2.1 Abelian differentials and translation surfaces

Let $g\geqslant 1$ and let $S=S_{g}$ be a fixed closed topological surface of genus  $g$ . Let $\unicode[STIX]{x1D6F4}=\{A_{1},\ldots ,A_{s}\}$ be a finite subset of  $S$ . An abelian differential on $(S,\unicode[STIX]{x1D6F4})$ is a pair $({\mathcal{J}},\unicode[STIX]{x1D714})$ where ${\mathcal{J}}$ is a complex structure on $S$ and $\unicode[STIX]{x1D714}$ is a holomorphic one form with respect to  ${\mathcal{J}}$ , and with zeros contained in  $\unicode[STIX]{x1D6F4}$ . As is well known, an abelian differential $\unicode[STIX]{x1D714}$ on $(S,\unicode[STIX]{x1D6F4})$ gives $S$ the structure of a translation surface with conical singularities in  $\unicode[STIX]{x1D6F4}$ ; the complex structure comes from integrating the differential. Hence we may speak about the area of an abelian differential as the area of the corresponding translation surface.

One may further specify that the abelian differential has a zero of order $\unicode[STIX]{x1D705}_{i}\in \mathbf{Z}_{+}$ at  $A_{i}$ . This is possible whenever $\sum \unicode[STIX]{x1D705}_{i}=2g-2$ . For such $\unicode[STIX]{x1D705}=(\unicode[STIX]{x1D705}_{1},\ldots ,\unicode[STIX]{x1D705}_{s})$ we let ${\mathcal{X}}(\unicode[STIX]{x1D705})$ denote the collection of abelian differentials on $(S,\unicode[STIX]{x1D6F4})$ with zeros of orders $\unicode[STIX]{x1D705}_{1},\ldots ,\unicode[STIX]{x1D705}_{s}$ at $A_{1},\ldots ,A_{s}$ , up to isotopies of $S$ preserving  $\unicode[STIX]{x1D6F4}$ . This Teichmüller space has a natural affine manifold structure arising through period coordinates as described in [Reference Avila, Gouëzel and YoccozAGY06, § 2.2.1]. Let ${\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})\subset {\mathcal{X}}(\unicode[STIX]{x1D705})$ be the abelian differentials whose corresponding translation surface has unit area, up to isotopy. Then ${\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})$ is a submanifold of ${\mathcal{X}}(\unicode[STIX]{x1D705})$ .

The modular group $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E4}(S,\unicode[STIX]{x1D6F4})$ is defined to be the homeomorphisms of $S$ that fix $\unicode[STIX]{x1D6F4}$ pointwise, modulo homeomorphisms that are isotopic to the identity relative to  $\unicode[STIX]{x1D6F4}$ . Thus $\unicode[STIX]{x1D6E4}$ acts on ${\mathcal{X}}(\unicode[STIX]{x1D705})$ , preserving ${\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})$ , and we define ${\mathcal{H}}(\unicode[STIX]{x1D705})$ to be $\unicode[STIX]{x1D6E4}\backslash {\mathcal{X}}(\unicode[STIX]{x1D705})$ and ${\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})=\unicode[STIX]{x1D6E4}\backslash {\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})$ . This ${\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})$ is often referred to as a stratum of the moduli space of unit area abelian differentials. The connected components of these strata have been classified by Kontsevich and Zorich [Reference Kontsevich and ZorichKZ03]. Throughout the paper we write ${\mathcal{M}}$ for a connected component of ${\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})$ .

Any connected component ${\mathcal{M}}$ of ${\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})$ inherits, from the manifold structure of ${\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})$ , the structure of an orbifold. We define ${\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705};q)=\unicode[STIX]{x1D6E4}_{g}(q)\backslash {\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})$ where $\unicode[STIX]{x1D6E4}_{g}(q)$ is the kernel of $\unicode[STIX]{x1D6F1}_{q}$ defined in (1). We thus have a covering map ${\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705};q)\rightarrow {\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})$ . We define ${\mathcal{M}}(q)$ to be the preimage of ${\mathcal{M}}$ under this map. For each $q$ the lift of ${\mathcal{M}}(q)$ to ${\mathcal{X}}(\unicode[STIX]{x1D705})$ is a submanifold.

Recall that a Finsler manifold is a smooth manifold together with a continuous assignment of norm on each tangent fibre. The norm is called a Finsler metric. As described in [Reference Avila, Gouëzel and YoccozAGY06, § 2.2.2] there is a $\unicode[STIX]{x1D6E4}$ -invariant Finsler metric on ${\mathcal{X}}(\unicode[STIX]{x1D705})$ arising from period coordinates making ${\mathcal{X}}(\unicode[STIX]{x1D705})$ into a Finsler manifold. This induces a Finsler manifold structure on ${\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})$ .

2.2 The Hodge bundle

The Hodge bundle is defined to be the fibred product

$$\begin{eqnarray}\displaystyle H_{1}({\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})):=\unicode[STIX]{x1D6E4}\backslash ({\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})\times H_{1}(S,\mathbf{R}))\rightarrow \unicode[STIX]{x1D6E4}\backslash {\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})={\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705}), & & \displaystyle \nonumber\end{eqnarray}$$

where the mapping class group $\unicode[STIX]{x1D6E4}$ acts diagonally. Let ${\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})^{0}$ be the complement of the orbifold points in ${\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})$ . The Hodge bundle restricts to a vector bundle $H_{1}({\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})^{0})$ over ${\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})^{0}$ . At any orbifold point $[({\mathcal{J}},\unicode[STIX]{x1D714})]$ of ${\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})$ the fibre degenerates to $\operatorname{Aut}({\mathcal{J}},\unicode[STIX]{x1D714})\backslash H_{1}(S,\mathbf{R})$ . Note that by Hurwitz’s automorphisms theorem, $\operatorname{Aut}({\mathcal{J}},\unicode[STIX]{x1D714})$ is a finite group.

The total space of the Hodge bundle contains as a discrete subset the lattice bundle

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}\backslash ({\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})\times H_{1}(S,\mathbf{Z})). & & \displaystyle \nonumber\end{eqnarray}$$

Then one may specify the Gauss–Manin connection on the Hodge bundle by the requirement that lattice-valued continuous sections be parallel. This gives a flat vector bundle connection on $H_{1}({\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})^{0})$ that extends to a flat connection on $H_{1}({\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705}))$ in the following sense. A section of $H_{1}({\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705}))$ can be viewed as a function $\unicode[STIX]{x1D70E}:{\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})\rightarrow H_{1}(S,\mathbf{R})$ that transforms according to

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6FE}.x)=\unicode[STIX]{x1D6FE}_{\ast }\unicode[STIX]{x1D70E}(x),\quad \unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}. & & \displaystyle \nonumber\end{eqnarray}$$

Then a local section is parallel by definition if it takes values in $H_{1}(S,\mathbf{Z})$ and this specifies the connection on general sections.

The action of $\unicode[STIX]{x1D6E4}$ on $H_{1}(S,\mathbf{Z})$ lies in the integral symplectic group $\operatorname{Sp}(H_{1}(S,\mathbf{Z}),\cap )$ where $\cap$ is the (symplectic) intersection form on integral homology. Therefore for any unitary representation $(\unicode[STIX]{x1D70C},{\mathcal{W}})$ of $\operatorname{Sp}(H_{1}(S,\mathbf{Z}),\cap )$ we obtain an associated orbifold vector bundle Footnote ¹¹   $H_{1}({\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705});\unicode[STIX]{x1D70C})$ . The total space of this bundle is

(2) $$\begin{eqnarray}\displaystyle H_{1}({\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705});\unicode[STIX]{x1D70C})=\unicode[STIX]{x1D6E4}\backslash ({\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})\times {\mathcal{W}}) & & \displaystyle\end{eqnarray}$$

where the action of $\unicode[STIX]{x1D6E4}$ on ${\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})\times {\mathcal{W}}$ is given by $\unicode[STIX]{x1D6FE}.(\unicode[STIX]{x1D714},v)=(\unicode[STIX]{x1D6FE}.\unicode[STIX]{x1D714},\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6FE}_{\ast }).v)$ , where $\unicode[STIX]{x1D6FE}_{\ast }\in \operatorname{Sp}(H_{1}(S,\mathbf{Z}),\cap )$ is the map induced by $\unicode[STIX]{x1D6FE}$ on homology. This bundle also has a flat connection, in the same sense as before, coming from the fibred product structure in (2).

Of course, for any connected component ${\mathcal{M}}$ of the stratum ${\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})$ we may restrict $H_{1}({\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705}))$ or $H_{1}({\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705});\unicode[STIX]{x1D70C})$ to  ${\mathcal{M}}$ . We denote by $H_{1}({\mathcal{M}};\unicode[STIX]{x1D70C})$ the obtained orbifold vector bundle.

For a lot of the rest of the paper we deal with abstract unitary $\unicode[STIX]{x1D70C}$ but in reality we are interested in the following specific examples. Recall the map $\unicode[STIX]{x1D6F1}_{q}$ from (1). Because the symplectic intersection product $\cap$ on $H_{1}(S,\mathbf{Z}/q\mathbf{Z})$ is preserved by the mapping class group, we have

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{q}:\unicode[STIX]{x1D6E4}\rightarrow \operatorname{Sp}(H_{1}(S;\mathbf{Z}/q\mathbf{Z}),\cap ). & & \displaystyle \nonumber\end{eqnarray}$$

We let $\unicode[STIX]{x1D6E4}_{q}=\operatorname{Sp}(H_{1}(S;\mathbf{Z}/q\mathbf{Z}),\cap )$ . Let $\ell _{0}^{2}(\unicode[STIX]{x1D6E4}_{q})$ be the subspace of functions in $\ell ^{2}(\unicode[STIX]{x1D6E4}_{q})$ that are orthogonal to constant functions with respect to the $\ell ^{2}$ inner product. This gives a subrepresentation $(\unicode[STIX]{x1D70C}_{q},\ell _{0}^{2}(\unicode[STIX]{x1D6E4}_{q}))$ of the action of $\unicode[STIX]{x1D6E4}$ on $\ell ^{2}(\unicode[STIX]{x1D6E4}_{q})$ by reduction mod $q$ and then left translation.Footnote ¹²

We will also consider the subspace of $\ell _{0}^{2}(\unicode[STIX]{x1D6E4}_{q})$ consisting of functions that are orthogonal to all functions lifted from $\unicode[STIX]{x1D6E4}_{q^{\prime }}$ with $q^{\prime }|q$ via the natural mapping of reduction modulo  $q^{\prime }$

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{q}\rightarrow \unicode[STIX]{x1D6E4}_{q^{\prime }}. & & \displaystyle \nonumber\end{eqnarray}$$

We denote by $\ell _{\text{new}}^{2}(\unicode[STIX]{x1D6E4}_{q})$ this new subspace of functions. This gives a subrepresentation $(\unicode[STIX]{x1D70C}_{q}^{\text{new}},\ell _{\text{new}}^{2}(\unicode[STIX]{x1D6E4}_{q}))$ of $(\unicode[STIX]{x1D70C}_{q},\ell _{0}^{2}(\unicode[STIX]{x1D6E4}_{q}))$ .

