Proof. For any $z \in Z \times Z$ , $g_t z g_{-t} \to _{t \to -\infty } 0$ . So the claim follows from the Mautner phenomenon; see, for example, [Reference Einsiedler and WardEW, Prop 11.18].

Proof. Let $\pi : {\mathcal L} \times {\mathcal L} \to {{\mathcal L}}$ be the projection onto the first factor, and let $\nu = \pi _* \theta .$ We have $\nu = m_{{{\mathcal L}}}$ by (3.1) and the properties of the ergodic decomposition. For each $x \in {{\mathcal L}}$ , let $\Omega _x {\, \stackrel {\mathrm {def}}{=}\, } \pi ^{-1}(x) = \{x\} \times {{\mathcal L}}$ be the fiber, and let $\theta _x$ be the fiber measure appearing in the disintegration $\theta = \int _{{{\mathcal L}} } \theta _x \, d\nu (x)$ . Then Z acts on $\Omega _x$ via the second factor in $Z \times Z$ , and $\theta _x$ is Z-invariant and ergodic by the definition of the ergodic decomposition and Proposition 3.1(i) (with $H=Z$ ).

It follows from Proposition 3.1(v) (with $\mathbb {G} = P \ltimes (Z \times Z)$ ) that $\theta $ is P-invariant. To prove ergodicity, let $f \in L^2({{\mathcal L}} \times {{\mathcal L}}, \theta )$ be a P-invariant function. By Proposition 4.1, f is $Z \times Z$ -invariant. For each $x \in {{\mathcal L}}$ , let $f_x {\, \stackrel {\mathrm {def}}{=}\, } f|_{\Omega _x}$ . There is ${{\mathcal L}}_0 \subset {{\mathcal L}}$ such that $m_{{{\mathcal L}}}({{\mathcal L}}_0)=1$ , and for every $x \in {{\mathcal L}}_0$ , $f_x $ belongs to $ L^2(\Omega _x, \theta _x)$ and is Z-invariant. Hence, by ergodicity, there is $\bar {f}: {{\mathcal L}}_0 \to {\mathbb {R}}$ such that for every $ x\in {{\mathcal L}}_0$ , $\bar {f}(x)$ is the $\theta _x$ -almost-sure value of $f_x$ . Since f is P-invariant for the diagonal action of P, $\bar {f}$ is P-invariant for the action of P on ${{\mathcal L}}$ . By ergodicity of $P \circlearrowright ({{\mathcal L}}, m_{{\mathcal L}})$ , $\bar f$ is $\nu $ -a.e. constant, and thus, f is $\theta $ -a.e. constant.

The last assertion follows from Proposition 3.4.

Proof of Theorem 1.2.

Let $Z_0 = \mathrm {span}_{{\mathbb {R}}}(z_0)$ be a one-dimensional connected real Rel subgroup. Assume that (1) fails, so that the action of $Z_0$ on $\left ({{\mathcal L}}, m_{\mathcal {L}} \right )$ is not mixing of all orders. Then, by Proposition 3.6, it is not ergodic. Let $\theta $ be the relatively independent self-joining over ${{\mathcal L}} /\!\!/ Z_0$ . Applying Propositions 3.3 and 3.4, we have that $\theta \neq m_{{{\mathcal L}}} \times m_{{{\mathcal L}}}$ and $\Delta _{{{\mathcal L}}} \subset \mathrm {supp} \, \theta $ . Applying Proposition 4.2 and Statement A, we have that there is a G-invariant open subset U of full $\theta $ -measure such that $U \cap \mathrm {supp} \, \theta $ is contained in the isomorphic image of an affine complex-linear manifold whose (real) dimension is strictly smaller than $2 \dim \bar {\mathcal {H}}$ , and $\theta $ is obtained from Lebesgue measure on this complex-linear manifold by the cone construction.

We claim that the set

$$ \begin{align*}U_1 {\, \stackrel{\mathrm{def}}{=}\, } \{q \in {\mathcal{H}}: (q,q) \in U\} \end{align*} $$

is of full measure for $(\pi _{1})_* \theta $ , where $\pi _1 : {{\mathcal L}} \times {{\mathcal L}} \to {{\mathcal L}}$ is the projection onto the first factor. Indeed, the measure $\theta $ is invariant under $Z_0 \times \{ \mathrm {Id} \}$ , and hence, so is its support. Since $Z_0$ acts by homeomorphisms where defined, and using property (v) in Theorem 5.1 and Statement A, we have that the set U is also $Z_{0} \times \{\mathrm {Id}\}$ -invariant. Thus, for any $Z_{0}$ -ergodic measure, it is either null or conull. Thus, if $q \notin U_1$ and q is generic for the measure $\mu _{T(q)}$ appearing in (3.1), then $\mu _{T(q),q}$ assigns measure zero to U, where $\mu _{T(q),q}$ is the measure on $\mathrm {supp}\, \theta $ defined by $\mu _{T(q),q}(A)=\mu _{T(q)}(\{q':(q',q) \in A\})$ . If this were to happen for a positive measure of q, it would follow from (3.1) and the fact that $\mu _{T(q)}\times \mu _{T(q)}=\int \mu _{T(q'),q'}d\mu _{T(q)}$ that U does not have full measure for $\theta $ .

