Exponential decay of correlations for finite horizon Sinai billiard flows

Abstract

We prove exponential decay of correlations for the billiard flow associated with a two-dimensional finite horizon Lorentz Gas (i.e., the Sinai billiard flow with finite horizon). Along the way, we describe the spectrum of the generator of the corresponding semi-group $\mathcal {L}_t$ of transfer operators, i.e., the resonances of the Sinai billiard flow, on a suitable Banach space of anisotropic distributions.

Appendices

Appendix A. Proofs of Lemmas 6.6 and 6.7 (approximate foliation holonomy)

This section contains the proofs of Lemmas 6.6 and 6.7 (Sects. 6.2 and 6.4) on the Hölder bounds of the Jacobians of the holonomy of the fake (approximate) foliation. These bounds are instrumental to get the four-point estimate (vii) for the fake unstable foliation.

Proof of Lemma 6.6

Letting $\mathbf {h}_{V^n_1, V^n_2}$ denote the holonomy map from $V^n_1 = T^{-n}V_1$ to $V^n_2 = T^{-n} V_2$, we begin by noting that

$$\begin{aligned} \begin{aligned}&\ln \frac{J\mathbf {h}_{12}( v_1(\bar{x}^s),\bar{x}^s)}{J\mathbf {h}_{12}( v_1(\bar{y}^s),\bar{y}^s)}\\&\qquad = \ln \frac{J^s_{V_1}T^{-n}( v_1(\bar{x}^s),\bar{x}^s)}{J^s_{V_2}T^{-n}(\mathbf {h}_{12}(v_1(\bar{x}^s),\bar{x}^s,))} - \ln \frac{J^s_{V_1}T^{-n}( v_1(\bar{y}^s),\bar{y}^s)}{J^s_{V_2}T^{-n}(\mathbf {h}_{12}(v_1(\bar{y}^s),\bar{y}^s))}\\&\quad + \ln \frac{J\mathbf {h}_{V^n_1, V^n_2}(x_n)}{J\mathbf {h}_{V^n_1, V^n_2}(y_n)} \\&\qquad = \sum _{j=0}^{n-1} \ln J^s_{T^{-j}V_1}T^{-1}(x_j) - \ln J^s_{T^{-j}V_2}T^{-1}(\tilde{x}_j)\\&\quad - \ln J^s_{T^{-j}V_1}T^{-1}(y_j) + \ln J^s_{T^{-j}V_2}T^{-1}(\tilde{y}_j) \\&\qquad \qquad \quad + \ln \frac{J\mathbf {h}_{V^n_1, V^n_2}(x_n)}{J\mathbf {h}_{V^n_1, V^n_2}(y_n)} , \end{aligned} \end{aligned}$$

(A.1)

where as in the proof of Lemma 6.4, $x_j = T^{-j}(v_1(\bar{x}^s),\bar{x}^s)$, $\tilde{x}_j = T^{-j}(\mathbf {h}_{12}(v_1(\bar{x}^s),\bar{x}^s))$, and similarly for $y_j$ and $\tilde{y}_j$.

We begin by estimating the difference of stable Jacobians in (A.1). The factors in each term involving the stable Jacobian are given by (6.4), and we can bound the differences by grouping together either the terms on the same stable curve (standard distortion bounds), or the terms on the same unstable curve (using Lemma 6.4). Assuming without loss of generality that $d(x_0, \tilde{x}_0) \ge d(y_0, \tilde{y}_0)$, this yields the following bound on the sum,

$$\begin{aligned} C \sum _{j=0}^{n-1} \min \{ d(x_{j+1}, y_{j+1}) k_{j+1}^2, d(x_{j+1}, \tilde{x}_{j+1}) k_{j+1}^2 + d(x_j, \tilde{x}_j) + \phi (x_j, \tilde{x}_j) + \phi (x_{j+1}, \tilde{x}_{j+1}) \} , \end{aligned}$$

(A.2)

where we have used the uniform bound on $J\mathbf {h}_{12}$ given by Lemma 6.4 and (6.7) to eliminate terms involving $d(\tilde{x}_j, \tilde{y}_j)$ and $d(y_j, \tilde{y}_j)$. We estimate each term using one of two cases.

Case 1. $d(x_{j+1}, y_{j+1}) \le d(x_{j+1}, \tilde{x}_{j+1})$. We write,

$$\begin{aligned} d(x_{j+1},y_{j+1})&= d(x_{j+1}, y_{j+1})^{\varpi } d(x_{j+1}, y_{j+1})^{1-\varpi }\nonumber \\&\le C \left( J^s_{V_1}T^{-j-1}(x_0)\right) ^\varpi d(x_0, y_0)^\varpi d(x_{j+1}, \tilde{x}_{j+1})^{1-\varpi } . \end{aligned}$$

(A.3)

On the one hand, we have

$$\begin{aligned} d(x_{j+1} , \tilde{x}_{j+1}) \le C d(x_0, \tilde{x}_0) \left( J^u_{T^{n-j}\ell }T^{j+1}(x_j)\right) ^{-1} , \end{aligned}$$

where $\ell$ is the curve so that $x_0 \in T^n(\ell ) =: \bar{\gamma }$. On the other hand, by the uniform transversality of the curves $T^{-j-1}(V_1)$ and $T^{-j-1}(\bar{\gamma })$, we have

$$\begin{aligned} \begin{aligned}&J^s_{V_1}T^{-j-1}(x_0)J^u_{\bar{\gamma }}T^{-j-1}(x_0) = C^{\pm 1} JT^{-j-1}(x_0) = C^{\pm 1} \tfrac{\cos \varphi (x_0)}{\cos \varphi (x_{j+1})} \\&\quad \implies J^s_{V_1}T^{-j-1}(x_0) = C^{\pm 1} k_W^{-2} k_{j+1}^{2} J^u_{T^{-j-1}(\bar{\gamma })}T^{j+1}(x_{j+1}) , \end{aligned} \end{aligned}$$

(A.4)

where $JT^{-j-1}$ is the full Jacobian of the map $T^{-j-1}$. This estimate together with (6.3) implies

$$\begin{aligned} J^s_{V_1}T^{-j-1}(x_0) \le C k_W^{-10/3} \rho ^{-2/3} \left( J^u_{T^{-j-1}\bar{\gamma }}T^{j+1}(x_{j+1})\right) ^{5/3} . \end{aligned}$$

(A.5)

Now using this estimate together with (A.3) and again (6.3) yields,

$$\begin{aligned} d(x_{j+1},y_{j+1})k_{j+1}^2\le & {} C d(x_0,y_0)^\varpi d(x_0, \tilde{x}_0)^{1-\varpi } \frac{k_W^{-(4+10\varpi )/3}}{\rho ^{2(1+\varpi )/3}}\nonumber \\&\times \frac{1}{(J^u_{T^{-j-1}\bar{\gamma }}T^{j+1}(x_{j+1}))^{(1-8\varpi )/3}}, \end{aligned}$$

(A.6)

and this will be summable over j as long as $\varpi < 1/8$.