2.3 The Teichmüller flow on moduli space

There is a postcomposition action of $\operatorname{SL}_{2}(\mathbf{R})$ on the space of abelian differentials on $S$ as follows. For $h\in \operatorname{SL}_{2}(\mathbf{R})$ we define

$$\begin{eqnarray}\displaystyle h.({\mathcal{J}},\unicode[STIX]{x1D714})=({\mathcal{J}}_{h},\unicode[STIX]{x1D714}_{h}), & & \displaystyle \nonumber\end{eqnarray}$$

where

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{h}=h\left(\begin{array}{@{}c@{}}\Re (w)\\ \Im (w)\end{array}\right) & & \displaystyle \nonumber\end{eqnarray}$$

and ${\mathcal{J}}_{h}$ is the unique complex structure on $S$ that makes $\unicode[STIX]{x1D714}_{h}$ holomorphic. This action preserves the area of abelian differentials. As this action also commutes with any homeomorphism of  $S$ , it descends to both the Teichmüller spaces ${\mathcal{X}}(\unicode[STIX]{x1D705})$ , ${\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})$ and ${\mathcal{H}}(\unicode[STIX]{x1D705})$ , ${\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})$ and  ${\mathcal{M}}$ . The Teichmüller geodesic flow on any of these objects is the restriction of the $\operatorname{SL}_{2}(\mathbf{R})$ action to the diagonal subgroup:

$$\begin{eqnarray}\displaystyle {\mathcal{T}}_{t}({\mathcal{J}},\unicode[STIX]{x1D714}):=\left(\begin{array}{@{}cc@{}}e^{t} & 0\\ 0 & e^{-t}\end{array}\right).({\mathcal{J}},\unicode[STIX]{x1D714}). & & \displaystyle \nonumber\end{eqnarray}$$

The Teichmüller flow also preserves each connected component  ${\mathcal{M}}$ . By results of Masur [Reference MasurMas82] and Veech [Reference VeechVee82] there is a unique probability measure $\unicode[STIX]{x1D708}_{{\mathcal{M}}}$ that is invariant and ergodic for the Teichmüller flow on  ${\mathcal{M}}$ . This measure is in the Lebesgue class with respect to period coordinates on  ${\mathcal{M}}$ . We pull back the measure $\unicode[STIX]{x1D708}_{{\mathcal{M}}}$ on  ${\mathcal{M}}$ , using the counting measure on the fibres of the covering map, to obtain a measure $\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}$ on ${\mathcal{M}}(q)$ . Note that $\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}$ is not a probability measure.

Since in § 2.2 we specified a connection on each of $H_{1}({\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705}))$ , $H_{1}({\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705});\unicode[STIX]{x1D70C})$ the Teichmüller flow acts on sections of each of these bundles by pullback along parallel transport. For example, viewing a section of $H_{1}({\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705});\unicode[STIX]{x1D70C})$ as a ${\mathcal{W}}$ -valued function $\unicode[STIX]{x1D70E}$ on ${\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})$ satisfying $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6FE}.x)=\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6FE})\unicode[STIX]{x1D70E}(x)$ for each $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}$ , we have the following defining equation for ${\mathcal{T}}_{t}^{\ast }$ :

(3) $$\begin{eqnarray}\displaystyle [{\mathcal{T}}_{t}^{\ast }\unicode[STIX]{x1D70E}]({\mathcal{J}},\unicode[STIX]{x1D714}):=\unicode[STIX]{x1D70E}({\mathcal{T}}_{t}({\mathcal{J}},\unicode[STIX]{x1D714})). & & \displaystyle\end{eqnarray}$$

This action also restricts to an action on sections of $H_{1}({\mathcal{M}};\unicode[STIX]{x1D70C})$ .

We now explain the relationship between sections of $H_{1}({\mathcal{M}};\unicode[STIX]{x1D70C}_{q})$ and functions on ${\mathcal{M}}(q)$ . Let $L_{\star }^{2}({\mathcal{M}}(q))$ be the subspace of functions in $L^{2}({\mathcal{M}}(q))$ orthogonal to lifts from $L^{2}({\mathcal{M}})$ , with respect to the measure $\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}$ . Let $L^{2}(H_{1}({\mathcal{M}};\unicode[STIX]{x1D70C}_{q}))$ denote the $L^{2}$ sections of $H_{1}({\mathcal{M}};\unicode[STIX]{x1D70C}_{q})$ with respect to the natural Hermitian fibre metric and measure $\unicode[STIX]{x1D708}_{{\mathcal{M}}}$ . We say that a function $f$ on ${\mathcal{M}}(q)$ or a section $\unicode[STIX]{x1D70E}$ of $H_{1}({\mathcal{M}};\unicode[STIX]{x1D70C})$ is $C^{1}$ if its lift to $\tilde{f}:{\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})\rightarrow \mathbf{C}$ (respectively $\tilde{\unicode[STIX]{x1D70E}}:{\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})\rightarrow {\mathcal{W}}$ ) is $C^{1}$ (bounded with bounded derivativeFootnote ¹³ ) with respect to the Finsler manifold structure on ${\mathcal{X}}^{(1)}(\unicode[STIX]{x1D705})$ . Define $\Vert f\Vert _{C^{1}}=\Vert \tilde{f}\Vert _{\infty }+\Vert D\tilde{f}\Vert _{\infty }$ and similarly $\Vert \unicode[STIX]{x1D70E}\Vert _{C^{1}}$ . Write $C^{1}({\mathcal{M}}(q))$ for the $C^{1}$ complex-valued functions on ${\mathcal{M}}(q)$ and $C^{1}(H_{1}({\mathcal{M}};\unicode[STIX]{x1D70C}))$ for the $C^{1}$ sections of $H_{1}({\mathcal{M}};\unicode[STIX]{x1D70C})$ . These are Banach spaces with respect to the respective $C^{1}$ norms.

2.4 Combinatorial data and Rauzy classes

Now we begin an account of the dynamics of the Teichmüller flow, viewed through the lens of Veech’s zippered rectangles construction. We draw in the following sections from the sources [Reference Avila, Gouëzel and YoccozAGY06, Reference VianaVia08] that both build on work of Marmi, Moussa and Yoccoz [Reference Marmi, Moussa and YoccozMMY05].

The relevant combinatorial objects are as follows. Let ${\mathcal{A}}$ denote a finite alphabet with $|{\mathcal{A}}|=d$ . Eventually, ${\mathcal{A}}$ will be chosen depending on  $g$ , $\unicode[STIX]{x1D705}$ and the component  ${\mathcal{M}}$ . We let $\mathfrak{S}({\mathcal{A}})$ denote the set of pairs

$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D70B}_{t},\unicode[STIX]{x1D70B}_{b}), & & \displaystyle \nonumber\end{eqnarray}$$

where each $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D716}}:{\mathcal{A}}\rightarrow \{1,\ldots ,d\}$ is a bijection. Henceforth, $\unicode[STIX]{x1D716}$ will index one of the symbols $t,b$ (‘top’ or ‘bottom’). As in [Reference Avila, Gouëzel and YoccozAGY06] it is convenient to visualize $(\unicode[STIX]{x1D70B}_{t},\unicode[STIX]{x1D70B}_{b})$ as a pair of rows each of which contains the elements of ${\mathcal{A}}$ in some order, where the top corresponds to $\unicode[STIX]{x1D70B}_{t}$ and the bottom to  $\unicode[STIX]{x1D70B}_{b}$ . We say $(\unicode[STIX]{x1D70B}_{t},\unicode[STIX]{x1D70B}_{b})$ is irreducible if there is no $d^{\prime }<d$ such that the set of the first $d^{\prime }$ elements of the top row is the same as the first $d^{\prime }$ elements of the bottom. Let $\mathfrak{S}^{0}({\mathcal{A}})\subset \mathfrak{S}({\mathcal{A}})$ denote the irreducible combinatorial data.

We now define ‘top’ and ‘bottom’ operations on $\mathfrak{S}^{0}({\mathcal{A}})$ . For the next paragraph, let $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ denote the last elements of the top and bottom rows of $\unicode[STIX]{x1D70B}\in \mathfrak{S}^{0}({\mathcal{A}})$ respectively. The top operation on $\unicode[STIX]{x1D70B}$ modifies the bottom row by moving the occurrence of $\unicode[STIX]{x1D6FD}$ to the immediate right of the occurrence of  $\unicode[STIX]{x1D6FC}$ . The bottom operation modifies the top row by moving $\unicode[STIX]{x1D6FC}$ to the right of  $\unicode[STIX]{x1D6FD}$ . As in [Reference Avila, Gouëzel and YoccozAGY06] we say that the last element of the unchanged row is the winner and the last element of the row of $\unicode[STIX]{x1D70B}$ that is to be changed the loser.

By adding directed ‘top’ and ‘bottom’ labeled edges according to these operations we obtain an edge-labeled directed graph on the vertex set of irreducible combinatorial data $\mathfrak{S}^{0}({\mathcal{A}})$ . Each vertex has exactly one incoming top (respectively bottom) and one outgoing top (respectively bottom) edge. A Rauzy diagram is a connected component of this graph and a Rauzy class is the vertex set of a Rauzy diagram.

2.5 Suspension data and zippered rectangles

Let $\mathfrak{R}$ be a Rauzy class. For each $\unicode[STIX]{x1D70B}\in \mathfrak{R}$ we form a cell

$$\begin{eqnarray}\displaystyle X_{\unicode[STIX]{x1D70B}}=\{\unicode[STIX]{x1D70B}\}\times \mathbf{R}_{+}^{{\mathcal{A}}}\times {\mathcal{K}}_{\unicode[STIX]{x1D70B}}, & & \displaystyle \nonumber\end{eqnarray}$$

where

$$\begin{eqnarray}\displaystyle {\mathcal{K}}_{\unicode[STIX]{x1D70B}}=\bigg\{\unicode[STIX]{x1D70F}\in \mathbf{R}^{{\mathcal{A}}}:\mathop{\sum }_{\unicode[STIX]{x1D70B}_{t}(\unicode[STIX]{x1D709})\leqslant k}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D709}}>0,\mathop{\sum }_{\unicode[STIX]{x1D70B}_{b}(\unicode[STIX]{x1D709})\leqslant k}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D709}}<0~\text{for all}~1\leqslant k\leqslant d-1\bigg\}. & & \displaystyle \nonumber\end{eqnarray}$$

The set ${\mathcal{K}}_{\unicode[STIX]{x1D70B}}$ is an open convex cone. Let $X_{\mathfrak{R}}=\cup _{\unicode[STIX]{x1D70B}\in \mathfrak{R}}X_{\unicode[STIX]{x1D70B}}$ . We may drop the dependence on $\mathfrak{R}$ since we usually view it as fixed. We associate to each $\unicode[STIX]{x1D70B}\in \mathfrak{R}$ a linear map $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70B}}:\mathbf{R}^{{\mathcal{A}}}\rightarrow \mathbf{R}^{{\mathcal{A}}}$ given by

$$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70B}}]_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}}=\left\{\begin{array}{@{}ll@{}}+1\quad & \text{if}~\unicode[STIX]{x1D70B}_{t}(\unicode[STIX]{x1D6FC})>\unicode[STIX]{x1D70B}_{t}(\unicode[STIX]{x1D6FD}),\unicode[STIX]{x1D70B}_{b}(\unicode[STIX]{x1D6FC})<\unicode[STIX]{x1D70B}_{b}(\unicode[STIX]{x1D6FD}),\\ -1\quad & \text{if}~\unicode[STIX]{x1D70B}_{t}(\unicode[STIX]{x1D6FC})<\unicode[STIX]{x1D70B}_{t}(\unicode[STIX]{x1D6FD}),\unicode[STIX]{x1D70B}_{b}(\unicode[STIX]{x1D6FC})>\unicode[STIX]{x1D70B}_{b}(\unicode[STIX]{x1D6FD}),\\ 0\quad & \text{else.}\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$

There is a construction due to Veech [Reference VeechVee82] that builds a point in the moduli space of translation surfaces from suspension data $(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})$ . This mapping is called the zippered rectangles construction that we denote by

$$\begin{eqnarray}\displaystyle \operatorname{zip}:X_{\mathfrak{R}}\rightarrow {\mathcal{H}}(\unicode[STIX]{x1D705}),\quad \unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}(\mathfrak{R}). & & \displaystyle \nonumber\end{eqnarray}$$

The explicit details of this construction are clearly described in lecture notes of Viana [Reference VianaVia08, ch. 2]. In the current paper it will be better to simply work with the properties of the map $\operatorname{zip}$ that we give below. Henceforth a superscript $^{(1)}$ on any set of suspension data refers to the subset whose associated zippered rectangles have unit area: for example $X_{\mathfrak{R}}^{(1)},X_{\unicode[STIX]{x1D70B}}^{(1)}$ etc.