For $q \in U_1$ , let $N_q$ denote the connected component of $ U \cap \pi _1^{-1}(q) \cap \mathrm {supp} \, \theta $ containing $(q,q)$ . Recall from item (ii) of Theorem 5.1 that ${{\mathcal L}}$ is locally the intersection of $\bar {{\mathcal L}} = \varphi (\mathcal {N})$ , a $\mathbb {C}$ -linear subset of $\bar {\mathcal {H}}$ , with the quadric hypersurface ${\mathcal {H}}$ consisting of surfaces of unit area. In this proof, we will call such a subset an affine submanifold of ${\mathcal {H}}$ . Similarly, we will say that a subset of ${{\mathcal L}} \times {{\mathcal L}}$ is an affine submanifold if it is locally the intersection of a $\mathbb {C}$ -linear subset of $\bar {\mathcal {H}} \times \bar {\mathcal {H}}$ with ${\mathcal {H}} \times {\mathcal {H}}$ . Since the fibers $\pi _1^{-1}(q)$ are also affine submanifolds of ${{\mathcal L}} \times {{\mathcal L}}$ , we have that the $N_q$ are affine submanifolds contained in $\pi _1^{-1}(q) \cong {{\mathcal L}}$ , so we can identify them with invariant submanifolds in ${{\mathcal L}}$ (which we continue to denote by $N_q)$ . With this notation, we have $q \in N_q$ .

The mapping $q \mapsto T_q(N_q)$ is locally constant; that is, letting $V \subset \bar {\mathcal {H}}$ and $\mathcal {V} \subset \bar {\mathcal {H}}_{\mathrm {m}}$ be open sets such that $\pi |_{\mathcal {V}}: \mathcal {V} \to V$ is a homeomorphism and $q \in V$ , the map $q \mapsto \mathrm {dev} \circ \pi |_{\mathcal {V}}^{-1} (q) $ sends a neighborhood of q in $N_q$ to an affine subspace $W = W_{q, \mathcal {V}}$ of $H^1(S, \Sigma; {\mathbb {R}}^2)$ , and the linear subspaces tangent to W are the same for all $q \in V$ . Since $m_{{{\mathcal L}}} \times m_{{{\mathcal L}}}$ is the unique P-invariant ergodic measure on ${{\mathcal L}} \times {{\mathcal L}}$ of full support, we have $\dim N_q < \dim {{\mathcal L}}$ for every $q \in U_1$ .

Let $\bar \pi _1: \bar {\mathcal {H}} \times \bar {\mathcal {H}}$ denote the projection on the first factor, let $\bar N_q$ denote the connected component of $U \cap \bar \pi _1^{-1}(q) \cap \bar {{\mathcal L}} $ containing $(q,q)$ , and let

$$ \begin{align*}\mathfrak{N}_q {\, \stackrel{\mathrm{def}}{=}\, } T_q(\bar N_q ) \end{align*} $$

(the tangent space to $\bar N_q$ at q, thought of as a subset of the tangent space $T_q(\bar {{\mathcal L}})$ ). That is, surfaces in $\bar N_q$ are obtained by moving locally in the affine space W defining $\theta $ , as in item (iv) of Theorem 5.1, without requiring that the surfaces in $\bar N_q$ have area one. Then we either have $\dim \bar N_q = \dim N_q$ or $\dim \bar N_q = \dim N_q+1$ , depending on whether or not moving in W can change the area of a surface. The assignment $q\mapsto \mathfrak {N}_q $ defines a proper flat sub-bundle of the tangent bundle $T(\bar {{\mathcal L}})$ . Flat sub-bundles of $T(\bar {{\mathcal L}})$ were classified in [Reference WrightWr2]. According to [Reference WrightWr2, Thm. 5.1], $\mathfrak {N}_q \subset \mathfrak {R}_{{{\mathcal L}}}$ for each q, and $\mathfrak {N}_q$ is a complex linear subspace which is locally constant. Since $\mathfrak {R}_{{{\mathcal L}}}$ is acted on trivially by monodromy, we in fact have that $\mathfrak {N}_q$ is independent of q, and we denote it by $\mathfrak {R}_1$ . The leaves $\mathfrak {R}_1(q)$ coincide with $\bar N_q$ for each q since $\bar N_q$ is connected and of the same dimension. Since Rel deformations do not affect the area of the surface, we see that $\bar {N}_q = N_q$ . In particular, $\mathfrak {R}(q)$ is closed for each q.

By (3.1), for a.e. q, $N_q$ is a connected component of the support of the ergodic component $(m_{{{\mathcal L}}})_q$ , which is a probability measure, and in particular,

$$ \begin{align*}(m_{{{\mathcal L}}})_q(N_q)<\infty, \ \ \text{ for a.e. } q. \end{align*} $$

Since $Z_0$ acts ergodically with respect to $(m_{{{\mathcal L}}})_q$ , we have that almost surely $N_q =\mathfrak {R}(q)$ . Since the measure $(m_{{{\mathcal L}}})_q$ is affine in charts, it is a scalar multiple of the translation-invariant measure on $\mathfrak {R}(q)$ , and thus, the volume $V_q$ of $\mathfrak {R}(q)$ (with respect to its translation-invariant measure) is almost surely finite. It is clear that the function $q \mapsto V_q$ is U-invariant, and by ergodicity, it is constant almost surely. Thus, assertion (2) of Theorem 1.2 holds, with $Z_1 = \mathfrak {R}_1 \cap Z_{{{\mathcal L}}}$ .