Case 2. $d(x_{j+1}, \tilde{x}_{j+1}) \le d(x_{j+1}, y_{j+1})$. In order to control the terms involving $\phi (x_j, \tilde{x}_j)$, we use the expressions for the Jacobians given by (6.4) and (A.4) together with [15, eq. (5.27) and following] to write,

$$\begin{aligned} \begin{aligned} \phi (x_j, \tilde{x}_j)&\le C\sum _{m=1}^j \frac{d(x_{j-m}, \tilde{x}_{j-m}) }{J^s_{T^{m-j-1}V_1}T^{-m+1}(x_{j-m+1}) J^u_{T^{-j}\bar{\gamma }}T^{m-1}(x_{j}) }\\&\quad + C \frac{\phi (x_0, \tilde{x}_0)}{J^s_{V_1}T^{-j}(x_0) J^u_{T^{-j}\bar{\gamma }}T^j(x_j) } \\&\le C d(x_0, \tilde{x}_0) \sum _{m=1}^j \frac{\Lambda _0^{-m+1}}{J^u_{T^{m-j}\bar{\gamma }}T^{j-m}(x_{j-m}) J^u_{T^{-j}\bar{\gamma }}T^{m-1}(x_{j})} \\&\quad + C \frac{\Lambda _0^{-j} d(x_0, \tilde{x}_0)}{J^u_{T^{-j}\bar{\gamma }}T^j(x_j)} , \end{aligned} \end{aligned}$$

(A.7)

where in the second line we have used the assumption that $\phi (x_0, \tilde{x}_0)$ is proportional to $d(x_0, \tilde{x}_0)$ and that $d(x_{j-m}, \tilde{x}_{j-m}) \le C d(x_0, \tilde{x}_0)/J^u_{T^{m-j}\bar{\gamma }}T^{j-m}(x_{j-m})$. Note that for terms with $m \ge 2$, there is a gap between the expansion factors of the unstable Jacobians in the denominator of the sum: The Jacobian $J^u_{T^{m-j-1}\bar{\gamma }}T(x_{j-m+1})$ is missing. We use the fact that this is proportional to $k_{j-m}^2$ to estimate,

$$\begin{aligned} \begin{aligned}&\frac{d(x_0, \tilde{x}_0)^{9\varpi }}{J^u_{T^{m-j}\bar{\gamma }}T^{j-m}(x_{j-m}) J^u_{T^{-j}\bar{\gamma }}T^{m-1}(x_{j})}\\&\quad \le \frac{C d(x_j, \tilde{x}_j)^{9\varpi } k_{j-m}^{18\varpi }}{\left( J^u_{T^{m-j}\bar{\gamma }}T^{j-m}(x_{j-m}) J^u_{T^{-j}\bar{\gamma }}T^{m-1}(x_{j})\right) ^{1-9\varpi }} \\&\quad \le \frac{C d(x_j, \tilde{x}_j)^{9\varpi } k_W^{-12\varpi } \rho ^{-6\varpi }}{\left( J^u_{T^{m-j}\bar{\gamma }}T^{j-m}(x_{j-m})\right) ^{1-15\varpi } \left( J^u_{T^{-j}\bar{\gamma }}T^{m-1}(x_{j})\right) ^{1-9\varpi }} \\&\quad \le C d(x_j, \tilde{x}_j)^{9\varpi } k_W^{-12\varpi } \rho ^{-6\varpi } \Lambda _0^{-(j-1)(1-15\varpi )}, \end{aligned} \end{aligned}$$

where we have used (6.3) in the second line. Notice that this estimate holds for $m=1$ as well. Putting this estimate together with (A.7) yields,

$$\begin{aligned} \phi (x_j , \tilde{x}_j) \le C d(x_0, \tilde{x}_0)^{1-9\varpi } d(x_j, \tilde{x}_j)^{9\varpi } k_W^{-12\varpi } \rho ^{-6\varpi } \Lambda _0^{-(j-1)(1-15\varpi )} . \end{aligned}$$

(A.8)

A similar estimate holds for $\phi (x_{j+1}, \tilde{x}_{j+1})$ with $d(x_{j+1}, \tilde{x}_{j+1})$ in place of $d(x_j, \tilde{x}_j)$.

Now since $d(x_j, \tilde{x}_j) = C^{\pm 1} d(x_{j+1}, \tilde{x}_{j+1}) J^u_{T^{-j-1}\bar{\gamma }}T(x_{j+1})$, we have

$$\begin{aligned} d(x_j,\tilde{x}_j) \le C d(x_{j+1}, y_{j+1}) k_j^2 \end{aligned}$$

by the assumption of this case as well. Since $d(x_{j+1}, \tilde{x}_{j+1}) \le C d(x_j, \tilde{x}_j)$, combining similar terms in (A.2), it remains to estimate the following expression in this case,

$$\begin{aligned} \begin{aligned}&d(x_{j+1}, \tilde{x}_{j+1}) k_{j+1}^2 + d(x_j, \tilde{x}_j)\\&\qquad + d(x_0, \tilde{x}_0)^{1-9\varpi } d(x_j, \tilde{x}_j)^{9\varpi } k_W^{-12\varpi } \rho ^{-6\varpi }\Lambda _0^{-(j-1)(1-15\varpi )} \\&\quad \le d(x_{j+1}, \tilde{x}_{j+1}) k_{j+1}^2 + d(x_{j+1}, \tilde{x}_{j+1}) k_j^2\\&\qquad + \frac{d(x_0, \tilde{x}_0)^{1-9\varpi } d(x_{j+1}, \tilde{x}_{j+1})^{9\varpi } k_j^{18\varpi }}{k_W^{12\varpi } \rho ^{6\varpi } \Lambda _0^{(j-1)(1-15\varpi )}} . \end{aligned} \end{aligned}$$

(A.9)

For the first term, we use the assumption of this case to write,

$$\begin{aligned} d(x_{j+1}, \tilde{x}_{j+1})&= d(x_{j+1}, \tilde{x}_{j+1})^\varpi d(x_{j+1}, \tilde{x}_{j+1})^{1-\varpi }\\&\le C \left( J^s_{V_1}T^{-j-1}(x_0)\right) ^\varpi d(x_0, y_0)^\varpi d(x_{j+1}, \tilde{x}_{j+1})^{1-\varpi } , \end{aligned}$$

which is the same expression as in (A.3). Thus the first term in (A.9) is bounded by (A.6).

For the second term in (A.9), we use (6.3) to bound $k_j^2$ and again (A.5) to estimate,

$$\begin{aligned} \begin{aligned} d(x_{j+1}, \tilde{x}_{j+1}) k_j^2&\le C \left( J^s_{V_1}T^{-j-1}(x_0)\right) ^{\varpi } d(x_0, y_0)^\varpi \\&\quad \times \frac{d(x_0, \tilde{x}_0)^{1-\varpi } }{\left( J^u_{T^{-j-1}\bar{\gamma }}T^{j+1}(x_{j+1})\right) ^{1-\varpi }} \frac{\left( J^u_{T^{-j}\bar{\gamma }}T^j(x_j)\right) ^{2/3}}{k_W^{4/3} \rho ^{2/3}} \\&\le C d(x_0, y_0)^\varpi d(x_0, \tilde{x}_0)^{1-\varpi }\\&\quad \times \frac{k_W^{-(4+ 10\varpi )/3}}{\rho ^{2(1+\varpi )/3}} \frac{1}{\left( J^u_{T^{-j-1}\bar{\gamma }}T^{j+1}(x_{j+1})\right) ^{(1-8\varpi )/3}}, \end{aligned} \end{aligned}$$

where in the second line we have used the fact that $J^u_{T^{-j}\bar{\gamma }}T^j(x_j) \le C J^u_{T^{-j-1}\bar{\gamma }}T^{j+1}(x_{j+1})$ as well as (A.4) to obtain a bound equivalent to (A.6).