There is a natural identification

(4) $$\begin{eqnarray}\displaystyle \mathbf{R}^{{\mathcal{A}}}/\text{ker}\,\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70B}}\cong H_{1}(\operatorname{zip}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F}),\mathbf{R}) & & \displaystyle\end{eqnarray}$$

for each $(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})\in X_{\unicode[STIX]{x1D70B}}$ . For a detailed discussion of this map see Viana [Reference VianaVia08, § 2.9]. The bilinear form

$$\begin{eqnarray}\displaystyle (v_{1},v_{2})\mapsto \langle v_{1},-\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70B}}v_{2}\rangle & & \displaystyle \nonumber\end{eqnarray}$$

descends to a non-degenerate symplectic form $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70B}}$ on $\mathbf{R}^{{\mathcal{A}}}/\text{ker}\,\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70B}}$ . Under the identification (4), the form $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70B}}$ is precisely the intersection form on homology. Furthermore, the map (4) arises from an isomorphism of integral symplectic lattices

(5) $$\begin{eqnarray}\displaystyle (\mathbf{Z}^{{\mathcal{A}}}/\text{ker}(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70B}}|_{\mathbf{Z}^{{\mathcal{A}}}}),\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70B}})\cong (H_{1}(S,\mathbf{Z}),\cap ). & & \displaystyle\end{eqnarray}$$

Therefore the pullback of the Hodge bundle to $X_{\unicode[STIX]{x1D70B}}$ via $\operatorname{zip}$ is naturally trivialized:

(6) $$\begin{eqnarray}\displaystyle [\operatorname{zip}^{\ast }H_{1}({\mathcal{H}}(\unicode[STIX]{x1D705}))]|_{X_{\unicode[STIX]{x1D70B}}}\cong X_{\unicode[STIX]{x1D70B}}\times \mathbf{R}^{{\mathcal{A}}}/\text{ker}\,\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70B}}. & & \displaystyle\end{eqnarray}$$

We also note here that the area of $\operatorname{zip}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})$ is given by

(7) $$\begin{eqnarray}\displaystyle \operatorname{area}(\operatorname{zip}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F}))=\langle \unicode[STIX]{x1D706},-\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70B}}\unicode[STIX]{x1D70F}\rangle . & & \displaystyle\end{eqnarray}$$

2.6 The Rauzy induction map

Given $\unicode[STIX]{x1D70B}$ , let $\unicode[STIX]{x1D6FC}$ be the last element of the top row of $\unicode[STIX]{x1D70B}$ and $\unicode[STIX]{x1D6FD}$ the last element of the bottom row. Say that a pair $(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706})$ has type top if $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FC}}>\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FD}}$ . Say it has type bottom if $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FD}}<\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FC}}$ . This splits each cell into two pieces of the form

$$\begin{eqnarray}\displaystyle X_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}=\{(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})\in \{\unicode[STIX]{x1D70B}\}\times \mathbf{R}_{+}^{{\mathcal{A}}}\times {\mathcal{K}}_{\unicode[STIX]{x1D70B}}:(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706})~\text{of type}~\unicode[STIX]{x1D716}\},\quad \unicode[STIX]{x1D716}\in \{t,b\} & & \displaystyle \nonumber\end{eqnarray}$$

together with a hyperplane. We also introduce $Y_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}=\{(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706})\in \{\unicode[STIX]{x1D70B}\}\times \mathbf{R}_{+}^{{\mathcal{A}}}~\text{of type}~\unicode[STIX]{x1D716}\}$ , so that

$$\begin{eqnarray}\displaystyle X_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}=Y_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}\times {\mathcal{K}}_{\unicode[STIX]{x1D70B}}. & & \displaystyle \nonumber\end{eqnarray}$$

We now give an assignment of a linear map $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}:\mathbf{R}^{{\mathcal{A}}}\rightarrow \mathbf{R}^{{\mathcal{A}}}$ to each pair $(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716})$ . This is given by [Reference VianaVia08, (1.9), (1.10)]

(8) $$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}]_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}}:=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if}~\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD},\\ 1\quad & \text{if}~\unicode[STIX]{x1D6FC}~\text{loses and}~\unicode[STIX]{x1D6FD}~\text{wins in type}~\unicode[STIX]{x1D716}~\text{move at}~\unicode[STIX]{x1D70B},\\ 0\quad & \text{else.}\end{array}\right. & & \displaystyle\end{eqnarray}$$

If $\unicode[STIX]{x1D70B}^{\prime }$ is obtained from $\unicode[STIX]{x1D70B}$ by a type $\unicode[STIX]{x1D716}$ move, then the mapFootnote ¹⁴   $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}^{\ast }$ maps $Y_{\unicode[STIX]{x1D70B}^{\prime }}:=\{\unicode[STIX]{x1D70B}^{\prime }\}\times \mathbf{R}_{+}^{{\mathcal{A}}}$ homeomorphically to $Y_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}$ . Furthermore $(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}^{\ast })^{-1}$ maps ${\mathcal{K}}_{\unicode[STIX]{x1D70B}}$ injectively into ${\mathcal{K}}_{\unicode[STIX]{x1D70B}^{\prime }}$ [Reference VianaVia08, Lemma 2.13]. We also have the intertwining relation

(9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70B}}\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}^{\ast }=\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70B}^{\prime }}. & & \displaystyle\end{eqnarray}$$

The Rauzy induction map on suspension data is given by

$$\begin{eqnarray}\displaystyle \hat{Q}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F}):=(\unicode[STIX]{x1D70B}^{\prime },(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}^{\ast })^{-1}\unicode[STIX]{x1D706},(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}^{\ast })^{-1}\unicode[STIX]{x1D70F}) & & \displaystyle \nonumber\end{eqnarray}$$

when $(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})\in X_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}$ ; here again $\unicode[STIX]{x1D70B}^{\prime }$ is obtained from $\unicode[STIX]{x1D70B}$ by an operation of type  $\unicode[STIX]{x1D716}$ . Using the same notation, notice that $\hat{Q}$ is a skew extension of the mapFootnote ¹⁵

$$\begin{eqnarray}\displaystyle Q(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706})=(\unicode[STIX]{x1D70B}^{\prime },(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}^{\ast })^{-1}\unicode[STIX]{x1D706}). & & \displaystyle \nonumber\end{eqnarray}$$

Equation (9) together with the area formula (7) shows that $\hat{Q}$ preserves the area of the associated zippered rectangles. Hence $\hat{Q}$ preserves  $X^{(1)}$ .

The zippered rectangles associated to $(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})$ define the same point in ${\mathcal{H}}(\unicode[STIX]{x1D705})$ as the zippered rectangles associated to $\hat{Q}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})$ , that is,

$$\begin{eqnarray}\displaystyle \operatorname{zip}\circ \hat{Q}=\text{zip}. & & \displaystyle \nonumber\end{eqnarray}$$

See Viana [Reference VianaVia08, § 2.8] for a clear explanation of this fact.

We now define cylinders for the Rauzy induction map. Let $\unicode[STIX]{x1D6FE}$ be a path in the Rauzy diagram associated to the class  $\mathfrak{R}$ . Throughout the rest of the paper, we consider oriented paths that follow the given direction of the edges.Footnote ¹⁶ Suppose that $\unicode[STIX]{x1D6FE}$ traverses vertices $\unicode[STIX]{x1D70B}(0),\unicode[STIX]{x1D70B}(1),\ldots ,\unicode[STIX]{x1D70B}(N)$ in that order. Then define

$$\begin{eqnarray}\displaystyle X_{\unicode[STIX]{x1D6FE}}:=X_{\unicode[STIX]{x1D70B}(0)}\cap \hat{Q}^{-1}(X_{\unicode[STIX]{x1D70B}(1)})\cap \hat{Q}^{-2}(X_{\unicode[STIX]{x1D70B}(2)})\cap \cdots \cap \hat{Q}^{-N}(X_{\unicode[STIX]{x1D70B}(N)}). & & \displaystyle \nonumber\end{eqnarray}$$

Notice that $X_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}$ is the same as $X_{\unicode[STIX]{x1D6FE}}$ where $\unicode[STIX]{x1D6FE}$ is the outgoing type $\unicode[STIX]{x1D716}$ arrow from  $\unicode[STIX]{x1D70B}$ . We then define $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}$ in terms of the $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}$ by stating that for $(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})\in X_{_{\unicode[STIX]{x1D6FE}}}$ we have

$$\begin{eqnarray}\displaystyle \hat{Q}^{N}(\unicode[STIX]{x1D70B}(0),\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})=(\unicode[STIX]{x1D70B}(N),(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}\unicode[STIX]{x1D706},(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}\unicode[STIX]{x1D70F}). & & \displaystyle \nonumber\end{eqnarray}$$

We define $Y_{\unicode[STIX]{x1D6FE}}=Y_{\unicode[STIX]{x1D70B}(0)}\cap \cdots \cap Q^{-N}(Y_{\unicode[STIX]{x1D70B}(N)})$ the analogous cylinder for  $Q$ . If $\unicode[STIX]{x1D6FE}$ begins at $\unicode[STIX]{x1D70B}$ , then we define the subcone of  ${\mathcal{K}}_{\unicode[STIX]{x1D70B}}$

$$\begin{eqnarray}\displaystyle {\mathcal{K}}_{\unicode[STIX]{x1D6FE}}:=(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}{\mathcal{K}}_{\unicode[STIX]{x1D70B}}. & & \displaystyle \nonumber\end{eqnarray}$$

2.7 The Rauzy–Veech group

Observe that $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}^{\ast }$ induces a map $\mathbf{Z}^{{\mathcal{A}}}/\text{ker}\,\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70B}^{\prime }}\rightarrow \mathbf{Z}^{{\mathcal{A}}}/\text{ker}\,\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70B}}$ in light of (9) and the fact that $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}$ is integral from (8). These facts are discussed by Viana in [Reference VianaVia08, § 2.8]. As a consequence, (9) implies that if $\unicode[STIX]{x1D6FE}$ begins and ends at  $\unicode[STIX]{x1D70B}$ , $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast }$ induces a symplectic endomorphism of $(\mathbf{Z}^{{\mathcal{A}}}/\text{ker}(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70B}}|_{\mathbf{Z}^{{\mathcal{A}}}}),\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70B}})\cong ^{(5)}(H_{1}(S,\mathbf{Z}),\cap )$ . In fact it is easy to check from (8) that $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast }$ is an automorphism. We therefore view each

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast }\in \operatorname{Sp}(\mathbf{Z}^{2g},\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70B}}). & & \displaystyle \nonumber\end{eqnarray}$$

For each $\unicode[STIX]{x1D70B}\in \mathfrak{R}$ let $G_{\unicode[STIX]{x1D70B}}$ be the subgroup of $\operatorname{Sp}(\mathbf{Z}^{2g},\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70B}})$ generated by the $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast }$ obtained as $\unicode[STIX]{x1D6FE}$ ranges over loops in $\mathfrak{R}$ beginning and ending at  $\unicode[STIX]{x1D70B}$ . This group $G_{\unicode[STIX]{x1D70B}}$ is called the Rauzy–Veech group at  $\unicode[STIX]{x1D70B}$ .

The key property of $G_{\unicode[STIX]{x1D70B}}$ that we rely on is the following recent theorem of Gutiérrez-Romo [Reference Gutiérrez-RomoGut19, Theorem 1.1] that was previously known for certain hyperelliptic components by work of Avila, Matheus and Yoccoz [Reference Avila, Matheus and YoccozAMY18, Theorem 1.1].

Theorem 2.3 resolved, in a strong form, the conjecture of Zorich [Reference ZorichZor99, Appendix A.3 Conjecture 5] that $G_{\unicode[STIX]{x1D70B}}$ should be Zariski-dense.