Proof. Since $\frac {c_1}{c_2} \notin {\mathbb {Q}}$ , the trajectory $\{{\mathrm {Rel}}_{tv}(x) : t \in {\mathbb {R}}\}$ is not closed, let $\mathcal {L}$ denote its closure. We claim that the tangent space to $\mathcal {L}$ is not contained in Z. Let $\sigma _1$ denote a saddle connection from p to q in $C_1$ and let $\sigma _2$ denote a saddle connection from q to p in $C_2$ . Let $\sigma $ be the concatenation. Then $\sigma $ represents an absolute homology class because it goes from p back to p, and it is nontrivial because the vertical component of its holonomy on x is nonzero. If we consider the restriction of the rel-action to $C_1 \cup C_2$ , then it only affects the twist parameters, which is a 2-dimensional space. This space can be generated by the horizontal holonomy of $\sigma _1$ and the horizontal holonomy of $\sigma _2$ . Since $\frac {c_1}{c_2} \notin {\mathbb {Q}}$ , this restricted action does not give a closed orbit. So the tangent space to $\mathcal {L}$ contains directions, which continuously affect the holonomy of $\sigma $ . Since $\sigma $ is an absolute period, we see that the tangent space to $\mathcal {L}$ is not contained in Z.

Proof. It will be easier to follow the proof while consulting Figures 1 (before) and 2 (after). Given a polygonal presentation for M, we give a polygonal presentation for $M'$ . Let M be a polygon representation for M in which the cylinder C is represented by a parallelogram P (in Figure 1, the large rectangle in the center of the presentation), with two horizontal sides of length c, nonhorizontal sides identified to each other, and the singular points $\xi _i$ , $\xi _j$ on adjacent corners of P. Thus, the nonhorizontal sides of P represent a saddle connection $\sigma $ on M connecting $\xi _i$ to $\xi _j$ . We consider the two nonhorizontal sides of P as distinct and label them by $\sigma _1, \sigma _2$ . Let $P'$ be a parallelogram with sides parallel to those of P, where the horizontal sides have length w and the nonhorizontal sides are longer than the ones on P (in Figure 2, $P'$ is to the right of P).

Figure 1 The surface M has a cylinder of circumference c, and its boundary components see only the singularities $\xi _i$ and $\xi _j$ (denoted by $\circ $ and $\bullet $ ). The edges not labeled by $\triangle $ are connected to $M \smallsetminus C$ .

Figure 2 To obtain $M'$ from M, glue in a torus (rectangle on the right). This transforms C into a cylinder $C^{\prime }_1$ of circumference $c+w$ , and adds a horizontal cylinder $C^{\prime }_2$ of circumference w. Edges not labeled by $\triangle $ , $\square $ , / or the color green are attached to $M' \smallsetminus (C^{\prime }_1 \cup C^{\prime }_2)$ .

Label the two horizontal sides of $P'$ by $h^{\prime }_1$ and $h^{\prime }_2$ , and identify them by a translation. Partition the nonhorizontal sides of $P'$ into two segments. The segments $\sigma ^{\prime }_1, \sigma ^{\prime }_2$ are parallel to each other and have the same length as $\sigma _1, \sigma _2$ and start at a corner of P. The segments $\gamma ^{\prime }_1, \gamma ^{\prime }_2$ comprise the remainder of the nonhorizontal sides of $P'$ (and in particular, have the same length). Identify $\gamma ^{\prime }_1$ to $\gamma ^{\prime }_2$ by a translation, and identify $\sigma ^{\prime }_1, \sigma ^{\prime }_2$ to $\sigma _1, \sigma _2$ by a translation so that each $\sigma ^{\prime }_i$ is attached to the $\sigma _j$ with the opposite orientation. Let $M'$ be the translation surface corresponding to this presentation. It is clear that $M'$ has the required properties.

Proof of Proposition 8.3.

The proof is by induction on $\sum a_i$ .

Base of induction: The base case is the stratum ${{\mathcal {H}}}(a_1, 0^{s})$ – that is, one singular point (removable or non-removable) of order $a_1$ , and some number $s \geq 1$ of removable singular points. In this case, we take a surface in ${{\mathcal {H}}}(a_1)$ which is made of one horizontal cylinder. We label the singular point by $\xi _1$ and place additional removable singular points $\xi _2, \ldots , \xi _{s+1}$ in the interior of the cylinder, at different heights (so that the resulting surface has no horizontal saddle connections between distinct singularities) and so that $\xi _i$ and $\xi _j$ are on opposite sides of a cylinder.

Inductive step: Suppose ${{\mathcal {H}}}' = {{\mathcal {H}}}(a_1, \ldots , a_k)$ is our stratum, where at least two of the singularities are non-removable. Let $p', q'$ be labels of singular points for surfaces in ${{\mathcal {H}}}'$ , corresponding to indices $i \neq j$ . To simplify notation, assume $i=1, j=2$ . There are three cases to consider: $a_i = a_j=0$ , or one of $a_i, a_j$ are positive, or both are positive.

If $a_i = a_j=0$ , then by assumption, $k \geq 4$ . We take a cylinder C on a fully stably completely periodic surface M in ${{\mathcal {H}}} = {{\mathcal {H}}}(a_1, \ldots , \hat {a}_i, \ldots , \hat {a}_j, \ldots , a_k)$ . The notation $\hat {a}_i$ means that the symbol should be ignored – that is, on a stratum of the same genus with $k-2 \geq 2$ singular points obtained by removing two removable singular points. We place two singular points marked $p', q'$ in the interior of C at different heights. If $a_i>0$ and $ a_j=0$ is zero, we take a fully stably periodic surface M in ${{\mathcal {H}}}(a_1, \ldots , a_i, \ldots , \hat {a}_j, \ldots , a_k)$ , find a cylinder C on M whose boundary component is made of saddle connections starting and ending at $\xi _i$ , and place a marked point labeled $\xi _j$ in the interior of C. If $a_i$ and $a_j$ are both positive, we use the induction hypothesis to find a surface $M \in {{\mathcal {H}}}(a_1, \ldots , a_i-1, \ldots , a_j-1, \ldots , a_k)$ with a cylinder whose boundary components see $\xi _i$ and $\xi _j$ , and we perform the surgery in Lemma 8.4 to this cylinder.