Finally, we estimate the third term in (A.9) following a similar strategy, using again (A.5),

$$\begin{aligned} \begin{aligned}&d(x_{j+1}, \tilde{x}_{j+1})^{9\varpi } k_j^{18\varpi } \\&\quad \le C \left( J^s_{V_1}T^{-j-1}(x_0)\right) ^{\varpi } d(x_0, y_0)^\varpi \\&\qquad \times \frac{d(x_0, \tilde{x}_0)^{8\varpi } }{\left( J^u_{T^{-j-1}\bar{\gamma }}T^{j+1}(x_{j+1})\right) ^{8\varpi }} \frac{\left( J^u_{T^{-j}\bar{\gamma }}T^j(x_j)\right) ^{6\varpi }}{k_W^{12\varpi } \rho ^{6\varpi }} \\&\quad \le C d(x_0, y_0)^\varpi d(x_0, \tilde{x}_0)^{8\varpi } \frac{k_W^{-46\varpi /3}}{\rho ^{20\varpi /3}} \frac{1}{\left( J^u_{T^{-j-1}\bar{\gamma }}T^{j+1}(x_{j+1})\right) ^{\varpi /3}} . \end{aligned} \end{aligned}$$

Putting these three estimates together in (A.9) implies that the minimum factor from (A.2) in this case is bounded by

$$\begin{aligned}&Cd(x_0,y_0)^\varpi d(x_0, \tilde{x}_0)^{1-\varpi } \Lambda _0^{-(j-1)(1-44\varpi /3)} \\&\quad \cdot \max \left\{ k_W^{-82\varpi /3} \rho ^{-38\varpi /3}, k_W^{-4/3 - 10\varpi /3} \rho ^{-2(1+\varpi )/3} \right\} , \end{aligned}$$

which will be summable over j as long as $\varpi \le 1/15$.

Finally, using Case 1 or Case 2 as appropriate for each term appearing in (A.2) and summing over j completes the required estimate on the difference of stable Jacobians appearing in (A.1).

It remains to estimate the term involving $J\mathbf {h}_{V^n_1, V^n_2}$, which is the holonomy of the seeding foliation $\{ \ell _z \}_{z \in \overline{W}_i}$. Since $\{ \ell _z \}$ is uniformly $\mathcal {C}^2$, we have on the one hand,

$$\begin{aligned} \ln \frac{J\mathbf {h}_{V^n_1, V^n_2}(x_n)}{J\mathbf {h}_{V^n_1, V^n_2}(y_n)} \le C(d(x_n, \tilde{x}_n) + \phi (x_n, \tilde{x}_n) + d(y_n, \tilde{y}_n) + \phi (y_n, \tilde{y}_n)) , \end{aligned}$$

while on the other hand, using the fact that the curvatures of $V^n_1$ and $V^n_2$ are uniformly bounded,

$$\begin{aligned} \ln \frac{J\mathbf {h}_{V^n_1, V^n_2}(x_n)}{J\mathbf {h}_{V^n_1, V^n_2}(y_n)} \le C ( d(x_n ,y_n) + d(\tilde{x}_n, \tilde{y}_n)) . \end{aligned}$$

Now $\ln \frac{J\mathbf {h}_{V^n_1, V^n_2}(x_n)}{J\mathbf {h}_{V^n_1, V^n_2}(y_n)}$ is bounded by the minimum of these two estimates, which we recognize as a simplified version of the expression (A.2) with $j=n$. Thus this quantity satisfies the same bounds as in Cases 1 and 2 above, completing the proof of Lemma 6.6. $\square$

Proof of Lemma 6.7

Let $V_0$ be the subcurve of $P^+(W)$ corresponding to the gap; in the coordinates $(\bar{x}^u, \bar{x}^s)$, $V_0$ is the vertical line segment identified by the interval [a, b]. The boundary curves of the gap from the surviving foliation on either side as described by $\{ (\overline{G}(\bar{x}^u, a), a) : |\bar{x}^u| \le k_W^2 \rho \}$ and $\{ (\overline{G}(\bar{x}^u, b), b) : |\bar{x}^u| \le k_W^2 \}$. Let $x_a = (0, a)$ and $y_b = (0,b)$. Fix $\bar{x}^u$ with $|\bar{x}^u| \le k_W^2 \rho$ and define $\tilde{x}_a = (\bar{x}^u, \overline{G}(\bar{x}^u, a))$ and $\tilde{y}_b = (\bar{x}^u, \overline{G}(\bar{x}^u, b))$. Let $V_1$ denote the stable curve (vertical in the $(\bar{x}^u, \bar{x}^s)$ coordinates) connecting $\tilde{x}_a$ and $\tilde{y}_b$.

Using similar notation to Sect. 6.3, let $j+1$ denote the least integer $j' \ge 1$ such that $T^{j'}(\mathcal {S}_0)$ intersects $V_0$. This implies in particular, that the surviving parts of the foliation immediately on either side of the gap lie in the same homogeneity strip for the first j interates of $T^{-1}$. Define $x_i = T^{-i}(x_a)$, $y_i = T^{-i}(y_b)$, $\tilde{x}_i = T^{-i}(\tilde{x}_a)$ and $\tilde{y}_i = T^{-i}(\tilde{y}_b)$, for $i = 0, 1, \ldots j$.

The following expressions for the quantities appearing in the slope will be useful,

$$\begin{aligned} \overline{G}(\bar{x}^u, b) - \overline{G}(\bar{x}^u, a)= & {} \int _{T^{-j}V_1} J^s_{T^{-j}V_1}T^j~dm_W, \\ b-a= & {} \int _{T^{-j}V_0} J^s_{T^{-j}V_0} T^j~dm_W. \end{aligned}$$

And similarly,

$$\begin{aligned} \partial _{\bar{x}^s} \overline{G}(\bar{x}^u, a) = \frac{J^s_{T^{-j}V_1}T^j(y_j)}{J^s_{T^{-j}V_0}T^j(x_j)} J \mathbf {h}_{-j}(x_j) , \end{aligned}$$

where $\mathbf {h}_{-j}$ is the holonomy map from $T^{-j}V_0$ to $T^{-j}V_1$ along the surviving foliation on the sides of the gap. Using these expressions, we estimate,

$$\begin{aligned} \begin{aligned} \ln \left| \frac{\overline{G}(\bar{x}^u, b) - \overline{G}(\bar{x}^u, a)}{(b-a) \partial _{\bar{x}^s} \overline{G}(\bar{x}^u, a)} \right| =&\ln \frac{\frac{1}{|T^{-j}V_1|} \int _{T^{-j}V_1} J^s_{T^{-j}V_1}T^j~dm_W}{J^s_{T^{-j}V_1}T^j(y_j)} - \ln J \mathbf {h}_{-j}(x_j) \\&\; \; + \ln \frac{J^s_{T^{-j}V_0}T^j(x_j)}{\frac{1}{|T^{-j}V_0|} \int _{T^{-j}V_0} J^s_{T^{-j}V_0}T^j~dm_W} + \ln \frac{|T^{-j}V_0|}{|T^{-j}V_1|} . \end{aligned} \end{aligned}$$

(A.10)

Since $J^s_{T^{-j}V_1}T^j$ is continuous on $T^{-j}V_1$, the average value of the function is equal to the function evaluated at some point on $T^{-j}V_1$. Thus by standard distortion bounds along stable curves, the first term on the right-hand side of (A.10) is bounded by $C_d |a-b|^{1/3}$. A similar bound holds for the third term above.