2.8 Relationship to the Hodge bundle

Let ${\mathcal{M}}$ be a connected component of ${\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})$ and let $\unicode[STIX]{x1D70E}$ be a section of the Hodge bundle $H_{1}({\mathcal{M}})$ . The pullback of $\unicode[STIX]{x1D70E}$ to any $X_{\unicode[STIX]{x1D70B}}^{(1)}$ under the zippered rectangles map can be naturally viewed as a

$$\begin{eqnarray}\displaystyle \mathbf{R}^{{\mathcal{A}}}/\text{ker}\,\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D70B}} & & \displaystyle \nonumber\end{eqnarray}$$

valued function $\tilde{\unicode[STIX]{x1D70E}}$ via the identifications (4) and (6). Since Rauzy induction does not change the modulus of zippered rectangles, the fibre of $\operatorname{zip}^{\ast }H_{1}({\mathcal{M}})$ at $(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})$ should be identified with the fibre at $\hat{Q}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})$ . In fact, the identification involves the previously defined map $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}$ and requires for $(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FE})\in X_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}^{(1)}$ that if $\unicode[STIX]{x1D70B}^{\prime }$ is the result of applying a type $\unicode[STIX]{x1D716}$ move to $\unicode[STIX]{x1D70B}$ then

(10) $$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B},\unicode[STIX]{x1D716}}^{\ast }\tilde{\unicode[STIX]{x1D70E}}(\hat{Q}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})). & & \displaystyle\end{eqnarray}$$

The iterated form of the compatibility equation (10) that we will use is the following. If $\unicode[STIX]{x1D6FE}$ is a path of $N$ edges in a Rauzy diagram that begins and ends at  $\unicode[STIX]{x1D70B}$ , then for $(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})\in X_{\unicode[STIX]{x1D6FE}}^{(1)}$ we have

$$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast }\tilde{\unicode[STIX]{x1D70E}}(\hat{Q}^{N}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})). & & \displaystyle \nonumber\end{eqnarray}$$

This is an important point of this paper as it describes the equivariance properties of sections of the Hodge bundle in the suspension model. We now extend this formula to the setting of associated orbifold vector bundles $H_{1}({\mathcal{M}};\unicode[STIX]{x1D70C})$ . After fixing  $\unicode[STIX]{x1D70B}$ , using the isomorphism (5) we identify

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{q}\cong \operatorname{Sp}((\mathbf{Z}/q\mathbf{Z})^{2g},\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70B}}) & & \displaystyle \nonumber\end{eqnarray}$$

so we may view $\unicode[STIX]{x1D70C}_{q}$ and $\unicode[STIX]{x1D70C}_{q}^{\text{new}}$ as representations of $\operatorname{Sp}(\mathbf{Z}^{2g},\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70B}})$ that are submodules of $\ell ^{2}(\operatorname{Sp}((\mathbf{Z}/q\mathbf{Z})^{2g},\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70B}}))$ . More generally, using (5) we may pull back any unitary representation $(\unicode[STIX]{x1D70C},{\mathcal{W}})$ of $\operatorname{Sp}(H_{1}(S;\mathbf{Z}),\cap )$ to a representation of $\operatorname{Sp}(\mathbf{Z}^{2g},\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D70B}})$ that we also call  $\unicode[STIX]{x1D70C}$ .

We may now argue by analogy with the Hodge bundle that if $\unicode[STIX]{x1D70E}$ is any section of the associated bundle $H_{1}({\mathcal{M}};\unicode[STIX]{x1D70C})$ , then the pullback $\tilde{\unicode[STIX]{x1D70E}}$ of this section to a ${\mathcal{W}}$ -valued function on $X_{\unicode[STIX]{x1D70B}}^{(1)}$ satisfies

(11) $$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })\tilde{\unicode[STIX]{x1D70E}}(\hat{Q}^{N}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})),\quad (\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})\in X_{\unicode[STIX]{x1D6FE}}^{(1)} & & \displaystyle\end{eqnarray}$$

for each path $\unicode[STIX]{x1D6FE}$ in $\mathfrak{R}$ of length $N$ beginning and ending at  $\unicode[STIX]{x1D70B}$ .

2.9 A fundamental domain

There is a nice fundamental domain for Rauzy induction on $X$ described in [Reference Avila, Gouëzel and YoccozAGY06, p. 159]. We let ${\mathcal{F}}={\mathcal{F}}_{\mathfrak{R}}$ denote the set of $(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})$ such that either:

1. (i) $\hat{Q}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})=(\unicode[STIX]{x1D70B}^{\prime },\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D70F}^{\prime })$ is defined and $\Vert \unicode[STIX]{x1D706}^{\prime }\Vert <1\leqslant \Vert \unicode[STIX]{x1D706}\Vert$ ; or

2. (ii) $\hat{Q}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})$ is not defined and $1\leqslant \Vert \unicode[STIX]{x1D706}\Vert$ ; or

3. (iii) $\hat{Q}^{-1}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})$ is not defined and $\Vert \unicode[STIX]{x1D706}\Vert <1$ .

The norm we use is $\Vert \unicode[STIX]{x1D706}\Vert :=\sum _{\unicode[STIX]{x1D6FC}\in {\mathcal{A}}}|\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FC}}|$ . The fibres of the zippered rectangles map

$$\begin{eqnarray}\displaystyle \operatorname{zip}:{\mathcal{F}}^{(1)}\rightarrow {\mathcal{M}},\quad {\mathcal{M}}={\mathcal{M}}(\mathfrak{R}) & & \displaystyle \nonumber\end{eqnarray}$$

are almost everywhere finite with constant cardinality depending on  ${\mathcal{M}}$ .

2.10 The Teichmüller flow on suspension data

Recall that ${\mathcal{M}}$ is a connected component of ${\mathcal{H}}^{(1)}(\unicode[STIX]{x1D705})$ and $\mathfrak{R}$ the associated Rauzy class. The Teichmüller flow is a one parameter flow on $X_{\mathfrak{R}}$ that commutes with $\hat{Q}$ and is given by $T_{t}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})=(\unicode[STIX]{x1D70B},e^{t}\unicode[STIX]{x1D706},e^{-t}\unicode[STIX]{x1D70F})$ . Note that this preserves each $X_{\unicode[STIX]{x1D70B}}^{(1)}$ and $X^{(1)}$ . The flow $T_{t}$ lifts the Teichmüller flow on  ${\mathcal{M}}$ , that is,

$$\begin{eqnarray}\displaystyle {\mathcal{T}}_{t}\circ \operatorname{zip}=\operatorname{zip}\circ T_{t}. & & \displaystyle \nonumber\end{eqnarray}$$

Evidently, $T_{t}$ preserves Lebesgue measure on  $X$ . The flow $T_{t}$ also preserves Lebesgue measure on $X^{(1)}$ , the pushforward of which under $\operatorname{zip}$ is a multiple of  $\unicode[STIX]{x1D708}_{{\mathcal{M}}}$ .

2.11 Time acceleration and renormalization

The approach of Avila, Gouëzel and Yoccoz [Reference Avila, Gouëzel and YoccozAGY06] to the Teichmüller flow is to consider the first return time to an appropriately chosen cross section. This cross section involves the choice of $\unicode[STIX]{x1D70B}\in \mathfrak{R}$ and a path $\unicode[STIX]{x1D6FE}_{0}$ that begins and ends at  $\unicode[STIX]{x1D70B}$ . We give details on the choice of $\unicode[STIX]{x1D6FE}_{0}$ in §§ 2.13 and 5.1. For now, assume we have chosen $\unicode[STIX]{x1D70B}$ and  $\unicode[STIX]{x1D6FE}_{0}$ .

We will use the notation ${\mathcal{F}}_{\unicode[STIX]{x1D70B}}={\mathcal{F}}\cap X_{\unicode[STIX]{x1D70B}}$ and ${\mathcal{F}}_{\unicode[STIX]{x1D6FE}}={\mathcal{F}}\cap X_{\unicode[STIX]{x1D6FE}}$ . We consider the regions

$$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D6EF}}:=\{(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})\in {\mathcal{F}}_{\unicode[STIX]{x1D6FE}_{0}}^{(1)}:\Vert \unicode[STIX]{x1D706}\Vert =1\}\cap (Y_{\unicode[STIX]{x1D6FE}_{0}}\times {\mathcal{K}}_{\unicode[STIX]{x1D6FE}_{0}}) & & \displaystyle \nonumber\end{eqnarray}$$

and the closely related

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}:=\{(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706})\in Y_{\unicode[STIX]{x1D6FE}_{0}}:\Vert \unicode[STIX]{x1D706}\Vert =1\}. & & \displaystyle \nonumber\end{eqnarray}$$

Let $\hat{m}$ (respectively $m$ ) denote the normalized natural Lebesgue measure on $\hat{\unicode[STIX]{x1D6EF}}$ (respectively  $\unicode[STIX]{x1D6EF}$ ). It is known that almost all orbits of the Teichmüller flow pass through $\hat{Q}^{\mathbf{Z}}(\unicode[STIX]{x1D6EF})$ , this is stated in [Reference Avila, Gouëzel and YoccozAGY06, 4.1.3] as a consequence of the ergodicity of the Veech flow.Footnote ¹⁷ For each $x\in \hat{\unicode[STIX]{x1D6EF}}$ we denote by $r(x)$ the first return time of $x$ to $\hat{Q}^{\mathbf{Z}}(\unicode[STIX]{x1D6EF})$ under the Teichmüller flow. That is, $r(x)$ is the smallest positive value such that

$$\begin{eqnarray}\displaystyle T_{r(x)}(x)\in \hat{Q}^{-n}(\hat{\unicode[STIX]{x1D6EF}}) & & \displaystyle \nonumber\end{eqnarray}$$

for some positiveFootnote ¹⁸ integer  $n$ . This means there is some value $\hat{Z}(x)\in \hat{\unicode[STIX]{x1D6EF}}$ such that

(12) $$\begin{eqnarray}\displaystyle T_{r(x)}\hat{Q}^{n}(x)=\hat{Q}^{n}T_{r(x)}(x)=\hat{Z}(x). & & \displaystyle\end{eqnarray}$$

Suppose that $x=(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})\in X_{\unicode[STIX]{x1D6FE}}$ with $\hat{Q}^{n}(x)=(\unicode[STIX]{x1D70B},(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}\unicode[STIX]{x1D706},(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}\unicode[STIX]{x1D70F})\in \hat{\unicode[STIX]{x1D6EF}}$ . Then

$$\begin{eqnarray}\displaystyle r(x)=-\log \Vert (\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}\unicode[STIX]{x1D706}\Vert . & & \displaystyle \nonumber\end{eqnarray}$$

Note here that $r(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F})$ depends only on the coordinates $(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706})$ and we can view $r$ also as a function on  $\unicode[STIX]{x1D6EF}$ .

We will write $\unicode[STIX]{x1D6FE}_{1}.\unicode[STIX]{x1D6FE}_{2}$ or just $\unicode[STIX]{x1D6FE}_{1}\unicode[STIX]{x1D6FE}_{2}$ for the concatenation of two oriented paths $\unicode[STIX]{x1D6FE}_{1}$ and $\unicode[STIX]{x1D6FE}_{2}$ in $\mathfrak{R}$ with compatible endpoints. In $\unicode[STIX]{x1D6FE}_{1}.\unicode[STIX]{x1D6FE}_{2}$ , $\unicode[STIX]{x1D6FE}_{1}$ is the first path traversed. Consider $\unicode[STIX]{x1D6FE}$ with the property that the $\unicode[STIX]{x1D6FE}_{0}$ subpaths of $\unicode[STIX]{x1D6FE}.\unicode[STIX]{x1D6FE}_{0}$ are precisely the beginning and the end segment. We say that such a $\unicode[STIX]{x1D6FE}$ is $\unicode[STIX]{x1D6FE}_{0}$ -adapted. For such a $\unicode[STIX]{x1D6FE}$ , if $x\in X_{\unicode[STIX]{x1D6FE}.\unicode[STIX]{x1D6FE}_{0}}\cap \hat{\unicode[STIX]{x1D6EF}}$ , then

$$\begin{eqnarray}\displaystyle \hat{Z}(x)=\biggl(\unicode[STIX]{x1D70B},\frac{(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}\unicode[STIX]{x1D706}}{\Vert (\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}\unicode[STIX]{x1D706}\Vert },\Vert (\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}\unicode[STIX]{x1D706}\Vert (\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}\unicode[STIX]{x1D70F}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$

The domain of $\hat{Z}$ is therefore $\cup _{\unicode[STIX]{x1D6FE}_{0}}$ -adapted $\unicode[STIX]{x1D6FE}\hat{\unicode[STIX]{x1D6EF}}_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FE}_{0}}$ where

$$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D6EF}}_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FE}_{0}}:=\hat{\unicode[STIX]{x1D6EF}}\cap (Y_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FE}_{0}}\times {\mathcal{K}}_{\unicode[STIX]{x1D6FE}_{0}}). & & \displaystyle \nonumber\end{eqnarray}$$

We extend this definition to $\hat{\unicode[STIX]{x1D6EF}}_{\unicode[STIX]{x1D6FE}_{1}\cdots \unicode[STIX]{x1D6FE}_{N}\unicode[STIX]{x1D6FE}_{0}}:=\hat{\unicode[STIX]{x1D6EF}}\cap (Y_{\unicode[STIX]{x1D6FE}_{1}\cdots \unicode[STIX]{x1D6FE}_{N}\unicode[STIX]{x1D6FE}_{0}}\times {\mathcal{K}}_{\unicode[STIX]{x1D6FE}_{0}})$ where $\unicode[STIX]{x1D6FE}_{1},\ldots ,\unicode[STIX]{x1D6FE}_{N}$ are a sequence of $\unicode[STIX]{x1D6FE}_{0}$ -adapted paths with both endpoints equal to  $\unicode[STIX]{x1D70B}$ .