Proof. Once again, we encourage the reader to consult Figures 3 and 4.

Figure 3 First option for $M'$ in Lemma 8.5. Attaching the subsurface on the right increases the genus by 2. Unlabeled edges are attached to $M' \smallsetminus (C_1 \cup C_2 \cup C_3)$ .

Figure 4 Second option for $M'$ , with a different spin.

In Lemma 8.4, we made a slit in M, running through P from top to bottom, and glued in a torus with a slit. In this case, we make an identical slit, this time gluing in a genus two surface with a slit. This surface is presented in Figures 3 and 4 as made up of three rectangles. It is straightforward to check that $M' \in {{\mathcal {H}}}'$ and that it has cylinders satisfying the desired properties. It remains to check the final assertion about the parity of the spin structure.

Recall from [Reference Kontsevich and ZorichKZ, eqn. (4)] that where defined, the spin structure of a surface M of genus g can be computed as follows. Let $\alpha _i, \beta _j $ (where $1 \leq i,j \leq g$ ) be a symplectic basis for $H_1(M)$ , realized explicitly as smooth curves on M. This means that all of these curves are disjoint, except for $\alpha _i$ and $\beta _i$ which intersect once. For each curve $\gamma $ , let $\mathrm {ind}(\gamma )$ be the turning index – that is, the total number of circles made by the tangent vector to $\gamma $ , as one goes around $\gamma $ . The parity of M is then the parity of the integer $\sum _{i=1}^g (1 +\mathrm {ind}(\alpha _i)) (1 +\mathrm {ind}(\beta _i)) $ . It is shown in [Reference Kontsevich and ZorichKZ] that this number is well defined (independent of the choice of the symplectic basis) when all the singular points have even order.

Figure 5 Modifying the symplectic basis. Gluings as in Figure 3.

Suppose M has genus g and is equipped with a symplectic basis. Since any non-separating simple closed curve can be completed to a symplectic basis, we can assume that $\alpha _1$ is the core curve of C, and the other curves in the basis do not intersect the saddle connection from p to q passing through C. We construct a symplectic basis for $M'$ in both cases, by modifying $\alpha _1$ , keeping $\alpha _2, \ldots , \alpha _g, \beta _1, \ldots , \beta _g$ , and adding new curves $\alpha _{g+1}, \alpha _{g+2}, \beta _{g+1}, \beta _{g+2}$ . The modified curves are shown in Figures 5 and 6, and the reader can easily check that these new curves still form a symplectic basis and that these two choices add two numbers of different parities to the spin structure.

Proof of Proposition 8.2.

The proof will be case-by-case. Here are the cases:

1. (i) ${{\mathcal {H}}}(1,1)$ .

2. (ii) All the $a_i$ are nonzero, and ${{\mathcal {H}}}$ is connected.

3. (iii) All the $a_i$ are nonzero, and ${{\mathcal {H}}}$ has two connected components distinguished by spin.

4. (iv) All the $a_i$ are nonzero, and ${{\mathcal {H}}}$ has two connected components distinguished by hyperellipticity.

5. (v) All the $a_i$ are nonzero, and ${{\mathcal {H}}}$ has three connected components.

6. (vi) Some of the $a_i$ are zero.

Case (i). There is just one connected component, and the desired surface is a Z-shaped surface, with three horizontal cylinders $C_1, C_2, C_3$ of circumferences $c_1, c_1+c_3, c_3$ , where $C_1, C_3$ are simple. We put all of the removable singular points in the interior of $C_3$ and choose $c_1, c_3$ so that $c_1/(c_1+c_3) \notin {\mathbb {Q}}$ . It is clear that with these choices, the conditions are satisfied.

Case (ii). The stratum ${{\mathcal {H}}}$ is connected, and we have at least two singularities of positive order. So with no loss of generality, they are labelled 1 and 2. The result follows from Lemma 8.4, applied to a surface in ${{\mathcal {H}}}(a_1-1, a_2-1, a_3, \ldots , a_k)$ and taking $w\notin c\mathbb {Q}$ , so that $w/(c+w) \notin {\mathbb {Q}}$ .

Case (iii). We apply the surgery in Lemma 8.5, with $w_1/w_2 \notin {\mathbb {Q}}$ . Namely, if p and q are labelled $i, j$ , we let $b_i = a_i-2$ , $b_j = a_j-2$ and $b_\ell = a_\ell $ for $\ell \neq i,j$ .

Case (iv). There are two connected components. One is hyperelliptic; one is not. This means that $a_1 = a_2$ and either $g=3$ (in which case $a_1 = a_2 = 2$ ) or $g \geq 4$ is even (in which case $a_1 = a_2 = g-1$ ). In this case, we give explicit surfaces, one in each connected component. The first surface (the ${{\mathcal {H}}}(2,2)$ case is shown in Figure 7) is a ‘staircase’ surface made of gluing $2g$ rectangles to each other. The rectangles are labelled $(k, B)$ and $(k, T)$ for $k = 1 , \ldots , g$ . The top (respectively, bottom) of $(k, B)$ is glued to the bottom (resp., top) of $(k,T)$ for $k=1, \ldots , g$ , and the left (resp., right) of $(k, T)$ is glued to the right (resp., left) of $(k+1, B)$ for $k=1, \ldots , g-1$ . The horizontal sides of $(1, B)$ are glued to each other, as are the horizontal sides of $(g, T)$ . This surface is hyperelliptic since it has a hyperelliptic involution rotating each rectangle around its midpoint, and this involution swaps the singularities (see [Reference Kontsevich and ZorichKZ, Remark 3]). The second surface is obtained as follows. We first construct a hyperelliptic surface in ${{\mathcal {H}}}(a_1-2, a_2-2)$ as in the previous paragraph. Then we perform the surgery described in Lemma 8.5. The resulting surface has a horizontal cylinder intersecting three vertical cylinders and thus, by [Reference LindseyLi, Lemma 2.1], is not hyperelliptic. See Figure 8 for an example in ${{\mathcal {H}}}(2,2)$ . In both of these constructions, there are no restrictions on the sidelengths of the rectangles, and we can easily arrange that two of the circumferences are incommensurable.