In order to estimate $J\mathbf {h}_{-j}(x_j)$, we note that this is part of the surviving foliation that originated at time $-n$. Applying Lemma 6.4 from time $-j$ to time $-n$, there exists $C>0$ such that

$$\begin{aligned} \ln J\mathbf {h}_{-j}(x_j) \le C \left( d(x_j, \tilde{x}_j) \rho ^{-2/3} k_{W}^{-4/3} + \phi (x_j, \tilde{x}_j)\right) , \end{aligned}$$

where $\phi (x_j, \tilde{x}_j)$ represents the angle formed by the tangent vectors to $T^{-j}V_0$ and $T^{-j}V_1$ at $x_j$ and $\tilde{x}_j$, respectively.

For the first term above, $d(x_j, \tilde{x}_j )\le C d(x_0 , \tilde{x}_0)/ J^u_{T^{-j}\bar{\gamma }}T^j(x_j)$, where $\bar{\gamma }$ denotes the element of the surviving foliation containing $x_a$. For the second term, we use (A.7) to write,

$$\begin{aligned} \phi (x_j, \tilde{x}_j)\le & {} C d(x_0, \tilde{x}_0) \sum _{m=1}^{j} \frac{\Lambda _0^{-m+1}}{J^u_{T^{m-j}\bar{\gamma }}T^{j-m}(x_{j-m}) J^u_{T^{-j}\bar{\gamma }}T^{m-1}(x_j)}\\&+ \,C \frac{\Lambda _0^{-j} d(x_0, \tilde{x}_0)}{J^u_{T^{-j}\bar{\gamma }}T^j(x_j)} , \end{aligned}$$

where we have used the fact that $\phi (x_0, \tilde{x}_0) \le C d(x_0, \tilde{x}_0)$ since both $V_0$ and $V_1$ are vertical segments and the unstable curves in the surviving foliation have uniformly bounded curvature. As in the proof of Lemma 6.6, there is a gap in the product of unstable Jacobians; the factor $J^u_{T^{m-j}\bar{\gamma }} T(x_{j-m+1})$ is missing. This is proportional to $k^2_{j-m}$, but due to (6.3), we do not ask for a full power of this factor; rather, we estimate,

$$\begin{aligned} \begin{aligned}&\frac{1}{J^u_{T^{m-j}\bar{\gamma }}T^{j-m}(x_{j-m})J^u_{T^{-j}\bar{\gamma }}T^{m-1}(x_j)}\\&\quad \le \frac{C k_{j-m}^{6/5}}{J^u_{T^{m-j}\bar{\gamma }}T^{j-m}(x_{j-m}) \left( J^u_{T^{m-j}\bar{\gamma }} T(x_{j-m+1})\right) ^{3/5} J^u_{T^{-j}\bar{\gamma }}T^{m-1}(x_j)} \\&\quad \le C \frac{k_W^{4/5} \rho ^{-2/5}}{\left( J^u_{T^{-j}\bar{\gamma }} T^j(x_j)\right) ^{3/5}} . \end{aligned} \end{aligned}$$

This, together with the above estimates implies,

$$\begin{aligned} \ln J\mathbf {h}_{-j}(x_j) \le C \left( \frac{d(x_0, \tilde{x}_0)}{\rho ^{2/3} k_W^{4/3} J^u_{T^{-j} \bar{\gamma }}T^j(x_j)} + \frac{d(x_0, \tilde{x}_0) k_W^{4/5}}{\rho ^{2/5} \left( J^u_{T^{-j}\bar{\gamma }}T^j(x_j)\right) ^{3/5}} \right) . \end{aligned}$$

(A.11)

$\square$

Sublemma A.1

(Relating the gap size and the unstable Jacobian) Let a, b define the gap in $P^+(W)$ as in the statement of the lemma and let $j, x_j$ be as defined above. There exists $C>0$, depending only on T, and not on n, j or W, such that

$$\begin{aligned} \frac{1}{J^u_{T^{-j}\bar{\gamma }}T^j(x_j) } \le C k_W^{-2} \rho ^{-26/35} |a-b|^{9/35}. \end{aligned}$$

Proof

We consider the size of the gap created in a neighborhood of $T^{-j-1}(x_a)$, depending on whether this gap is created by an intersection with homogeneity strips of high index or not.

Case 1. The connected component of $T^{-j-1}(V_1)$ containing $T^{-j-1}(x_a)$ intersects $\mathcal {S}_0$. Then using (6.12) and (6.13), we have,

$$\begin{aligned} |a-b| \ge \frac{C^{\pm 1} k_{j+1}^{-4}}{J^s_{V_1}T^{-j}(x_a)} \ge \frac{C^{\pm 1} k_W^{8/3} \rho ^{4/3}}{\left( J^u_{T^{-j-1}\bar{\gamma }}T^{j+1}(x_{j+1})\right) ^{4/3} J^s_{V_1}T^{-j}(x_a)} , \end{aligned}$$

where we used (6.3) in the second inequality. Now $J^u_{T^{-j-1}\bar{\gamma }}T^{j+1}(x_{j+1}) = C^{\pm 1} k_j^2 J^u_{T^{-j}\bar{\gamma }}T^j(x_j)$. Using this together with (A.4) to convert between stable and unstable Jacobians yields,

$$\begin{aligned} |a-b| \ge \frac{C^{\pm 1} k_W^{8/3} \rho ^{4/3}}{\left( J^u_{T^{-j}\bar{\gamma }}T^j(x_j)\right) ^{7/3} k_j^{14/3} k_W^{-2}} \ge \frac{C^{\pm 1} k_W^{70/9} \rho ^{26/9} }{\left( J^u_{T^{-j}\bar{\gamma }}T^j(x_j)\right) ^{35/9} } , \end{aligned}$$

where in the second inequality we have used (6.3). This proves the sublemma in this case.