Notice that the mapping $\hat{Z}$ has the following properties.

1. (i) The map $\hat{Z}$ is a skew extension of the mapping $Z:\unicode[STIX]{x1D6EF}\rightarrow \unicode[STIX]{x1D6EF}$ defined Lebesgue almost everywhere by

   $$\begin{eqnarray}\displaystyle Z(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706})=\biggl(\unicode[STIX]{x1D70B},\frac{(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}\unicode[STIX]{x1D706}}{\Vert (\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}\unicode[STIX]{x1D706}\Vert }\biggr),\quad (\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706})\in Y_{\unicode[STIX]{x1D6FE}.\unicode[STIX]{x1D6FE}_{0}}. & & \displaystyle \nonumber\end{eqnarray}$$
   The connected components of the domain of $Z$ are the sets
   $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FE}_{0}}:=\unicode[STIX]{x1D6EF}\cap Y_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FE}_{0}}. & & \displaystyle \nonumber\end{eqnarray}$$

2. (ii) The maps $\hat{Z}$ and $Z$ preserve $\Vert \unicode[STIX]{x1D706}\Vert =1$ . This is usually referred to as renormalization.

3. (iii) The maps $\hat{Z}$ and $Z$ involve many iterations of Rauzy induction and this is usually referred to as time acceleration. This is first due to Zorich [Reference ZorichZor96]; see also [Reference ZorichZor06, § 5.3] for further discussion.

4. (iv) The map $\hat{Z}$ (respectively  $Z$ ) preserves the Lebesgue measure $\hat{m}$ (respectively  $m$ ).

Following [Reference Avila, Gouëzel and YoccozAGY06, § 4.2.1], in order to enforce hyperbolicity of the map $\hat{Z}$ (cf. Proposition 3.1 and Lemma 3.2) one puts adapted metrics on $\unicode[STIX]{x1D6EF}$ and  $\hat{\unicode[STIX]{x1D6EF}}$ . On $\unicode[STIX]{x1D6EF}$ we put the Hilbert metric $d_{\unicode[STIX]{x1D6EF}}$ coming from the inclusion $\unicode[STIX]{x1D6EF}\rightarrow Y_{\unicode[STIX]{x1D70B}}$ and on $\hat{\unicode[STIX]{x1D6EF}}$ we consider the product metric

$$\begin{eqnarray}\displaystyle d_{\hat{\unicode[STIX]{x1D6EF}}}((\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706},\unicode[STIX]{x1D70F}),(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D70F}^{\prime })):=d_{\unicode[STIX]{x1D6EF}}((\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706}),(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706}^{\prime }))+d_{{\mathcal{K}}_{\unicode[STIX]{x1D70B}}}(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70F}^{\prime }), & & \displaystyle \nonumber\end{eqnarray}$$

where $d_{{\mathcal{K}}_{\unicode[STIX]{x1D70B}}}$ is the Euclidean distance in ${\mathcal{K}}_{\unicode[STIX]{x1D70B}}$ . These metrics induce Finsler metric structures on $\unicode[STIX]{x1D6EF}$ and $\hat{\unicode[STIX]{x1D6EF}}$ that make them into Finsler manifolds.

2.12 Flow on sections of associated bundles in the suspension model

We may now map

$$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D6EF}}_{r}:=\{(x,s):x\in \hat{\unicode[STIX]{x1D6EF}},s\in [0,r(x))\} & & \displaystyle \nonumber\end{eqnarray}$$

homeomorphically to a part of $X_{\unicode[STIX]{x1D70B}}^{(1)}$ by the map

(13) $$\begin{eqnarray}\displaystyle P:(x,s)\mapsto T_{s}(x). & & \displaystyle\end{eqnarray}$$

The image $X_{\unicode[STIX]{x1D70B}}^{\prime (1)}$ of $P$ is up to a Lebesgue-null set, a fundamental domain for the action of $\hat{Q}$ on $X^{(1)}$ . Given a section of $H_{1}({\mathcal{M}};\unicode[STIX]{x1D70C})$ , its pullback to $X^{(1)}$ is therefore determined (up to a zero measure set) by its values on $X_{\unicode[STIX]{x1D70B}}^{\prime (1)}\subset X_{\unicode[STIX]{x1D70B}}^{(1)}$ .

The pushforward of Lebesgue measure under the mapping in (13) is Lebesgue measure. We write $\hat{m}_{r}=\hat{m}\otimes \operatorname{Leb}$ for the Lebesgue measure on  $\hat{\unicode[STIX]{x1D6EF}}_{r}$ .

As explained in § 2.3, ${\mathcal{T}}_{t}$ acts by ${\mathcal{T}}_{t}^{\ast }$ on sections of $H_{1}({\mathcal{M}};\unicode[STIX]{x1D70C})$ . If (after pullback) we view a section $\tilde{\unicode[STIX]{x1D70E}}$ as a ${\mathcal{W}}$ -valued function satisfying (11) and then view $\tilde{\unicode[STIX]{x1D70E}}$ as a ${\mathcal{W}}$ -valued function $\hat{\unicode[STIX]{x1D70E}}$ on $\hat{\unicode[STIX]{x1D6EF}}_{r}$ by the mapping in (13) then the action of $T_{t}^{\ast }$ on $\hat{\unicode[STIX]{x1D70E}}$ will be denoted by $\hat{T}_{t}^{\ast }$ and defined as follows. Let $\unicode[STIX]{x1D6FE}$ be $\unicode[STIX]{x1D6FE}_{0}$ -adapted with $l(\unicode[STIX]{x1D6FE})=n$ . If $x\in X_{\unicode[STIX]{x1D6FE}.\unicode[STIX]{x1D6FE}_{0}}^{(1)}\cap \hat{\unicode[STIX]{x1D6EF}}$ and $t+s\in [r(x),r(x)+r(\hat{Z}(x)))$ , then

$$\begin{eqnarray}\displaystyle [\hat{T}_{t}^{\ast }\hat{\unicode[STIX]{x1D70E}}](x,s) & = & \displaystyle \hat{\unicode[STIX]{x1D70E}}(x,t+s)\nonumber\\ \displaystyle & = & \displaystyle \tilde{\unicode[STIX]{x1D70E}}(T_{t+s}x)\stackrel{(11)}{=}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })\tilde{\unicode[STIX]{x1D70E}}(\hat{Q}^{n}T_{t+s}x)\nonumber\\ \displaystyle & \stackrel{(12)}{=} & \displaystyle \unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })\tilde{\unicode[STIX]{x1D70E}}(T_{t+s-r(x)}\hat{Z}(x))=\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })\hat{\unicode[STIX]{x1D70E}}(\hat{Z}(x),t+s-r(x)).\nonumber\end{eqnarray}$$

Let

$$\begin{eqnarray}\displaystyle r^{(N)}(x):=r(x)+r(\hat{Z}(x))+\cdots +r(\hat{Z}^{N-1}(x)). & & \displaystyle \nonumber\end{eqnarray}$$

For $\unicode[STIX]{x1D6FE}_{1},\unicode[STIX]{x1D6FE}_{2},\ldots ,\unicode[STIX]{x1D6FE}_{N}$ each $\unicode[STIX]{x1D6FE}_{0}$ -adapted, $x\in X_{\unicode[STIX]{x1D6FE}_{1}.\unicode[STIX]{x1D6FE}_{2}\cdots \unicode[STIX]{x1D6FE}_{N}.\unicode[STIX]{x1D6FE}_{0}}^{(1)}\cap \hat{\unicode[STIX]{x1D6EF}}$ , and $t+s\in [r^{(N)}(x),r^{(N+1)}(x))$ , we have then

(14) $$\begin{eqnarray}\displaystyle [\hat{T}_{t}^{\ast }\hat{\unicode[STIX]{x1D70E}}](x,s)=\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}_{1}}^{\ast })\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}_{2}}^{\ast })\cdots \unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}_{N}}^{\ast }).\hat{\unicode[STIX]{x1D70E}}(\hat{Z}^{N}(x),t+s-r^{(N)}(x)). & & \displaystyle\end{eqnarray}$$

This is the master equation for the Teichmüller flow on sections of $H_{1}({\mathcal{M}};\unicode[STIX]{x1D70C})$ in our suspension model. Notice that the argument of $\hat{\unicode[STIX]{x1D70E}}$ in the right-hand side of (14) defines a mapping we call

$$\begin{eqnarray}\displaystyle \hat{T}_{t}:\hat{\unicode[STIX]{x1D6EF}}_{r}\rightarrow \hat{\unicode[STIX]{x1D6EF}}_{r},\quad \hat{T}_{t}(x,s):=(\hat{Z}^{N}(x),t+s-r^{(N)}(x)) & & \displaystyle \nonumber\end{eqnarray}$$

for $x\in X_{\unicode[STIX]{x1D6FE}_{1}.\unicode[STIX]{x1D6FE}_{2}\cdots \unicode[STIX]{x1D6FE}_{N}.\unicode[STIX]{x1D6FE}_{0}}^{(1)}\cap \hat{\unicode[STIX]{x1D6EF}}$ and $t+s\in [r^{(N)}(x),r^{(N+1)}(x))$ . Then $\hat{T}_{t}$ is the suspension flow over $\hat{Z}$ with roof function  $r$ . The flow $\hat{T}_{t}$ lifts the Teichmüller flow under the mapping in (13) and as a consequence, Lebesgue measure $\hat{m}_{r}$ on $\hat{\unicode[STIX]{x1D6EF}}_{r}$ is invariant under  $\hat{T}_{t}$ .

Since the roof function $r$ depends only on a coordinate in $\unicode[STIX]{x1D6EF}$ we may also define

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}_{r}=\{(y,s):y\in \unicode[STIX]{x1D6EF},s\in [0,r(y))\}. & & \displaystyle \nonumber\end{eqnarray}$$

We write $m_{r}$ for the Lebesgue measure on $\unicode[STIX]{x1D6EF}_{r}$ . We also define for $r\in Z^{-(N-1)}(\unicode[STIX]{x1D6EF})$

$$\begin{eqnarray}\displaystyle r^{(N)}(y):=r(y)+r(Z(y))+\cdots +r(Z^{N-1}(y)). & & \displaystyle \nonumber\end{eqnarray}$$

We may define a similar operator to $\hat{T}_{t}^{\ast }$ that we will call $T_{t}^{\ast }$ that will act on ${\mathcal{W}}$ -valued functions on  $\unicode[STIX]{x1D6EF}_{r}$ . For $\unicode[STIX]{x1D70E}:\unicode[STIX]{x1D6EF}_{r}\rightarrow {\mathcal{W}}$ , $\unicode[STIX]{x1D6FE}_{1},\unicode[STIX]{x1D6FE}_{2},\ldots ,\unicode[STIX]{x1D6FE}_{N}$ each $\unicode[STIX]{x1D6FE}_{0}$ -adapted, $t+s\in [r^{(N)}(y),r^{(N+1)}(y))$ and $y\in \unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FE}_{1}.\unicode[STIX]{x1D6FE}_{2}\cdots \unicode[STIX]{x1D6FE}_{N}.\unicode[STIX]{x1D6FE}_{0}}$ we define

(15) $$\begin{eqnarray}\displaystyle [T_{t}^{\ast }\unicode[STIX]{x1D70E}](y,s)=\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}_{1}}^{\ast })\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}_{2}}^{\ast })\cdots \unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}_{N}}^{\ast }).\unicode[STIX]{x1D70E}(Z^{N}(y),t+s-r^{(N)}(y)). & & \displaystyle\end{eqnarray}$$

We give $\unicode[STIX]{x1D6EF}_{r}$ and $\hat{\unicode[STIX]{x1D6EF}}_{r}$ Finsler metrics that are the product of the Finsler metric on $\unicode[STIX]{x1D6EF}$ (respectively  $\hat{\unicode[STIX]{x1D6EF}}$ ) with the usual metric in the $s$ direction.