Figure 7 A surface in ${\mathcal {H}}^{hyp}(2,2)$ .

Figure 8 A surface in ${\mathcal {H}}^{nonhyp}(2,2)$ .

Case (v). In this case, $a_1 = a_2 = g-1$ for $g \geq 5$ odd. Applying the argument in Case (iii), we obtain the required surfaces in the odd and even connected components. To obtain the required surface in the hyperelliptic component, we use the ‘staircase surface’ described in Case (iv).

Case (vi). Assume with no loss of generality that the removable singularities are labelled $k'+1, \ldots , k$ for some $k' \geq 2$ , and let ${{\mathcal {H}}}' = {{\mathcal {H}}}(a_1, \ldots , a_{k'})$ . Note that the singularities p and q have label in $\{1, \ldots , k'\}$ . Apply the preceding considerations to obtain a surface in ${{\mathcal {H}}}'$ with the required cylinders. By examining the proof in all preceding cases, one sees that the number of horizontal cylinders on this surface is at least three; that is, there is at least one cylinder $C_3$ which is distinct from the cylinders $C_1, C_2$ , and we modify M by adding $k-k'$ points in general position in the interior of $C_3$ , to obtain the desired surface.

Proof of Proposition 9.2.

Write $B {\, \stackrel {\mathrm {def}}{=}\, } B\left (q, \frac {1}{L^{5}} \right ), \, A {\, \stackrel {\mathrm {def}}{=}\, } g_{-\ell }(B)$ (note that A and B both depend on L and q, but we suppress this from the notation). Let $K'$ be the 1-neighborhood of K, which is a pre-compact subset of ${\mathcal {H}}$ by Proposition 9.3(b). Since $\mathrm {diam}(B) \leq \frac {2}{L^{5}}$ , Proposition 9.3(a) implies that

(9.4) $$ \begin{align} \mathrm{diam}(A) \leq \frac{2}{L^3}. \end{align} $$

It follows from (9.3) that $A \cap K \neq \varnothing $ , and therefore, $A \subset K'$ . Since

(9.5) $$ \begin{align} \max_{j=1, \ldots, L} \|je^{-\ell} z_0 \| \leq \frac{1}{L} \|z_0\| \to_{L \to \infty} 0, \end{align} $$

for all large enough L (depending on $K'$ ), we have that $\mathrm {Rel}_{je^{-\ell }z_0}(q')$ is defined for $q' \in K'$ . Since $q^{\prime }_1 {\, \stackrel {\mathrm {def}}{=}\, } \mathrm {Rel}_{je^{-\ell }z_0}\circ g_{-\ell }(q_1)$ is defined for $q_1 \in B$ , we have from (2.3) that $T^j(q_1) =\mathrm {Rel}_{jz_0}(q_1) = g_{\ell }(q^{\prime }_1) $ is also defined. This proves that the maps $T, T^2, \ldots , T^L$ are all defined on B.

Furthermore, this computation shows that $T^j(B) = g_\ell (\mathrm {Rel}_{je^{-\ell }z_0} (A))$ , and so by Proposition 9.3(a), it suffices to show that

$$ \begin{align*}L^2 \cdot \mathrm{diam} \left(\mathrm{Rel}_{je^{-\ell}z_0} (A) \right) \to_{L \to \infty } 0.\end{align*} $$

Taking into account (9.4) and (9.5), it suffices to show that for any compact $K'$ , there are positive $\varepsilon , C$ such that for any $q_0, q_1 \in K'$ with $\mathrm {dist}(q_0, q_1) < \varepsilon $ , and any $z \in Z$ with $\|z\|< \varepsilon $ , we have

(9.6) $$ \begin{align} \mathrm{dist}(\mathrm{Rel}_{z}(q_0, q_1)) \leq C \, \mathrm{dist}(q_0, q_1). \end{align} $$

Informally, this is a uniform local Lipschitz estimate for the family of maps defined by small elements of Z.

To see (9.6), let $\varepsilon _1$ be small enough so that for any $q \in K'$ , the ball $B(q, 2\varepsilon _1)$ is contained in a neighborhood which is evenly covered by the map $\pi : {\mathcal {H}}_{\mathrm {m}} \to {\mathcal {H}}$ , and let C be a bound as in Proposition 9.3(c), corresponding to the compact set which is the $2\varepsilon _1$ -neighborhood of $K'$ . Let $\varepsilon < \varepsilon _1$ so that for any $z \in Z$ with $\|z\|< \varepsilon $ and any $q \in {\mathcal {H}}$ , $\mathrm {dist}(q, \mathrm {Rel}_{z}(q))< \varepsilon _1$ . If $\mathrm {dist}(q_0, q_1)<\varepsilon $ , then the path $\gamma $ realizing their distance (see Proposition 9.3(d)) is contained in a connected component $\mathcal {V}$ of $\pi ^{-1}\left (B(q_0, \varepsilon _1) \right )$ . Let