Case 2. The component of $T^{-j-1}(V_1)$ containing $T^{-j-1}(x_a)$ does not intersect $\mathcal {S}_0$. In this case, by the uniform transversality of $\mathcal {S}_1$ with unstable curves, the gap is bounded below by

$$\begin{aligned} \begin{aligned} |a-b|&\ge \frac{C^{\pm 1} \rho k_W^2}{J^u_{T^{-j-1}\bar{\gamma }}T^{j+1}(x_{j+1}) J^s_{V_1}T^{-j-1}(x_a)} \ge \frac{C^{\pm 1} \rho k_W^4}{\left( J^u_{T^{-j-1}\bar{\gamma }}T^{j+1}(x_{j+1})\right) ^2 k_{j+1}^2 } \\&\ge \frac{C^{\pm 1} \rho k_W^4}{\left( J^u_{T^{-j}\bar{\gamma }}T^j(x_j)\right) ^2}, \end{aligned} \end{aligned}$$

where in the last step, we have used the fact that in Case 2, $k_{j+1}$ is of order 1. This is clearly a greater lower bound on $|a-b|$ than in Case 1, proving the sublemma. $\square$

With the sublemma proved, we return to our estimate (A.11),

$$\begin{aligned} \ln J\mathbf {h}_{-j}(x_j)\le & {} C \left( \frac{d(x_0, \tilde{x}_0)}{\rho ^{2/3} k_W^{4/3} J^u_{T^{-j} \bar{\gamma }}T^j(x_j)} + \frac{d(x_0, \tilde{x}_0) k_W^{4/5}}{\rho ^{2/5} \left( J^u_{T^{-j}\bar{\gamma }}T^j(x_j)\right) ^{3/5}} \right) \\\le & {} C d(x_0, \tilde{x}_0)\left( k_W^{-10/3} \rho ^{-148/105} |a-b|^{9/35} + k_W^{-2/5} \rho ^{-148/175} |a-b|^{27/175} \right) \\\le & {} C \left( k_W^{-4/3} \rho ^{-43/105} |a-b|^{9/35} + k_W^{8/5} \rho ^{27/175} |a-b|^{27/175} \right) \\\le & {} C \left( \rho ^{-43/105} |a-b|^{9/35} + \rho ^{-29/175} |a-b|^{27/175} \right) \le C \rho ^{-31/105} |a-b|^{1/7} , \end{aligned}$$

where in the third line we have used the fact that $d(x_0, \tilde{x}_0) \le C \rho k_W^2$ and in the last line that $k_W \le C \rho ^{-1/5}$. We have also opted to take simpler (slightly less than optimal) exponents, taking the power 1 / 7 rather than 27 / 175 in the second term of the last line and converting $|a-b|^{4/35} \le C\rho ^{4/35}$ in the first term. Note that $|a-b| \le C\rho$ follows from Sect. 6.1.

Finally, for the fourth term on the right-hand side of (A.10), we note that the boundary curves for the gap containing $T^{-j}V_0$ and $T^{-j}V_1$ both lie in the unstable cone. Since $T^{-j}V_0$ and $T^{-j}V_1$ have bounded curvature, we have

$$\begin{aligned} \ln \frac{|T^{-j}V_0|}{|T^{-j}V_1|} \le C d(x_j, \tilde{x}_j) \le \frac{C d(x_0, \tilde{x}_0)}{J^u_{T^{-j} \bar{\gamma }}T^j(x_j)} . \end{aligned}$$

Using Sublemma A.1, this quantity is bounded by

$$\begin{aligned} C d(x_0, \tilde{x}_0) k_W^{-2} \rho ^{-26/35} |a-b|^{9/35} \le C \rho ^{9/35} |a-b|^{9/35}, \end{aligned}$$

where again we have used the fact that $d(x_0, \tilde{x}_0) \le k_W^2 \rho$. Using the estimates for these four terms in (A.10) ends the proof of Lemma 6.7 since both the average slope and $\partial _{\bar{x}^s} \overline{G}$ are uniformly bounded away from 0 and infinity. $\square$

Appendix B: Estimates for the Dolgopyat bound (Lemmas 8.7, 8.8, and 8.9)

This appendix contains several crucial, but technical, estimates used in the Dolgopyat type cancellation argument developed in Sect. 8.

Proof of Lemma 8.7

Recalling condition (ii) on the foliation from Sect. 6, the first identity in (8.35) together with (8.34) gives

$$\begin{aligned} \begin{aligned} x^s&= G_{i,j,\varkappa }(M_A(x^s),0)+\int _0^{\mathbf {h}^s_A(x^s)}\!\!\!\!\!\!\!\!ds\; \partial _{x^s} G_{i,j,\varkappa }(M_A(x^s),s)\\&= \int _0^{M_A(x^s)}\!\!\!\!\!\!\!\! du\; \partial _{x^u}G_{i,j,\varkappa }(u,0)+ \int _0^{\mathbf {h}^s_A(x^s)}\!\!\!\!\!\!\!\! ds\left[ 1+\int _0^{M_A(x^s)}\!\!\!\!\!\!\!\! du\; \partial _{x^u}\partial _{x^s}G_{i,j,\varkappa }(u,s)\right] \\&=\mathbf {h}^s_A(x^s)+\int _0^{M_A(x^s)}\!\!\!\!\!\!\!\!\!\! du\; \partial _{x^u}G_{i,j,\varkappa }(u,0)+ \int _0^{\mathbf {h}^s_A(x^s)}\!\!\!\!\!\!\!\! ds\int _0^{M_A(x^s)}\!\!\!\!\!\!\!\!\!\! du\; \partial _{x^u}\partial _{x^s}G_{i,j,\varkappa }(u,s) , \end{aligned} \end{aligned}$$

(B.1)

where we have used the fact that $\partial _{x^s} G_{i,j,\varkappa }(0,s) =1$ for each s by property (ii) of the foliation. Thus, combining the bound (vi) from Sect. 6, that is $|\partial _{x^u}\partial _{x^s} G_{i,j,\varkappa }|\le C\rho ^{-4/5}=C{\varepsilon }^{-4\varsigma /5}$, the bound $|M'_A|\le C_{\#}{\varepsilon }^{1-\frac{4}{5}\theta }$ from (8.22) (which implies $|M_A(x^s)|\le C_{\#}{\varepsilon }$), and the condition $|\mathbf {h}^s_A(x^s)|\le C_{\#}{\varepsilon }^\theta$ from (8.34), we get $\mathbf {h}^s_A(x^s)=x^s(1+\mathcal {O}({\varepsilon }^{1-4\varsigma /5}))+\mathcal {O}({\varepsilon })$.

Next, differentiating the first identity in (8.35), we find

$$\begin{aligned} (\mathbf {h}^s_A)'(x^s)=\frac{1-\partial _{x^u} G_{i,j,\varkappa }(M_A(x^s),\mathbf {h}^s_A(x^s)) M'_A(x^s)}{\partial _{x^s} G_{i,j,\varkappa }(M_A(x^s),\mathbf {h}^s_A(x^s))}. \end{aligned}$$

(B.2)

We have

$$\begin{aligned} \begin{aligned} \partial _{x^s} G_{i,j,\varkappa }\left( M_A(x^s),\mathbf {h}^s_A(x^s)\right)&=\partial _{x^s} G_{i,j,\varkappa }\left( 0,\mathbf {h}^s_A(x^s)\right) \\&\quad + \int _0^{M_A(x^s)}\!\!\!\!\!\!\! \partial _{x^u}\partial _{x^s} G_{i,j,\varkappa }(u,\mathbf {h}^s_A(x^s)) du\\&=1+\mathcal {O}({\varepsilon }^{1-4\varsigma /5}), \end{aligned} \end{aligned}$$

while $|\partial _{x^u} G_{i,j,\varkappa }|_\infty \le C_{\#}$, hence we have the second inequality of Lemma 8.7.

$$\begin{aligned} \left| 1-\left( \mathbf {h}^s_A\right) '\right| \le C_{\#}{\varepsilon }^{1-\frac{4}{5}\varsigma } , \end{aligned}$$