2.13 Preliminary choice of $\unicode[STIX]{x1D6FE}_{0}$

Recall $\unicode[STIX]{x1D6FE}_{0}$ is a path in $\mathfrak{R}$ beginning and ending in  $\unicode[STIX]{x1D70B}$ . We now explain the choice of $\unicode[STIX]{x1D6FE}_{0}$ that is made in [Reference Avila, Gouëzel and YoccozAGY06]. Avila, Gouëzel and Yoccoz require that the following hold.

(Strongly Positive)

$\unicode[STIX]{x1D6FE}_{0}$ is a strongly positive path, meaning that all the entries of $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast }$ are positive and moreover $(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}_{0}}^{\ast })^{-1}$ maps $\overline{{\mathcal{K}}}_{\unicode[STIX]{x1D70B}}-\{0\}$ into ${\mathcal{K}}_{\unicode[STIX]{x1D70B}}$ .

(Neat)

$\unicode[STIX]{x1D6FE}_{0}$ is neat, meaning that $\unicode[STIX]{x1D6FE}_{0}=\unicode[STIX]{x1D6FE}^{\prime }\unicode[STIX]{x1D6FE}_{e}=\unicode[STIX]{x1D6FE}_{s}\unicode[STIX]{x1D6FE}^{\prime }$ implies $\unicode[STIX]{x1D6FE}^{\prime }$ is trivial or $\unicode[STIX]{x1D6FE}^{\prime }=\unicode[STIX]{x1D6FE}_{0}$ . This means in any path, occurrences of $\unicode[STIX]{x1D6FE}_{0}$ are (edge) disjoint. Therefore $\unicode[STIX]{x1D6FE}_{0}$ -adapted $\unicode[STIX]{x1D6FE}$ are precisely those of the form

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{0}\unicode[STIX]{x1D6FE}^{\prime }, & & \displaystyle \nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}^{\prime }$ does not contain $\unicode[STIX]{x1D6FE}_{0}$ as a subpath.

According to [Reference Avila, Gouëzel and YoccozAGY06, § 4.13], such a choice of $\unicode[STIX]{x1D6FE}_{0}$ is possible. However, in the present paper, we must choose $\unicode[STIX]{x1D6FE}_{0}$ more carefully, while still making sure $\unicode[STIX]{x1D6FE}_{0}$ is strongly positive and neat. This is done in § 5.1. For now, assume that $\unicode[STIX]{x1D6FE}_{0}$ is strongly positive and neat.

3.1 Dynamical setup

The following definitions and results are from [Reference Avila, Gouëzel and YoccozAGY06]. Recall the maps $\hat{Z}$ and $Z$ introduced in § 2.11. Throughout we use the Finsler metric on the tangent bundle to $\unicode[STIX]{x1D6EF}$ defined in § 2.11. We write $D$ for the total derivative of a function. We write $C^{0}(\unicode[STIX]{x1D6EF})$ for the uniform norm. For a ${\mathcal{W}}$ -valued function  $F$ , $\Vert DF\Vert$ refers to operator norm with respect to the Finsler metric on the fibres and the Hilbert space metric on  ${\mathcal{W}}$ . When we write $\bigcup _{\unicode[STIX]{x1D6FE}}^{\ast }$ or $\sum _{\unicode[STIX]{x1D6FE}}^{\ast }$ it means that we restrict the indexing to $\unicode[STIX]{x1D6FE}_{0}$ -adapted  $\unicode[STIX]{x1D6FE}$ . We assume here that $\unicode[STIX]{x1D6FE}_{0}$ is strongly positive and neat as in § 2.13, since these conditions are required for the results of Avila, Gouëzel and Yoccoz [Reference Avila, Gouëzel and YoccozAGY06].

Theorem 3.4 [Reference Avila, Gouëzel and YoccozAGY06, Theorem 4.6].

The roof function $r$ has exponential tails. This means there is $\unicode[STIX]{x1D70E}_{0}>0$ such that

$$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x1D6EF}}\exp (\unicode[STIX]{x1D70E}_{0}r)\,dm<\infty . & & \displaystyle \nonumber\end{eqnarray}$$

3.2 The main technical results

The following will be the precise version of Theorem 1.4(C). Recall the definition of ${\mathcal{M}}(q)$ from § 2.1.

Theorem 3.5. There exists $\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D702}>0$ and $Q_{0}\in \mathbf{Z}_{+}$ such that for all $q$ coprime to  $Q_{0}$ , for all $u$ , $v\in C^{1}({\mathcal{M}}(q))$ whose supports project into a compact set $K\subset {\mathcal{M}}$ , there exists $C=C(K)>0$ such that for all $t\geqslant 0$

$$\begin{eqnarray}\displaystyle \biggl|\int u.v\circ {\mathcal{T}}_{t}\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}-|\unicode[STIX]{x1D6E4}_{q}|^{-1}\biggl(\int u\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}\biggr)\biggl(\int v\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}\biggr)\biggr|\leqslant C(K)\Vert u\Vert _{C^{1}}\Vert v\Vert _{C^{1}}q^{\unicode[STIX]{x1D702}}e^{-\unicode[STIX]{x1D6FF}t}. & & \displaystyle \nonumber\end{eqnarray}$$

The key feature of this estimate is that $\unicode[STIX]{x1D6FF}$ does not depend on  $q$ .

For any Finsler manifold $X$ and Hilbert space ${\mathcal{W}}$ we may define the Banach space of $C^{1}$   ${\mathcal{W}}$ -valued functions on $X$ as in § 2.3. Recall from §§ 2.11 and 2.12 that there are Finsler metric structures on $\unicode[STIX]{x1D6EF}$ , $\hat{\unicode[STIX]{x1D6EF}}$ , $\unicode[STIX]{x1D6EF}_{r}$ , $\hat{\unicode[STIX]{x1D6EF}}_{r}$ . If $(\unicode[STIX]{x1D70C},{\mathcal{W}})$ is a unitary representation we write e.g. $C^{1}(\unicode[STIX]{x1D6EF};\unicode[STIX]{x1D70C})$ for the $C^{1}$   ${\mathcal{W}}$ -valued functions on  $\unicode[STIX]{x1D6EF}$ , with respect to the Finsler metric. We make a reduction of Theorem 3.5 to the following that is analogous to [Reference Avila, Gouëzel and YoccozAGY06, Theorem 2.7].

Theorem 3.6. There exists $C,\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D702}>0$ and $Q_{0}\in \mathbf{Z}_{+}$ such that for all $q$ coprime to  $Q_{0}$ , for all $U,V\in C^{1}(\hat{\unicode[STIX]{x1D6EF}}_{r};\unicode[STIX]{x1D70C}_{q})$ and all $t\geqslant 0$

$$\begin{eqnarray}\displaystyle \biggl|\int \langle U,T_{t}^{\ast }V\rangle \,d\hat{m}_{r}\biggr|\leqslant C\Vert U\Vert _{C^{1}}\Vert V\Vert _{C^{1}}q^{\unicode[STIX]{x1D702}}e^{-\unicode[STIX]{x1D6FF}t}. & & \displaystyle \nonumber\end{eqnarray}$$

We now explain how Theorem 3.5 reduces to Theorem 3.6.

Passage from Theorem 3.6 to Theorem 3.5.

Note that in the context of Theorem 3.5, we can write $u=\tilde{u} _{0}+u^{\prime }$ , $v=\tilde{v}_{0}+v^{\prime }$ with $u^{\prime },v^{\prime }\in L_{\star }^{2}({\mathcal{M}}(q))$ and $\tilde{u} _{0},\tilde{v}_{0}$ given by lifts of functions from  ${\mathcal{M}}$ . In other words, if $\operatorname{cover}_{q}:{\mathcal{M}}(q)\rightarrow {\mathcal{M}}$ is the covering map, there are functions $u_{0}$ and $v_{0}$ such that $\tilde{u} _{0}=u_{0}\circ \operatorname{cover}_{q}$ and $\tilde{v}_{0}=v_{0}\circ \operatorname{cover}_{q}$ . Since $\tilde{u} _{0}$ (respectively $\tilde{v}_{0}$ ) is obtained from $u$ (respectively  $v$ ) by averaging over $\unicode[STIX]{x1D6E4}_{q}$ , and the Finsler metric on ${\mathcal{X}}(\unicode[STIX]{x1D705})$ is $\unicode[STIX]{x1D6E4}$ -invariant, we have estimates

$$\begin{eqnarray}\displaystyle \Vert \tilde{u} _{0}\Vert _{C^{1}}=\Vert u_{0}\Vert _{C^{1}}\leqslant \Vert u\Vert _{C^{1}},\quad \Vert \tilde{v}_{0}\Vert _{C^{1}}=\Vert v_{0}\Vert _{C^{1}}\leqslant \Vert v\Vert _{C^{1}} & & \displaystyle \nonumber\end{eqnarray}$$

and hence also by the triangle inequality

$$\begin{eqnarray}\displaystyle \Vert u^{\prime }\Vert _{C^{1}}\leqslant 2\Vert u\Vert _{C^{1}},\quad \Vert v^{\prime }\Vert _{C^{1}}\leqslant 2\Vert v\Vert _{C^{1}}. & & \displaystyle \nonumber\end{eqnarray}$$

Also note that $\int u^{\prime }\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}=\int v^{\prime }\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}=0$ . Moreover since the supports of  $u$ , $v$ project to $K$ in  ${\mathcal{M}}$ , the same holds for $\tilde{u} _{0},\tilde{v}_{0},u^{\prime },v^{\prime },u_{0},v_{0}$ . Since ${\mathcal{T}}_{t}$ preserves $L_{\star }^{2}({\mathcal{M}}(q))$ and its orthogonal complement, we have

$$\begin{eqnarray}\displaystyle \int u.v\circ {\mathcal{T}}_{t}\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}=\int \tilde{u} _{0}.\tilde{v}_{0}\circ {\mathcal{T}}_{t}\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}+\int u^{\prime }.v^{\prime }\circ {\mathcal{T}}_{t}\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}. & & \displaystyle \nonumber\end{eqnarray}$$

We can replace the first term by

$$\begin{eqnarray}\displaystyle |\unicode[STIX]{x1D6E4}_{q}|\int u_{0}.v_{0}\circ {\mathcal{T}}_{t}\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}} & & \displaystyle \nonumber\end{eqnarray}$$

which by exponential mixing on ${\mathcal{M}}$ [Reference Avila, Gouëzel and YoccozAGY06, Theorem 2.14] is for some $\unicode[STIX]{x1D6FF}^{\prime }>0$

$$\begin{eqnarray}\displaystyle |\unicode[STIX]{x1D6E4}_{q}|\biggl(\int u_{0}\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}}\biggr)\biggl(\int v_{0}\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}}\biggr)+O_{K}(|\unicode[STIX]{x1D6E4}_{q}\Vert \Vert u\Vert _{C^{1}}\Vert v\Vert _{C^{1}}e^{-\unicode[STIX]{x1D6FF}^{\prime }t}). & & \displaystyle \nonumber\end{eqnarray}$$

Notice that since $\unicode[STIX]{x1D6FF}^{\prime }$ depends only on ${\mathcal{M}}$ and, for some $\unicode[STIX]{x1D702}>0$ , $|\unicode[STIX]{x1D6E4}_{q}|\ll q^{\unicode[STIX]{x1D702}}$ for all  $q$ , the error term here is of the form as in Theorem 3.5. This also explains why the error term of Theorem 3.5 must contain a $q^{\unicode[STIX]{x1D702}}$ factor.