$$ \begin{align*}\bar \gamma: [0,1] \to H^1(S, \Sigma; {\mathbb{R}}^2) , \ \ \ \bar \gamma (t) {\, \stackrel{\mathrm{def}}{=}\, } \mathrm{dev}(\gamma(t)) - \mathrm{dev}(\gamma(0)),\end{align*} $$

let

$$ \begin{align*}\gamma_1 {\, \stackrel{\mathrm{def}}{=}\, } \mathrm{Rel}_{z}\circ \gamma,\end{align*} $$

and analogously define

$$ \begin{align*}\bar \gamma_1 : [0,1] \to H^1(S, \Sigma; {\mathbb{R}}^2), \ \ \ \bar \gamma_1(t) {\, \stackrel{\mathrm{def}}{=}\, } \mathrm{dev}(\gamma_1(t)) - \mathrm{dev}(\gamma_1(0)).\end{align*} $$

By choice of $\varepsilon $ and $\varepsilon _1$ , the curve $\gamma _1$ also has its image in $\mathcal {V}$ . Since $\mathrm {Rel}_{z}$ is expressed by $\mathrm {dev}|_{\mathcal {V}}$ as a translation map, the curves $\bar \gamma , \bar \gamma _1$ are identical maps. When computing $\mathrm {dist}(\mathrm {Rel}_{z}(q_0), \mathrm {Rel}_{z}(q_1))$ via (9.2), an upper bound is given by computing the path integral along the curve $\gamma _1$ . We compare this path integral along $\gamma _1$ , with the path integral along $\gamma $ giving $\mathrm {dist}(q_0, q_1)$ . In these two integrals, for any t, the tangent vectors $\gamma '(t), \gamma ^{\prime }_1(t)$ are identical elements of $H^1(S, \Sigma; {\mathbb {R}}^2))$ for all t, but the norms are evaluated using different basepoints. Since these basepoints are all in the $2\varepsilon _1$ -neighborhood of $K'$ , by choice of C, we have $\|\gamma _1'(t)\|_{\gamma _1(t)} \leq C \|\gamma '(t) \|_{\gamma (t)}$ for all t. This implies (9.6).

Proof. The argument is standard; we sketch it for lack of a suitable reference.

We first claim that given a compact K, there is N so that for any $r \in (0, 1)$ , the largest r-separated subset of any ball of radius $2r$ has cardinality at most N. Indeed, this property is true for Euclidean space (for any r) by a simple volume argument and is invariant under biLipschitz maps (up to changing the constant N). For any compact K, let $K_1$ be the $1$ -neighborhood of K in ${\mathcal {H}}$ , which is also compact by Proposition 9.3(b) and contains all balls in ${\mathcal {H}}$ which are centered in K and have radius $r < 1$ . Now the claim holds by Proposition 9.3(c).

We now inductively choose the $\mathcal {F}_i$ . Let $\mathcal {F}_1$ be a maximal collection of disjoint balls of radius r with centers in G. For $i \geq 2$ , suppose $\mathcal {F}_1, \ldots , \mathcal {F}_{i-1}$ have been chosen, let $G_i {\, \stackrel {\mathrm {def}}{=}\, } G \smallsetminus \bigcup _{j=1}^{i-1} \bigcup \mathcal {F}_j$ , and let $\mathcal {F}_i$ be the maximal collection of disjoint balls of radius r with centers in $G_i$ . Clearly, $G \supset G_1 \supset \cdots \supset G_N$ , and we need to show that $G_{N+1} = \varnothing $ . Since $\mathcal {F}_i$ is maximal, for any $x \in G_i$ there is $x'$ which is the center of one of the balls of $\mathcal {F}_i$ , so that $d(x,x') < 2r$ . If $x \in G_{N+1} \neq \varnothing $ , then the ball $B(x, 2r)$ contains $x_1, \ldots , x_N$ such that $x_i$ is the center of one of the balls of $\mathcal {F}_i$ . For $i'>i$ , $d(x_i, x_{i'}) \geq r$ since $x_{i'} \in G_{i'}$ . This contradicts the property of N from the preceding paragraph.

Proof. Sketch of proof.

We fix n. Observe that (9.7) says that for almost every point x, for all $\varepsilon>0$ , $e^{-\varepsilon r^{-\frac 1 n}}$ is $o(\mu (B(x,r)))$ , as $r \to 0$ . Let K be a compact subset of ${\mathcal {H}}$ , and let $d = \dim {\mathcal {H}}$ . For all small enough r, we can cover K by $O\left (r^{-d} \right )$ balls of radius r; we let $\mathcal {B}_r=\{B(x_i,r)\}$ denote such a finite collection of balls. If $B = B(x,r) $ , then any element of $\mathcal {B}_{\frac {r}3}$ that contains x is contained in B and hence has measure at most $\mu (B)$ . For $k \in {\mathbb {N}}$ , let

$$ \begin{align*}\mathcal{E}_k=\left\{B \in \mathcal{B}_{\frac{2^{-k}}{3}}:\mu(B)<e^{-{2^{k/(2n)}}} \right\},\end{align*} $$

so that

(9.8) $$ \begin{align} \mu\left(\bigcup_{B \in \mathcal{E}_k} B\right) = O\left( 2^{kd} e^{-2^{k/(2n)}}\right). \end{align} $$

Since K was arbitrary, it suffices to show that a.e. $x \in K$ belongs to at most finitely many of the sets $\mathcal {E}_k$ ; this follows from (9.8) and Borel-Cantelli.

