)}_{{\bar{T}}_0^{-n} \mathcal {W}^s}(e^{iu\cdot S_n\Phi _0}\cdot \psi \circ {\bar{T}}_0^n) \\&\quad \le |e^{i u \cdot S_n \Phi _0}|_\infty C^{(p)}_{\bar{T}_0^{-n} \mathcal {W}^s}(\psi \circ {\bar{T}}_0^n) + |\psi \circ {\bar{T}}_0^n|_\infty C^{(p)}_{{\bar{T}}_0^{-n}\mathcal {W}^s}(e^{i u \cdot S_n \Phi _0}) \\&\quad \le C \Lambda ^{-pn} C^{(p)}_{\mathcal {W}^s}(\psi ) . \end{aligned} \end{aligned}$$

Using this estimate in (56) and taking the limit as $n \rightarrow \infty$ yields $|h(\psi )| = 0$ if $|\lambda |>1$ and $|h(\psi )| \le C| h |_w |\psi |_\infty$ for all $\psi \in \mathcal {C}^p(\mathcal {W}^s)$ if $|\lambda |=1$. From this we conclude that the spectrum of $\mathcal {L}_{u,0}$ is always contained in the unit disk. Furthermore, when $|\lambda |=1$, then h is a signed measure. For the remainder of the proof, we assume $|\lambda |=1$.

Let ${\mathbb {V}}_{u,0}$ be the eigenspace of ${\mathcal {L}}_{u,0}$ corresponding to eigenvalue $\lambda _{u,0}$, and $\Pi _{u,0}$ the eigenprojection operator. Since we are assuming ${\mathbb {V}}_{u,0}$ is non-empty, Lemma 3.14 implies that $\mathcal {L}_{u,0}$ is quasi-compact with essential spectral radius bounded by $\tau < 1$. Moreover, Lemma 3.14 implies that $\Vert \mathcal {L}_{u,0}^n \Vert _{L(\mathcal {B}, \mathcal {B})}$ remains bounded for all $n \ge 0$, so using [15, Lemma 5.1], we conclude that $\mathcal {L}_{u,0}$ has no Jordan blocks corresponding to its peripheral spectrum.

Using these facts, $\Pi _{u,0}$ has the representation

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n} \sum _{j=1}^n \lambda ^{-j} \mathcal L_{u,0}^j =\Pi _{u,0} . \end{aligned}$$

In addition, for $f\in \mathcal {C}^1({\bar{M}}_0)$, $\psi \in \mathcal {C}^p(\mathcal {W}^s)$,

$$\begin{aligned} \left| \Pi _{u,0} f(\psi ) \right| =\left| \lim _{n\rightarrow \infty }\frac{1}{n} \sum _{j=1}^n \lambda ^{-j} f((e^{iu \cdot S_j \Phi _0}\psi \circ {\bar{T}}_0^j) \right| \le |f|_{\infty }|\psi |_{\infty } . \end{aligned}$$

Since $\Pi _{u,0} \mathcal {C}^1({\bar{M}}_0)$ is dense in the finite dimensional space $\Pi _{u,0}\mathcal {B}$, therefore $\Pi _{u,0} \mathcal {C}^1(\bar{M}_0)=\Pi _{u,0}\mathcal {B}={\mathbb {V}}_{u,0}$. So for $h \in {\mathbb {V}}_{u,0}$, there exists $f \in \mathcal {C}^1({\bar{M}}_0)$ such that $\Pi _{u,0} f = h$. Now for each $\psi \in \mathcal {C}^p({\bar{M}}_0)$,

$$\begin{aligned} |h(\psi )| = |\Pi _{u,0} f(\psi )| \le |f|_\infty \Pi _0 1(|\psi |) = |f|_\infty {\bar{\mu }}_0(|\psi |) . \end{aligned}$$

Thus h is absolutely continuous with respect to ${\bar{\mu }}_0$. For simplicity, we identify h and its density with respect to ${\bar{\mu }}_0$; then $h \in L^\infty ({\bar{M}}_0, {\bar{\mu }}_0)$. Now for any $\psi \in \mathcal {C}^p(\mathcal {W}^s)$, we have

$$\begin{aligned} \lambda \int _{{\bar{M}}_0} h \psi \, d\mu _0&=\int _{{\bar{M}}_0} \mathcal {L}_0(e^{iu \cdot \Phi _0} h)\cdot \psi \, d{\bar{\mu }}_0\\&=\int _{{\bar{M}}_0} (e^{iu\cdot \Phi _0} h)\circ {\bar{T}}_0^{-1}\cdot \psi \,d{\bar{\mu }}_0. \end{aligned}$$

Accordingly, $\lambda \, h=(e^{iu \cdot \Phi _0} h)\circ \bar{T}_0^{-1}$, ${\bar{\mu }}_0$-a.e. Or equivalently, we have $\lambda \, h\circ {\bar{T}}_0=e^{iu \cdot \Phi _0}h$. Hence $\lambda ^n\, h\circ {\bar{T}}_0^n=e^{iu\cdot S_n \Phi _0}h$.