Since $\int u_{0}\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}}=|\unicode[STIX]{x1D6E4}_{q}|^{-1}\int \tilde{u} _{0}\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}$ and similarly for $v_{0},\tilde{v}_{0}$ , we have by putting the previous arguments together

$$\begin{eqnarray}\displaystyle \int u.v\circ {\mathcal{T}}_{t}\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)} & = & \displaystyle |\unicode[STIX]{x1D6E4}_{q}|^{-1}\biggl(\int u\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}\biggr)\biggl(\int v\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}\biggr)+O_{K}(q^{\unicode[STIX]{x1D702}}\Vert u\Vert _{C^{1}}\Vert v\Vert _{C^{1}}e^{-\unicode[STIX]{x1D6FF}^{\prime }t})\nonumber\\ \displaystyle & & \displaystyle +\,\int u^{\prime }.v^{\prime }\circ {\mathcal{T}}_{t}\,d\unicode[STIX]{x1D708}_{{\mathcal{M}}(q)}.\nonumber\end{eqnarray}$$

This reduces Theorem 3.5 to the case of $u=u^{\prime }$ , $v=v^{\prime }\in L_{\star }^{2}({\mathcal{M}}(q))$ . Now assume this is the case.

We apply Lemma 2.1 to obtain sections $u^{\ast },v^{\ast }\in L^{2}(H_{1}({\mathcal{M}};\unicode[STIX]{x1D70C}_{q}))$ that have the same $C^{1}$ norms as $u$ and  $v$ . To apply Theorem 3.6 to the correlation function of $u^{\ast }$ and $v^{\ast }$ and conclude the proof, one needs to use the correspondence from § 2.12 to lift $u^{\ast }$ and $v^{\ast }$ to continuously differentiable ${\mathcal{W}}$ -valued functions $u^{\ast \ast }$ and $v^{\ast \ast }$ on $\hat{\unicode[STIX]{x1D6EF}}_{r}$ . However, $u^{\ast \ast }$ and $v^{\ast \ast }$ may not have bounded $C^{1}$ norms, because of distortion between the Finsler metric structures on $\hat{\unicode[STIX]{x1D6EF}}_{r}$ and  ${\mathcal{M}}$ . So one needs to perform some ‘chopping’ and ‘smoothing’ to conclude the result and it is at this stage that the condition on the support of $u^{\ast }$ and $v^{\ast }$ must be used. One may obtain estimates for $L^{p}$ norms of $u^{\ast }$ and $v^{\ast }$ in terms of their $C^{1}$ norms and the compact set  $K$ . Once this is done, the rest of the argument is as in [Reference Avila, Gouëzel and YoccozAGY06, pp. 166–169]. It applies in the same way to vector-valued functions as to scalar-valued functions.◻

3.3 Entrance of the transfer operator

We now recall the definition of the spaces ${\mathcal{B}}_{0}$ and ${\mathcal{B}}_{1}$ from [Reference Avila, Gouëzel and YoccozAGY06].

Definition 3.7. A function $U:\unicode[STIX]{x1D6EF}_{r}\rightarrow {\mathcal{W}}$ is in ${\mathcal{B}}_{0}(\unicode[STIX]{x1D6EF}_{r};\unicode[STIX]{x1D70C})$ if it is bounded, continuously differentiable on each set

$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D6EF}_{r})_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FE}_{0}}:=\{(y,t):y\in \unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FE}_{0}},t\in (0,r(y))\}\quad \unicode[STIX]{x1D6FE}~\text{is}~\unicode[STIX]{x1D6FE}_{0}\text{-}\text{adapted} & & \displaystyle \nonumber\end{eqnarray}$$

and also $\sup _{(y,t)\in \bigcup ^{\ast }(\unicode[STIX]{x1D6EF}_{r})_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FE}_{0}}}\Vert DU(y,t)\Vert <\infty$ . Define the norm

$$\begin{eqnarray}\displaystyle \Vert U\Vert _{{\mathcal{B}}_{0}(\unicode[STIX]{x1D6EF}_{r};\unicode[STIX]{x1D70C})}:=\sup _{(y,t)\in \bigcup ^{\ast }(\unicode[STIX]{x1D6EF}_{r})_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FE}_{0}}}\Vert U(y,t)\Vert +\sup _{(y,t)\in \bigcup ^{\ast }(\unicode[STIX]{x1D6EF}_{r})_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FE}_{0}}}\Vert DU(y,t)\Vert . & & \displaystyle \nonumber\end{eqnarray}$$

Definition 3.8. A function $U:\unicode[STIX]{x1D6EF}_{r}\rightarrow {\mathcal{W}}$ is in ${\mathcal{B}}_{1}(\unicode[STIX]{x1D6EF}_{r};\unicode[STIX]{x1D70C})$ if it is bounded and there exists a constant $C>0$ such that for all fixed $y\in \cup ^{\ast }\unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FE}_{0}}$ , the function $t\mapsto U(y,t)$ is of bounded variationFootnote ¹⁹ on the interval $(0,r(y))$ and its variation $\operatorname{Var}_{(0,r(y))}(t\mapsto U(y,t))$ is bounded by $Cr(y)$ . Let

$$\begin{eqnarray}\displaystyle \Vert U\Vert _{{\mathcal{B}}_{1}}=\sup _{(y,t)\in \bigcup ^{\ast }(\unicode[STIX]{x1D6EF}_{r})_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FE}_{0}}}\Vert U(y,t)\Vert +\sup _{y\in \bigcup ^{\ast }(\unicode[STIX]{x1D6EF})_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FE}_{0}}}\frac{\operatorname{Var}_{(0,r(y))}(t\mapsto U(y,t))}{r(y)}. & & \displaystyle \nonumber\end{eqnarray}$$

As in [Reference Avila, Gouëzel and YoccozAGY06] we reduce to decay of correlations for the $\unicode[STIX]{x1D70C}$ -skew extension of $\unicode[STIX]{x1D6EF}_{r}$ rather than  $\hat{\unicode[STIX]{x1D6EF}}_{r}$ .

Theorem 3.9 (Decay of correlations).

There exists $C,\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D702}>0$ and $Q_{0}\in \mathbf{Z}_{+}$ such that for all $q$ coprime to  $Q_{0}$ , for all $U\in {\mathcal{B}}_{0}(\unicode[STIX]{x1D6EF}_{r};\unicode[STIX]{x1D70C}_{q})$ and $V\in {\mathcal{B}}_{1}(\unicode[STIX]{x1D6EF};\unicode[STIX]{x1D70C}_{q})$ , for all $t\geqslant 0$ ,

$$\begin{eqnarray}\displaystyle \biggl|\int \langle U,T_{t}^{\ast }V\rangle \,dm_{r}\biggr|\leqslant Cq^{\unicode[STIX]{x1D702}}\Vert U\Vert _{{\mathcal{B}}_{0}}\Vert V\Vert _{{\mathcal{B}}_{1}}e^{-\unicode[STIX]{x1D6FF}t}. & & \displaystyle \nonumber\end{eqnarray}$$

This is proved for scalar-valued functions in [Reference Avila, Gouëzel and YoccozAGY06, Theorem 7.3]. The key point of Theorem 3.9 is the uniformity in  $q$ . The passage from Theorem 3.9 to Theorem 3.6 is handled as in [Reference Avila, Gouëzel and YoccozAGY06, § 8]. In fact, the arguments of [Reference Avila, Gouëzel and YoccozAGY06, § 8] are followed closely and extended to the skew setting by Oh and Winter in [Reference Oh and WinterOW16, Proof of Theorem 1.5]. So we have presently explained the reduction of Theorem 3.5 to Theorem 3.9 whose proof we now take up.

From now on, all integrals are taken with respect to the relevant Lebesgue measure. Following [Reference Avila, Gouëzel and YoccozAGY06] let

$$\begin{eqnarray}\displaystyle A_{t}=\{(y,a)\in \unicode[STIX]{x1D6EF}_{r}:a+t\geqslant r(y)\} & & \displaystyle \nonumber\end{eqnarray}$$

and $B_{t}=\unicode[STIX]{x1D6EF}_{r}\backslash A_{t}$ . We bound

$$\begin{eqnarray}\displaystyle \int _{B_{t}}\langle U,T_{t}^{\ast }V\rangle \leqslant \Vert U\Vert _{{\mathcal{B}}_{0}}\Vert V\Vert _{{\mathcal{B}}_{1}}\int _{y\in \unicode[STIX]{x1D6EF}}\max (r(y)-t,0)\leqslant \Vert U\Vert _{{\mathcal{B}}_{0}}\Vert V\Vert _{{\mathcal{B}}_{1}}\int _{y:r(y)\geqslant t}r(y). & & \displaystyle \nonumber\end{eqnarray}$$

By the Cauchy–Schwarz inequality and since $r$ has exponential tails (Theorem 3.4) the above contributes ${\leqslant}C^{\prime }\Vert U\Vert _{{\mathcal{B}}_{0}}\Vert V\Vert _{{\mathcal{B}}_{1}}\exp (-\unicode[STIX]{x1D6FF}^{\prime }t)$ for some $\unicode[STIX]{x1D6FF}^{\prime }>0$ and $C^{\prime }>0$ that do not depend on $U,V$ or  $\unicode[STIX]{x1D70C}$ . Therefore the proof of Theorem 3.9 reduces to estimating the quantity

$$\begin{eqnarray}\displaystyle I(t):=\int _{A_{t}}\langle U,T_{t}^{\ast }V\rangle & & \displaystyle \nonumber\end{eqnarray}$$

on the order of

(16) $$\begin{eqnarray}\displaystyle I(t)\leqslant Cq^{\unicode[STIX]{x1D702}}\Vert U\Vert _{{\mathcal{B}}_{0}}\Vert V\Vert _{{\mathcal{B}}_{1}}\exp (-\unicode[STIX]{x1D6FF}t) & & \displaystyle\end{eqnarray}$$

for some absolute constants $C,\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D702}>0$ .

We now begin the proof of (16). We will estimate the Laplace transform

(17) $$\begin{eqnarray}\displaystyle \hat{I} (s):=\int _{0}^{\infty }\exp (-st)I(t)\,dt. & & \displaystyle\end{eqnarray}$$

This is convergent for $\Re (s)>0$ since $I$ is bounded using the finiteness of  $m_{r}$ . The estimation of $\hat{I} _{s}(t)$ is closely related to certain skew transfer operators as follows. Using notation of [Reference Avila, Gouëzel and YoccozAGY06], if $F:\unicode[STIX]{x1D6EF}_{r}\rightarrow {\mathcal{W}}$ and $s\in \mathbf{C}$ , let

$$\begin{eqnarray}\displaystyle \hat{F}_{s}(y):=\int _{0}^{r(y)}F(y,\unicode[STIX]{x1D70F})\exp (-s\unicode[STIX]{x1D70F})\,d\unicode[STIX]{x1D70F}. & & \displaystyle \nonumber\end{eqnarray}$$

Then following the proof of [Reference Avila, Gouëzel and YoccozAGY06, Lemma 7.17] and adapting to our $\unicode[STIX]{x1D70C}$ -skew setting we have

(18) $$\begin{eqnarray}\displaystyle \hat{I} (s) & = & \displaystyle \int _{y\in \unicode[STIX]{x1D6EF}}\int _{\unicode[STIX]{x1D70F}=0}^{r(y)}\int _{t+\unicode[STIX]{x1D70F}\geqslant r(y)}e^{-st}\langle U(y,\unicode[STIX]{x1D70F}),[T_{t}^{\ast }V](y,\unicode[STIX]{x1D70F})\rangle \,dt\,d\unicode[STIX]{x1D70F}\,dy\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{k=1}^{\infty }\int _{y\in \unicode[STIX]{x1D6EF}}\int _{\unicode[STIX]{x1D70F}=0}^{r(y)}\int _{\unicode[STIX]{x1D70F}^{\prime }=0}^{r(Z^{k}y)}e^{-s(r^{(k)}(y)+\unicode[STIX]{x1D70F}^{\prime }-\unicode[STIX]{x1D70F})}\langle U(y,\unicode[STIX]{x1D70F}),[T_{r^{(k)}(y)+\unicode[STIX]{x1D70F}^{\prime }-\unicode[STIX]{x1D70F}}^{\ast }V](y,\unicode[STIX]{x1D70F})\rangle \,d\unicode[STIX]{x1D70F}^{\prime }\,d\unicode[STIX]{x1D70F}\,dy.\qquad\end{eqnarray}$$