Proof of Theorem 9.1.

We assume that the entropy $h = h_\mu (T)$ satisfies $h>0$ , and we will derive a contradiction. Using Proposition 9.6(3), we choose a partition $\mathcal {P}=\{P_i\}_{i=1}^k$ so that $\mu (\partial P_i)=0$ for each i and $h_\mu (T, \mathcal {P})> \frac {h}{2}$ . Choose K compact so that $\mu (K)>\frac 3 4 $ and

(9.9) $$ \begin{align}\underset{t\to \infty}{\limsup} \, \mu\left(g_{t}(K)\right)>\frac 3 4. \end{align} $$

A compact set with this property exists by the nondivergence assumption (9.1). Let N and $r_0$ be as in Proposition 9.4 for this choice of K. Using the Shannon-McMillan-Breiman theorem, let $L_0$ be large enough so that for all $L> L_0$ , the set

$$ \begin{align*}W {\, \stackrel{\mathrm{def}}{=}\, } \left\{q: \frac{-\log (\mu(P_L(q)))}{L} \leq \frac{h}{2} \right\} \end{align*} $$

satisfies

(9.10) $$ \begin{align} \mu(W) < \frac{1}{6N}. \end{align} $$

Our goal will be to choose some $L>L_0$ for which we have a contradiction to (9.10).

Below, we will simplify notation by writing $ B\left (q, \frac {1}{L^{5}} \right )$ as $B_{q,L}$ or simply as B. Let

$$ \begin{align*}G_L {\, \stackrel{\mathrm{def}}{=}\, } \left\{ q : \mu \left( B_{q, L} \cap W \right)> \frac{\mu(B_{q, L})}{2} \right\}. \end{align*} $$

We will show below that

(9.11) $$ \begin{align} \text{ there are arbitrarily large }L\text{ for which } \mu(K \cap G_{L})> \frac{1}{3}. \end{align} $$

We first explain why (9.11) leads to a contradiction with (9.10). Let $L>L_0$ be large enough so that $\mathrm {diam}(B) \leq \frac {2}{L^5} < r_0$ and $\mu (K \cap G_L)> \frac {1}{3}$ , and let

$$ \begin{align*}\mathcal{C} {\, \stackrel{\mathrm{def}}{=}\, } \left\{ B_{q, L} : q \in G_{L} \right\}. \end{align*} $$

By Proposition 9.4, there is a subcollection $\mathcal {F} \subset \mathcal {C}$ , consisting of disjoint balls, so that

$$ \begin{align*}\mu\left(K \cap G_L \cap \bigcup \mathcal{F} \right) \geq \frac{1}{N} \, \mu(K \cap G_L)> \frac{1}{3N}. \end{align*} $$

Then we have

$$ \begin{align*}\mu(W) \geq \mu\left(W \cap \bigcup \mathcal{F} \right) = \sum_{B \in \mathcal{F}} \mu(W \cap B)> \sum_{B \in \mathcal{F}} \frac{\mu(B)}{2} \geq \frac{\mu \left(\bigcup \mathcal{F} \right)}{2}\geq \frac{1}{6N}, \end{align*} $$

where the equality follows from the disjointness of $\mathcal {F}$ and the strict inequality follows from the definitions of $G_L$ and $\mathcal {C}.$ This gives the desired contradiction to (9.10).

It remains to show (9.11). Choose $\varepsilon>0$ so that

(9.12) $$ \begin{align} 21 \, \varepsilon \, \log(k)< \frac{h }{2}. \end{align} $$

Given any $L_1$ , let $L>L_1$ , and let $X_0 = X_0(L) \subset K$ such that $\mu (X_0) \geq \frac {1}{2},$ and so that for any $q \in X_0$ , we have (9.3). Such L and $X_0$ exist by (9.9). Using Lemma 9.5 with $n=5$ , we can take L large enough so that

(9.13) $$ \begin{align} \mu\left( X_1\right)>\frac {99}{100}, \ \ \ \text{ where } X_1{\, \stackrel{\mathrm{def}}{=}\, } \left\{q :\mu\left(B_{q,L} \right ) >k^{-\varepsilon L} \right \} , \end{align} $$

and by making L even larger, we can assume that

(9.14) $$ \begin{align} k^{-10 \varepsilon L} < \frac{1}{2}. \end{align} $$

Now choose $r>0$ so that

(9.15) $$ \begin{align} \mu(V ) < \varepsilon, \ \ \text{ where } \ V {\, \stackrel{\mathrm{def}}{=}\, } \left\{y: \mathrm{dist}\left(y,\bigcup_{i=1}^k \partial P_i\right)<r\right\} .\end{align} $$

This is possible because $\mu \left ( \bigcup _i \partial P_i \right )=0. $

We claim that

(9.16) $$ \begin{align} \mu(X_2)> \frac 2 5, \ \ \ \text{ where } X_2 {\, \stackrel{\mathrm{def}}{=}\, } \left\{q \in X_0 : |\{0\leq i \leq L: T^iq \in V\}|<10\varepsilon L \right\}. \end{align} $$

To see this, define

$$ \begin{align*}E {\, \stackrel{\mathrm{def}}{=}\, } \{q:|\{0\leq i \leq L: T^iq \in V\}|\geq 10\varepsilon L\},\end{align*} $$

and let $\mathbf {1}_V$ denote the indicator function of V. Using (9.15), and since $\mu $ is T-invariant,

$$ \begin{align*}\varepsilon L>L\mu(V)=\sum_{i=1}^L\int \mathbf{1}_V\left(T^iq\right) d\mu\geq 10 \varepsilon L \mu(E).\end{align*} $$

Dividing through by $10L \varepsilon $ , we have $\mu (E)<\frac {1}{10}$ , giving (9.16).