(B.3)

while the bound on $\left| \left( \mathbf {h}^s_A\right) (x^s)-x^s\right|$ follows by integration. Finally, $\mathbf {h}^s_A$ is invertible and the claimed bound on $\left| \left( \mathbf {h}^s_A\right) ^{-1}(x^s)-x^s\right|$ holds using $\left( (\mathbf {h}^s_A)^{-1}(x^s) - x^s\right) ' = \frac{1}{\left( \mathbf {h}^s_A\right) ' \circ \left( \mathbf {h}^s_A\right) ^{-1}} - 1$ and integrating as before. $\square$

Proof of Lemma 8.8

To start with, note that formula [5, (E.1)] is obtained by a purely geometric argument and uses only that the strong manifolds are in the kernel of the contact form and the weak manifolds in the kernel of its differential. Since the exact same situation holds here, we have, for all relevant manifolds $W_A$, the formula

$$\begin{aligned} {\varvec{\omega }}_A(x^s)=\int _0^{x^s}ds\int _0^{M_A((\mathbf {h}_A^s)^{-1}(s))}du\;\partial _{x^s}G_{i,j\varkappa }(u,s). \end{aligned}$$

(B.4)

Note that this implies

$$\begin{aligned} |{\varvec{\omega }}_A|_{\infty }\le C_{\#}{\varepsilon }^{1+\theta } . \end{aligned}$$

(B.5)

To obtain the formula we are interested in for each pair A, B, it suffices to differentiate. Remembering properties (ii) and (vi) of the foliation constructed in Sect. 6, we have

$$\begin{aligned} \begin{aligned} \partial _{x^s} {\varvec{\omega }}_B(x^s)-\partial _{x^s} {\varvec{\omega }}_A(x^s)&=\int _{M_A\left( \left( \mathbf {h}_A^s\right) ^{-1}(x^s)\right) }^{M_B\left( \left( \mathbf {h}_B^s\right) ^{-1}(x^s)\right) }\!du\;\partial _{x^s} G_{i,j,\varkappa }(u,x^s)\\&=\int _{M_A\left( \left( \mathbf {h}_A^s\right) ^{-1}(x^s)\right) }^{M_B\left( \left( \mathbf {h}_B^s\right) ^{-1}(x^s)\right) }\!du\left[ 1+\int _0^u du_1\; \partial _{x^u} \partial _{x^s}G_{i,j,\varkappa }(u_1,x^s)\right] \\&=\left[ M_B\left( \left( \mathbf {h}_B^s\right) ^{-1}(x^s)\right) {-}M_A\left( \left( \mathbf {h}_A^s\right) ^{-1}(x^s)\right) \right] \left( 1{+}\mathcal {O}\left( {\varepsilon }^{1-\frac{4}{5}\varsigma }\right) \right) . \end{aligned} \end{aligned}$$

(B.6)

Next, the distance between $W_A^0\cap \{x^s=0\}$ and $W_B^0\cap \{x^s=0\}$ is given by $|M_A(0)-M_B(0)|=:d(W_A,W_B)$. Then, by the argument developed in (8.18) (proof of Lemma 8.5), we have

$$\begin{aligned} |~|M_A(0)-M_B(0)|-|M_A(x^s)-M_B(x^s)|~|\le C_{\#}d(W_A,W_B){\varepsilon }^{\theta /5} . \end{aligned}$$

(B.7)

On the other hand, by (B.1) and property (vi) of the foliation, we have

$$\begin{aligned} \begin{aligned} \left| \left( \mathbf {h}^s_B\right) ^{-1}(x^s) -\left( \mathbf {h}^s_A\right) ^{-1}(x^s)\right|&= \left| \int _{M_A\circ \left( \mathbf {h}^s_A\right) ^{-1}(x^s)}^{M_B\circ \left( \mathbf {h}^s_B\right) ^{-1}(x^s)} \!\!\!\!\!\!\! du \; \partial _{x^u} G_{i,j,\varkappa }(u,0) \right| \\&\quad + \left| \int _{M_A\circ \left( \mathbf {h}^s_A\right) ^{-1}(x^s)}^{M_B\circ \left( \mathbf {h}^s_B\right) ^{-1}(x^s)}\!\!\!\!\!\!\!du\int _0^{x^s}\! ds\; \partial _{x^u}\partial _{x^s}G_{i,j,\varkappa }(u,s)\right| \\&\le C_{\#}\left( 1+ {\varepsilon }^{\theta -\frac{4}{5} \varsigma }\right) \left| M_B\circ \left( \mathbf {h}^s_B\right) ^{-1}(x^s)-M_A\circ \left( \mathbf {h}^s_A\right) ^{-1}(x^s)\right| . \end{aligned} \end{aligned}$$

(B.8)

To conclude recall that we are working in coordinates in which $|M'_{A}|\le C_{\#}{\varepsilon }^{1-\frac{4}{5}\theta }$, cf. the proof of Lemma 8.5, hence

$$\begin{aligned} \begin{aligned}&\left| M_B\circ \left( \mathbf {h}^s_B\right) ^{-1}-M_A\circ \left( \mathbf {h}^s_A\right) ^{-1}\right| \ge \left| M_B\circ \left( \mathbf {h}^s_B\right) ^{-1}-M_A\circ \left( \mathbf {h}^s_B\right) ^{-1}\right| \\&\qquad -C_{\#}{\varepsilon }^{1-\frac{4}{5}\theta }\left| \left( \mathbf {h}^s_B\right) ^{-1}-\left( \mathbf {h}^s_A\right) ^{-1}\right| \\&\quad \ge \left| M_B\circ \left( \mathbf {h}^s_B\right) ^{-1}-M_A\circ \left( \mathbf {h}^s_B\right) ^{-1}\right| \\&\qquad -C_{\#}\left( {\varepsilon }^{1-\frac{4}{5}\theta } + {\varepsilon }^{1-\frac{4}{5} \varsigma + \frac{1}{5} \theta }\right) \left| M_B\circ \left( \mathbf {h}^s_B\right) ^{-1}-M_A\circ \left( \mathbf {h}^s_A\right) ^{-1}\right| , \end{aligned} \end{aligned}$$

which, together with (B.6) and (B.7), proves (8.41).