Let $G_\lambda$ be the closed multiplicative group generated by $\lambda$ and let $m_{\lambda }$ be the normalized Haar measure on $G_\lambda$. ($G_\lambda$ is finite if $\lambda$ is a root of unity; it is $\{z\in {\mathbb {C}}\, :\, |z|=1\}$ otherwise.) The dynamical system $(G_\lambda ,m_{\lambda },T_\lambda )$ is ergodic, where $T_\lambda$ denotes multiplication by $\lambda$ in $G_\lambda$. Due to [28], the dynamical system $(M_0 \times G_\lambda ,\mu _0\otimes m_{\lambda }, T_0 \times T_\lambda )$ in infinite measure is conservative and ergodic. But the function $H: M_0 \times G_\lambda \rightarrow {\mathbb {C}}$ defined as follows is $(T_0 \times T_\lambda )$-invariant:

$$\begin{aligned} \forall ({\bar{x}},\ell ,y)\in {\bar{M}}_0\times {\mathbb {Z}}^2\times G_\lambda ,\quad H({\bar{x}}+\ell ,y):=y h({\bar{x}})e^{-i u\cdot \ell }. \end{aligned}$$

Indeed, for $\mu _0\otimes m_{\lambda }$-a.e. $({\bar{x}}+\ell ,y)\in M_0 \times G_\lambda$,

$$\begin{aligned} H((T_0 \times T_\lambda )({\bar{x}}+\ell ,y))= & {} H({\bar{T}}_0(\bar{x})+\ell +\Phi _0({\bar{x}}),\lambda y) =\lambda y h({\bar{T}}_0({\bar{x}}))e^{-i u\cdot (\ell +\Phi _0({\bar{x}}))}\\= & {} ye^{-i u\cdot \ell }(\lambda h({\bar{T}}_0({\bar{x}})) e^{-i u\cdot \Phi _0({\bar{x}})})\\= & {} ye^{-i u\cdot \ell } h({\bar{x}})\, , \end{aligned}$$

due to our assumption on h. We conclude that H is a.e. equal to a constant, which implies that $u\in 2\pi {\mathbb {Z}}^2$, $\lambda =1$, and h is ${\bar{\mu }}_0$-a.s. constant. $\quad \square$

Proposition C.2

Given $\beta > 0$, there exists $C>1$ and $\alpha \in (0,1)$ such that

$$\begin{aligned} \forall n\in {\mathbb {N}}^*,\quad \sup _{\beta \le |u|\le \pi }\Vert {\mathcal {P}}_u^n\Vert _{L({\widetilde{\mathcal {B}}},{\widetilde{\mathcal {B}}})}\le C\alpha ^n\, . \end{aligned}$$

Proof

Fix $\beta > 0$. Due to [1, Lemma 4.3], Lemma C.1, and the continuity in u provided by [17, Lemma 5.4] (see also Lemma 3.16 applied to $\mathcal {L}_{u,0}$ rather than $P_u$), we know that there exists $C>1$ and $\alpha \in (0,1)$ such that

$$\begin{aligned} \forall n\in {\mathbb {N}}^*,\quad \sup _{\beta \le |u|\le \pi }\Vert {\mathcal {L}}_{u,0}^n\Vert _{L(\mathcal {B},\mathcal {B})}\le C\alpha ^n\, . \end{aligned}$$

Therefore, for every $f\in {\widetilde{\mathcal {B}}}$, we have

$$\begin{aligned}&\sup _{\underline{\omega }\in E^{{\mathbb {N}}}}\left\| \mathcal P_u^nf(x,\underline{\omega })\right\| _{\mathcal {B}}\\&\quad =\sup _{\underline{\omega }\in E^{{\mathbb {N}}}}\left\| \int _{E^n}{\mathcal {L}}_{u,0}^nf(\cdot ,({{\tilde{\omega }}},\underline{\omega }))\, d\eta ^{\otimes n}({{\tilde{\omega }}}) \right\| _{\mathcal {B}}\\&\quad \le \sup _{\underline{\omega }\in E^{{\mathbb {N}}}} \int _{E^n}\left\| \mathcal L_{u,0}^nf(\cdot ,({{\tilde{\omega }}},\underline{\omega }))\right\| _{\mathcal {B}}\, d\eta ^{\otimes n}({{\tilde{\omega }}})\\&\quad \le \sup _{\underline{\omega }\in E^{{\mathbb {N}}}} C\alpha ^n\sup _{\underline{\omega '}}\left\| f(\cdot ,\underline{\omega '})\right\| _{\mathcal {B}}\, \end{aligned}$$

where we used Lemma 3.7 to obtain the second line. Analogously,

$$\begin{aligned}&\sup _{\underline{\omega }\ne \underline{\omega '}}\frac{\left\| \mathcal P_u^nf(x,\underline{\omega })-\mathcal P_u^nf(x,\underline{\omega '})\right\| _{\mathcal {B}}}{d(\underline{\omega },\underline{\omega '})}\\&\quad =\sup _{\underline{\omega }\ne \underline{\omega '}}\frac{\left\| \int _{E^n}\mathcal L_{u,0}^n\left( f(\cdot ,({{\tilde{\omega }}},\underline{\omega }))-f(\cdot ,({{\tilde{\omega }}},\underline{\omega '}))\right) \, d\eta ^{\otimes n}({{\tilde{\omega }}}) \right\| _{\mathcal {B}}}{d(\underline{\omega },\underline{\omega '})}\\&\quad \le \sup _{\underline{\omega }\ne \underline{\omega '}} \int _{E^n}\frac{\left\| \mathcal L_{u,0}^n\left( f(\cdot ,({{\tilde{\omega }}},\underline{\omega }))-f(\cdot ,({{\tilde{\omega }}},\underline{\omega '}))\right) \right\| _{\mathcal {B}}}{d(\underline{\omega },\underline{\omega '})}\, d\eta ^{\otimes n}({{\tilde{\omega }}})\\&\quad \le C\alpha ^n\varkappa ^n\sup _{\underline{\omega }\ne \underline{\omega '}} \frac{\left\| f(\cdot ,\underline{\omega '})-f(\cdot ,\underline{\omega '})\right\| _{\mathcal {B}}}{d(\underline{\omega },\underline{\omega '})}\, . \end{aligned}$$

We conclude by putting these two estimates together. $\quad \square$