Let

$$ \begin{align*}\beta {\, \stackrel{\mathrm{def}}{=}\, } k^{-20 \varepsilon L}, \ \ \ \text{ and write } \ \ \mathcal{P}^{(L)} {\, \stackrel{\mathrm{def}}{=}\, } \bigvee_{i=1}^LT^{-i}(\mathcal{P}).\end{align*} $$

For each q, we let $\mathcal {P}^{(L, B)}$ be the elements of $\mathcal {P}^{(L)}$ which intersect $B=B_{q,L}$ and partition $\mathcal {P}^{(L,B)}$ into two subcollections defined by

$$ \begin{align*}\mathcal{P}^{(q, L)}_{\mathrm{big}} {\, \stackrel{\mathrm{def}}{=}\, } \left\{P \in\mathcal{P}^{(L,B)} : \mu(P) \geq \beta \mu(B) \right \} \ \ \text{ and } \ \ \mathcal{P}^{(q, L)}_{\mathrm{small}} {\, \stackrel{\mathrm{def}}{=}\, } \mathcal{P}^{(L,B)} \smallsetminus \mathcal{P}^{(q, L)}_{\mathrm{big}}. \end{align*} $$

We claim that if $q \in X_2$ , then

(9.17) $$ \begin{align} \mu \left(B \cap \bigcup \mathcal{P}^{(q,L)}_{\mathrm{big}} \right)> \left(1-k^{-10 \varepsilon L} \right) \, \mu \left(B\right) \stackrel{(9.14)}{>} \frac{\mu(B)}{2}. \end{align} $$

To see this, we note that for q satisfying the conclusion of Proposition 9.2, the cardinality of $\mathcal {P}^{(L, B)}$ is at most $k^{|\{0 \leq i \leq L \, : \, T^i q \in V\} |}$ . Indeed, for such q, whenever $T^i q \notin V$ , $T^i\left (B\right )$ is contained in one of the $P_i$ (and for the other i, we use the obvious bound that $T^iq \in V$ , $T^i\left (B \right )$ could intersect all of the $P_i$ ). For $q \in X_2$ , we also have that $\beta ^{-1/2} \geq k^{|\{0 \leq i \leq L \, : \, T^i q \in V\} |}$ , and this implies that

$$ \begin{align*}\mu\left(B \cap \bigcup \mathcal{P}^{(q,L)}_{\mathrm{small}} \right) < \beta^{-1/2} \beta \mu(B) = k^{-10\varepsilon L} \mu(B), \end{align*} $$

and this proves (9.17).

If $q \in X_1 \cap X_2$ and $q' \in B_{q, L} \cap \bigcup \mathcal {P}^{(q,L)}_{\mathrm {big}} $ , then we have

$$ \begin{align*}\mu(P_L(q')) \geq \beta \mu(B_{q,L}) \geq k^{-21 \varepsilon L}, \end{align*} $$

and this implies via (9.12) that $q' \in W.$ This and (9.17) shows that $X_1 \cap X_2 \subset G_L$ . Thus,

(9.18) $$ \begin{align} \mu(G_L) \geq \mu(X_1 \cap X_2) \geq \frac{2}{5} - \frac{1}{100}> \frac 1 3, \end{align} $$

and we have shown (9.11).

Proof of Theorem 1.6.

Denote by T the map defined by $\mathrm {Rel}_{z_0}$ (where defined). Since $m_{{{\mathcal L}}}$ is G-invariant, it is rotation invariant, and thus $m_{{{\mathcal L}}}$ -a.e. q has no horizontal saddle connections. In particular, for such q, $\mathrm {Rel}_z (q)$ is defined for all $z \in Z$ .

Assume first that $m_{{{\mathcal L}}}$ is ergodic. By G-invariance of $m_{{\mathcal L}}$ , we have (9.1), so the hypotheses of Theorem 9.1 are satisfied for $\mu = m_{{{\mathcal L}}}$ . Now suppose $m_{{{\mathcal L}}}$ is not ergodic, and let $\mu = \int _{{\mathrm {Prob}}(X)^T} \nu \ d\theta $ be the ergodic decomposition of $\mu $ , where $\theta $ is a probability measure on ${\mathrm {Prob}}(X)^T$ such that $\theta $ -a.e. $\nu $ is ergodic for T. By Proposition 9.6(2), it suffices to show that the entropy of $\nu $ is zero for $\theta $ -a.e. $\nu $ , and thus, we only need to show that assumption (9.1) holds for $\theta $ -a.e. $\nu $ . This follows from the $g_t$ -invariance of $m_{{{\mathcal L}}}$ . Indeed, by invariance and regularity of $m_{{{\mathcal L}}}$ , for any $\varepsilon>0$ , there exists a compact K, so that for all t, $m_{{{\mathcal L}}}(g_t(K)) =m_{{{\mathcal L}}}(K)>1-\varepsilon ^2$ . Thus, for every t,

$$ \begin{align*}\theta(\{\nu :g_{-t*}\nu(K) \geq 1-\varepsilon\}) \geq 1-\varepsilon.\end{align*} $$

Thus, for any $\varepsilon>0$ , there is K so that the set of $\nu $ , for which $(g_{-t_i})_*\nu (K) \geq 1-\varepsilon $ for a sequence $t_i \to \infty $ , has $\theta $ -measure at least $1-\varepsilon $ . Since $\varepsilon $ was arbitrary, we have (9.1) for $\theta $ -a.e. $\nu $ .