To prove the second statement, let us introduce the shorthand notation ${\varvec{\omega }}_{A,B}(x^s)={\varvec{\omega }}_A(x^s)-{\varvec{\omega }}_B(x^s)$ and ${\varvec{\eta }}_A=\left( \mathbf {h}_A^s\right) ^{-1}$, ${\varvec{\eta }}_B=\left( \mathbf {h}_B^s\right) ^{-1}$. By (B.6) we have

$$\begin{aligned}&\left| \partial _{x^s}{\varvec{\omega }}_{A,B}(x^s)-\partial _{x^s}{\varvec{\omega }}_{A,B}(y^s)\right| \\&\quad =\left| \int _{M_A\left( {\varvec{\eta }}_A(x^s)\right) }^{M_B\left( {\varvec{\eta }}_B(x^s)\right) }du\;\partial _{x^s} G_{i,j,\varkappa }(u,x^s)-\int _{M_A\left( {\varvec{\eta }}_A(y^s)\right) }^{M_B\left( {\varvec{\eta }}_B(y^s)\right) }du\;\partial _{x^s} G_{i,j,\varkappa }(u,y^s)\right| \\&\quad \le \left| \int _{M_A({\varvec{\eta }}_A(x^s))}^{M_B({\varvec{\eta }}_B(x^s))}\!du\;\partial _{x^s} G_{i,j,\varkappa }(u,x^s)- \partial _{x^s}G_{i,j,\varkappa }(u,y^s)\right| \\&\qquad +\left| \int _{M_B({\varvec{\eta }}_B(y^s))}^{M_B({\varvec{\eta }}_B(x^s))}\!du\;\partial _{x^s} G_{i,j,\varkappa }(u,y^s)\right| \\&\qquad +\left| \int _{M_A({\varvec{\eta }}_A(x^s))}^{M_A({\varvec{\eta }}_B(y^s))}\!du\;\partial _{x^s} G_{i,j,\varkappa }(u,y^s)\right| . \end{aligned}$$

Notice that (ii) of the foliation implies

$$\begin{aligned} \begin{aligned}&\partial _{x^s}G_{i,j,\varkappa }(u,x^s)- \partial _{x^s}G_{i,j,\varkappa }(u,y^s)\\&\quad =\partial _{x^s}G_{i,j,\varkappa }(u,x^s)-\partial _{x^s} G_{i,j,\varkappa }(u,y^s)-\partial _{x^s} G_{i,j,\varkappa }(0,x^s)\\&\qquad +\partial _{x^s} G_{i,j,\varkappa }(0,y^s) . \end{aligned} \end{aligned}$$

Thus, the four-point property (vii) (applied to the points $(u,x^s)$ and $(0,y^s)$) implies

$$\begin{aligned} |\partial _{x^s}G_{i,j,\varkappa }(u,x^s)- \partial _{x^s}G_{i,j,\varkappa }(u,y^s)|\le C_{\#}{\varepsilon }^{-\left( \frac{4}{5}+\frac{11{\varpi }}{15}\right) \varsigma +1-7{\varpi }}|x^s-y^s|^{\varpi }. \end{aligned}$$

(B.9)

On the other hand, Eq. (8.22) and Lemma 8.7 imply

$$\begin{aligned} |M_B({\varvec{\eta }}_B(x^s))-M_B({\varvec{\eta }}_B(y^s))|\le C_{\#}{\varepsilon }^{1-\frac{4}{5}\theta }|x^s-y^s|, \end{aligned}$$

and the same for A. Remembering that $|x^s|, |y^s|\le c{\varepsilon }^\theta$, property (v) of the foliation, and our conditions on ${\varpi }, \theta ,\varsigma$ from (8.42), the above facts yield

$$\begin{aligned} \left| \partial _{x^s}{\varvec{\omega }}_{A,B}(x^s)-\partial _{x^s}{\varvec{\omega }}_{A,B}(y^s)\right|\le & {} C_{\#}{\varepsilon }|x^s-y^s|^{\varpi }+{\varepsilon }^{1-\frac{4}{5}\theta }|x^s-y^s|\\\le & {} C_{\#}{\varepsilon }|x^s-y^s|^{\varpi }\end{aligned}$$

which, together with (B.5), proves the second statement of Lemma 8.8.

To continue, let

$$\begin{aligned} {\varvec{L}}_{A,B}(x^s, x^0)=\frac{[(m-1)!]^2}{(\ell {\tau }_-)^{2m-2}} e^{-2a\ell {\tau }_-} \mathbf{G}^*_{\ell ,m,i,A}(x^s, x^0)\overline{\mathbf{G}^*_{\ell ,m,i,B}(x^s, x^0)} . \end{aligned}$$

Next we introduce a sequence $\{w_j\}_{j=0}^M\subset {\mathbb R}$ such that $w_0=-cr^\theta$ and $\partial _{x^s}{\varvec{\omega }}_{A,B}(w_j)(w_{j+1}-w_j)=2\pi b^{-1}$ and let $M\in {\mathbb N}$ be such that $w_M\le c {\varepsilon }^\theta$ and $w_{M+1}> c{\varepsilon }^\theta$.⁴⁷ Also, we set ${\varvec{\delta }}_j=w_{j+1}-w_j$. By Lemma 8.7 it follows that, for each $x^s\in [w_j,w_{j+1}]$,

$$\begin{aligned} |{\varvec{\omega }}_{A,B}(x^s)-{\varvec{\omega }}_{A,B}(w_j)-\partial _{x^s}{\varvec{\omega }}_{A,B}(w_j)(x^s-w_j)|\le C_{\#}{\varepsilon }{\varvec{\delta }}_j^{1+{\varpi }}. \end{aligned}$$

In addition, the bounds in (8.39) imply

$$\begin{aligned} |{\varvec{L}}_{A,B}(x^s,x^0)-{\varvec{L}}_{A,B}(w_j,x^0)|\le C_{\#}{\varvec{\delta }}_j {\varepsilon }^{-3\theta } . \end{aligned}$$

Then, using the first part of the Lemma,

$$\begin{aligned} \begin{aligned}&\left| \int _{w_j}^{w_{j+1}} e^{-ib{\varvec{\omega }}_{A,B}(x^s)} {\varvec{L}}_{A,B}(x^s,x^0) d x^s\right| \\&\quad =\left| \int _{w_j}^{w_{j+1}} e^{-ib[\partial _{x^s}{\varvec{\omega }}_{A,B}(w_i)(x^s-w_j)+\mathcal {O}({\varepsilon }{\varvec{\delta }}_j^{1+{\varpi }})]} [{\varvec{L}}_{A,B}(w_j, x^0)+\mathcal {O}({\varepsilon }^{-3\theta }{\varvec{\delta }}_j)] dx^s\right| \\&\quad \le C_{\#}\left( b{\varvec{\delta }}_j^{1+{\varpi }}{\varepsilon }^{-2\theta +1}+{\varepsilon }^{-3\theta }{\varvec{\delta }}_j\right) {\varvec{\delta }}_j\\&\quad \le C_{\#}\left( \frac{{\varepsilon }^{-2\theta +1}}{d(W_A,W_B)^{1+{\varpi }} b^{\varpi }}+\frac{{\varepsilon }^{-3\theta }}{d(W_A,W_B) b}\right) {\varvec{\delta }}_j . \end{aligned} \end{aligned}$$

We may be left with the integral over the interval $[w_M,cr^\theta ]$ which is trivially bounded by $C_{\#}{\varepsilon }^{-2\theta }{\varvec{\delta }}_M\le C_{\#}[{\varepsilon }^{2\theta }bd(W_A, W_B)]^{-1}$. The statement follows since the manifolds we are considering have length at most $c{\varepsilon }^\theta$, hence $\sum _{j=0}^{M-1}{\varvec{\delta }}_j \le c{\varepsilon }^\theta$. $\square$

Proof of Lemma 8.9

We start by introducing a function $\bar{R}:W\rightarrow {\mathbb N}$ such that $\bar{R}(\xi )$ is the first $t\in {\mathbb N}$ at which $\Phi _{-t{\tau }_-}\xi$ belongs to a component of $\Phi _{-t{\tau }_-}W$ of size larger than $\kappa _* L_{0}$, $\kappa _*<1/3$, and distant more than $L_0$ from $\partial \Omega _0$. $\square$

Remark B.1

We choose $\kappa _*$ such that, if $\Phi _{-t{\tau }_-}W$ is a regular piece of size larger than $L_{0}/3$ but in an $L_0$ neighborhood of $\partial \Omega _0$, then either $\Phi _{-(t+k){\tau }_-}W$ or $\Phi _{-(t-k){\tau }_-}W$ will satisfy our requirement for some $C_{\#}>k{\tau }_->L_0$.

We define then $R(\xi )=\min \{\bar{R}(\xi ),\ell \}$. Let $\mathcal {P}=\{J_{i}\}$ be the coarser partition of W in intervals on which $R$ is constant. Note that, for each $W_{B,i}$, $\Phi _{\ell {\tau }_-}(W_{B,i})\subset J_{j}$ for some $J_{j}\in \mathcal {P}$.

Let

$$\begin{aligned} \Sigma _{\ell ,j}=\{(B,i)\;:\; i\in {\mathbb N},~ B\in E_{\ell ,i},~ d\left( W_{B,i}, W^0_{A,i}\right) \le \rho _{*},\; \Phi _{\ell {\tau }_-}W_{B,i}\subset J_{j}\}. \end{aligned}$$

Then, by Lemma 3.5, for each $(B,i)\in \Sigma _{\ell ,j}$

$$\begin{aligned} Z_{\ell ,B,i} = \int _{W_{B,i}}J^s_{\ell {\tau }_-}\le C_{\#}\frac{|J_{j}|}{|\overline{W}_{j}|}\int _{W_{B,i}} J^s_{(\ell -R_j) {\tau }_-} \le C_{\#}\frac{|J_{j}|}{|\overline{W}_{j}|}|\Phi _{(\ell -R_{j}){\tau }_-}W_{B,i}|~, \end{aligned}$$

(B.10)

where $R_{j}=R(J_{j})$ and $\overline{W}_{j}=\Phi _{-R_{j}{\tau }_-}J_{j}$. Note that, by construction, either $|\overline{W}_{j}|\ge \kappa _* L_{0}$ or $R_j=\ell$ and $\overline{W}_{j}=W_{B,i}$. Next, consider the local weak stable surfaces $W_{B,i}^{0} := \cup _{t \in [-cr^\theta , cr^\theta ]} \Phi _t W_{B,i}$ and $\overline{W}^{0}_j := \cup _{t \in [-cr^\theta , cr^\theta ]} \Phi _t \overline{W}_{j}$.

Let us analyze first the case in which $R_j<\ell$. Let $\rho \le C_{\#}L_0^{\frac{5}{3}}$. Then, by assumption, $\overline{W}_{j}$ is a manifold with satisfies condition (a) at the beginning of Sect. 6. Indeed $\overline{W}_{j}$ is too long to belong to $\mathbb {H}_k$ with $k\ge C\rho ^{-\frac{1}{5}}$, so that $\overline{W}_j$ satisfies condition (b) of that section as well. We can then use the construction in Sect. 6 to define an approximate unstable foliation in a $\rho$ neighborhood of the surface $\overline{W}^{0}_j$. Let $\Gamma _{B,i}$ be the set of leaves that intersect $\Delta _{\ell -R_{j}}\cap \Phi _{(\ell -R_{j}){\tau }_-}W^{0}_{B,i}$. By the construction of the covering $B_{c{\varepsilon }^\theta }(x_i)$, the $\Gamma _{B,i}$ can have at most $C_{\#}$ overlaps and since $W_B$ completely crosses $B_{c{\varepsilon }^\theta }(x_i)$, there can be no gaps between the curves in $\Gamma _{B,i}$. Now using the uniform transversality between the stable, unstable and flow directions,

$$\begin{aligned} \sum _{(B,i)\in \Sigma _{\ell ,j}}m(\Gamma _{B,i}) \ge C_{\#}\!\!\sum _{(B,i)\in \Sigma _{\ell ,j}}|\Phi _{(\ell -R_{j}){\tau }_-}W_{B,i}|\rho {\varepsilon }^\theta . \end{aligned}$$

Accordingly, for each j such that $R_j<\ell$, remembering (B.10),

$$\begin{aligned} \sum _{(B,i)\in \Sigma _{\ell ,j}}Z_{\ell ,B,i}\le C_{\#}|J_{j}| \rho ^{-1} {\varepsilon }^{-\theta } \sum _{(B,i)\in \Sigma _{\ell ,j}}m(\Phi _{-(\ell -R_{j}){\tau }_-}\Gamma _{B,i}) ~, \end{aligned}$$

(B.11)

where we have used the invariance of the volume. Remembering that the $\Phi _{-(\ell -R_{j}){\tau }_-}\Gamma _{B,i}$ have a fixed maximal number of overlaps and since they must be all contained in a box containing $\widetilde{O}$ of length $C_{\#}{\varepsilon }^\theta$ in the flow direction, of length $C_{\#}{\varepsilon }^\theta$ in the stable direction and of length $C_{\#}(\rho _{*} + \Lambda ^{-(\ell -R_j){\tau }_-}\rho )$ in the unstable direction, we have,⁴⁸

$$\begin{aligned} \begin{aligned} \sum _j\sum _{(B,i)\in \Sigma _{\ell ,j}}Z_{B,i}\le&\sum _{\left\{ j\;:\; R_{j}\le \frac{\ell }{2}\right\} } C_{\#}|J_{j}|~\rho ^{-1} {\varepsilon }^\theta (\rho _{*}+\Lambda ^{-\frac{\ell }{2}{\tau }_-} \rho )\\&+\sum _{\left\{ j\;:\; R_{j}> \frac{\ell }{2}\right\} }\sum _{(B,i)\in \Sigma _{\ell ,j}}Z_{B,i}. \end{aligned} \end{aligned}$$

(B.12)

To estimate the sum on $R_{j}> \frac{\ell }{2}$ we use the growth Lemma 3.8(a), with ${1/q_0}=0$, which, remembering Remark B.1, implies

$$\begin{aligned} \begin{aligned} \sum _{\left\{ j\;:\; R_{j}>\frac{\ell }{2}\right\} }\sum _{(B,i)\in \Sigma _{\ell ,j}}Z_{B,i}&\le \sum _{\left\{ j:\; R_{j}> \frac{\ell }{2}\right\} } \sum _{(B,i)\in \Sigma _{\ell ,j}}C_{\#}\int _{\Phi _{\ell {\tau }_-/2}W_{B,i}}J^s_{\Phi _{\ell {\tau }_-/2}W_{B,i}}\Phi _{\ell {\tau }_-/2}\\&\le \sum _{W_B\in \mathcal {I}_{\ell {\tau }_-/2}(W)} C_{\#}\int _{W_{B}}J^s_{W_{B}}\Phi _{\ell {\tau }_-/2} \\&\le \sum _{W_B\in \mathcal {I}_{\ell {\tau }_-/2}(W)} C_{\#}L_0|J^s_{W_{B}}\Phi _{\ell {\tau }_-/2}|_{\mathcal {C}^0}\le C_{\#}\lambda ^{\ell {\tau }_-/2}~. \end{aligned} \end{aligned}$$

Since $\lambda > \Lambda ^{-1}$, the lemma follows by choosing $\rho = \rho _{*}^{1/2}$.