The manipulation above follows from writing for each $y$ , $t+\unicode[STIX]{x1D70F}=r^{(k)}(y)+\unicode[STIX]{x1D70F}^{\prime }$ with $\unicode[STIX]{x1D70F}^{\prime }\in [0,r(Z^{k}x))$ . For each $y$ and $t$ there is a unique $k$ and $\unicode[STIX]{x1D70F}^{\prime }$ for which this is possible. Supposing more specifically that $y\in \unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FE}_{1}\cdots \unicode[STIX]{x1D6FE}_{k}\unicode[STIX]{x1D6FE}_{0}}$ with each $\unicode[STIX]{x1D6FE}_{i}$ $\unicode[STIX]{x1D6FE}_{0}$ -adapted, we get from (15) that

(19) $$\begin{eqnarray}\displaystyle [T_{r^{(k)}(y)+\unicode[STIX]{x1D70F}^{\prime }-\unicode[STIX]{x1D70F}}^{\ast }V](y,\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}_{1}}^{\ast })\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}_{2}}^{\ast })\cdots \unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}_{k}}^{\ast }).V(Z^{k}(y),\unicode[STIX]{x1D70F}^{\prime }). & & \displaystyle\end{eqnarray}$$

Inserting this into (18) gives that (throwing out a measure zero set)

(20) $$\begin{eqnarray}\displaystyle \hat{I} (s) & = & \displaystyle \mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{\unicode[STIX]{x1D6FE}_{1},\ldots ,\unicode[STIX]{x1D6FE}_{k}}^{\ast }\int _{y\in \unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FE}_{1}\cdots \unicode[STIX]{x1D6FE}_{k}\unicode[STIX]{x1D6FE}_{0}}}\int _{\unicode[STIX]{x1D70F}=0}^{r(y)}\int _{\unicode[STIX]{x1D70F}^{\prime }=0}^{r(Z^{k}y)}e^{-s(r^{(k)}(y)+\unicode[STIX]{x1D70F}^{\prime }-\unicode[STIX]{x1D70F})}\langle U(y,\unicode[STIX]{x1D70F}),(19)\rangle \,d\unicode[STIX]{x1D70F}^{\prime }\,d\unicode[STIX]{x1D70F}\,dy\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{k=1}^{\infty }\mathop{\sum }_{\unicode[STIX]{x1D6FE}_{1},\ldots ,\unicode[STIX]{x1D6FE}_{k}}^{\ast }\int _{y\in \unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FE}_{1}\cdots \unicode[STIX]{x1D6FE}_{k}\unicode[STIX]{x1D6FE}_{0}}}e^{-sr^{(k)}(y)}\langle \hat{U} _{-s}(y),\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}_{1}}^{\ast })\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}_{2}}^{\ast })\cdots \unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}_{k}}^{\ast })\hat{V}_{s}(Z^{k}(y))\rangle \,dy.\qquad\end{eqnarray}$$

Here, we write a $\sum ^{\ast }$ to indicate that the $\unicode[STIX]{x1D6FE}_{i}$ being summed over are all $\unicode[STIX]{x1D6FE}_{0}$ -adapted. The expression (20) is best understood by the skew transfer operator that we now introduce. Recall that $y\in \unicode[STIX]{x1D6EF}$ can be written $y=(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706})$ . The inverse branches of $Z$ are indexed by $\unicode[STIX]{x1D6FE}_{0}$ -adapted $\unicode[STIX]{x1D6FE}$ and are given explicitly by

(21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}:(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D706})\mapsto \biggl(\unicode[STIX]{x1D70B},\frac{\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast }\unicode[STIX]{x1D706}}{\Vert \unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast }\unicode[STIX]{x1D706}\Vert }\biggr),\quad \unicode[STIX]{x1D6EF}\rightarrow \unicode[STIX]{x1D6EF}_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FE}_{0}}. & & \displaystyle\end{eqnarray}$$

The skew transfer operator ${\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}$ is defined for arbitrary unitary $(\unicode[STIX]{x1D70C},{\mathcal{W}})$ and $f:\unicode[STIX]{x1D6EF}\rightarrow {\mathcal{W}}$ by

$$\begin{eqnarray}\displaystyle {\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}[f](y):=\mathop{\sum }_{\unicode[STIX]{x1D6FE}}^{\ast }e^{-sr\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}(y)}J\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}(y)\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}.f\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}(y). & & \displaystyle \nonumber\end{eqnarray}$$

Recall that $J$ is the inverse of the Jacobian of $Z$ with respect to Lebesgue measure. By results of [Reference Avila, Gouëzel and YoccozAGY06] the summation involved in ${\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}$ is convergent (cf. Theorem 3.10 and the discussion afterwards). With the operator ${\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}$ in hand, by making a change of variables of the form $y\mapsto Z^{k}(y)$ one obtains from (20)

(22) $$\begin{eqnarray}\displaystyle \hat{I} (s)=\mathop{\sum }_{k=1}^{\infty }\int _{y\in \unicode[STIX]{x1D6EF}}\langle {\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}^{k}[\hat{U} _{-s}](y),\hat{V}_{s}(y)\rangle \,dy,\quad \Re (s)>0. & & \displaystyle\end{eqnarray}$$

It is clear from inspection of the above that spectral bounds for the operator ${\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}$ will be helpful in estimating  $\hat{I}$ . More precisely, we will aim to analytically continue $\hat{I} (s)$ to a strip $\Re (s)>-\unicode[STIX]{x1D70E}^{\prime }$ with $\unicode[STIX]{x1D70E}^{\prime }>0$ .

3.4 Spectral bounds for transfer operators

It will be useful at times to compare ${\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}$ to the operator on scalar functions on $\unicode[STIX]{x1D6EF}$ given by

$$\begin{eqnarray}\displaystyle {\mathcal{L}}_{s}[f](y):=\mathop{\sum }_{\unicode[STIX]{x1D6FE}}^{\ast }e^{-sr\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}(y)}J\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}(y)f(\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}(y)) & & \displaystyle \nonumber\end{eqnarray}$$

that features in [Reference Avila, Gouëzel and YoccozAGY06, formula (7.13)]. Recall that $\unicode[STIX]{x1D70E}_{0}$ is such that $\int \exp (\unicode[STIX]{x1D70E}_{0}r)\,dm<\infty$ given by Theorem 3.4. The following is given in [Reference Avila, Gouëzel and YoccozAGY06, p. 188].

As a corollary to Theorem 3.10 we may note that for real $\unicode[STIX]{x1D70E}$ with $|\unicode[STIX]{x1D70E}|<\unicode[STIX]{x1D70E}_{1}$ , the infinite sum

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\unicode[STIX]{x1D6FE}}^{\ast }e^{-\unicode[STIX]{x1D70E}r\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}(y)}J\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}(y)={\mathcal{L}}_{\unicode[STIX]{x1D70E}}[1](y) & & \displaystyle \nonumber\end{eqnarray}$$

converges to a $C^{1}$ function of $y\in \unicode[STIX]{x1D6EF}$ . Moreover (see [Reference Avila, Gouëzel and YoccozAGY06, paragraph following Proposition 7.8]) since for $\unicode[STIX]{x1D70E}<\unicode[STIX]{x1D70E}_{1}$ , ${\mathcal{L}}_{\unicode[STIX]{x1D70E}}$ is a continuous perturbation of  ${\mathcal{L}}_{0}$ , by possibly decreasing  $\unicode[STIX]{x1D70E}_{1}$ , we can ensure the sum above is uniformly bounded for all $y\in \unicode[STIX]{x1D6F4}$ and all $\unicode[STIX]{x1D70E}\in (-\unicode[STIX]{x1D70E}_{1},\unicode[STIX]{x1D70E}_{1})$ . This will be useful later.

We now give spectral estimates for ${\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}$ in two regimes: for large imaginary part of $s$ (corresponding to high frequency aspects of the dynamics) and small (bounded) imaginary part of $s$ (corresponding to low frequencies).

(a) $|\Im (s)|\gg 1$ . Here we give spectral bounds for transfer operators ${\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}$ , where $\unicode[STIX]{x1D70C}$ is an arbitrary unitary representation, that come from the method of Dolgopyat [Reference DolgopyatDol98]. In the case of scalar-valued functions on $\unicode[STIX]{x1D6EF}$ these bounds were obtained by Avila, Gouëzel and Yoccoz in [Reference Avila, Gouëzel and YoccozAGY06] by adapting Dolgopyat’s argument to the Teichmüller setting.

To state the next result we introduce the warped norm on $C^{1}(\unicode[STIX]{x1D6EF};\unicode[STIX]{x1D70C})$ by

$$\begin{eqnarray}\displaystyle \Vert u\Vert _{1,t}=\sup _{y\in \unicode[STIX]{x1D6EF}}\Vert u(y)\Vert +\frac{1}{\max (1,|t|)}\sup _{y\in \unicode[STIX]{x1D6EF}}\Vert Du(y)\Vert . & & \displaystyle \nonumber\end{eqnarray}$$

Proposition 3.11. There is $0<\unicode[STIX]{x1D70E}_{0}^{\prime }\leqslant \unicode[STIX]{x1D70E}_{0}$ , $T_{0}>0$ , $C>0$ and $\unicode[STIX]{x1D6FD}<1$ such that for all $s=\unicode[STIX]{x1D70E}+it$ with $|\unicode[STIX]{x1D70E}|\leqslant \unicode[STIX]{x1D70E}_{0}^{\prime }$ and $|t|\geqslant T_{0}$ , for any unitary $(\unicode[STIX]{x1D70C},{\mathcal{W}})$ , $u\in C^{1}(\unicode[STIX]{x1D6EF};\unicode[STIX]{x1D70C})$ and for all $k\in \mathbf{N}$

$$\begin{eqnarray}\displaystyle \Vert {\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}^{k}u\Vert _{L^{2}(\unicode[STIX]{x1D6EF})}\leqslant C\unicode[STIX]{x1D6FD}^{k}\Vert u\Vert _{1,t}. & & \displaystyle \nonumber\end{eqnarray}$$

The version of Proposition 3.11 with no twist by $\unicode[STIX]{x1D70C}$ can be found in [Reference Avila, Gouëzel and YoccozAGY06, Proposition 7.7]. We prove Proposition 3.11 in § 4.

(b) $|\Im (s)|\ll 1$ . Here we give spectral bounds for ${\mathcal{L}}_{s,\unicode[STIX]{x1D70C}_{q}}$ that are good when $|\Im (s)|$ is below a fixed constant.

Proposition 3.12. Let $s=\unicode[STIX]{x1D70E}+it$ . For all $t_{0}>0$ there are constants $C,\unicode[STIX]{x1D702},\unicode[STIX]{x1D716}>0$ , $Q_{0}\in \mathbf{Z}_{+}$ , and $0<\unicode[STIX]{x1D70E}_{1}^{\prime }<\unicode[STIX]{x1D70E}_{0}$ such that when $|\unicode[STIX]{x1D70E}|<\unicode[STIX]{x1D70E}_{1}^{\prime }$ and $|t|<t_{0}$ then for all $u\in C^{1}(\unicode[STIX]{x1D6EF};\unicode[STIX]{x1D70C}_{q})$ , all $k\in \mathbf{N}$ , all $q$ coprime to  $Q_{0}$ , for $s=\unicode[STIX]{x1D70E}+it$ ,

$$\begin{eqnarray}\displaystyle \Vert {\mathcal{L}}_{s,\unicode[STIX]{x1D70C}_{q}}^{k}u\Vert _{C^{1}}\leqslant C(1-\unicode[STIX]{x1D716})^{k}q^{\unicode[STIX]{x1D702}}\Vert u\Vert _{C^{1}}. & & \displaystyle \nonumber\end{eqnarray}$$

Proposition 3.12 is proved in § 5. Propositions 3.11 and 3.12 together with the expression (22) imply Theorem 3.9 by the arguments that we give now.

Proof of Theorem 3.9.

In the first part of the argument we follow [Reference Avila, Gouëzel and YoccozAGY06, Lemma 7.18], and extend the argument to vector-valued functions. Recall we aim to prove (16). We write $s=\unicode[STIX]{x1D70E}+it$ . Suppose $|\unicode[STIX]{x1D70E}|\leqslant \unicode[STIX]{x1D70E}_{1}/4$ for $\unicode[STIX]{x1D70E}_{1}>0$ as in Theorem 3.10. By integration by parts in the flow direction, we have for some $c>0$ and all $x\in \unicode[STIX]{x1D6EF}$

(23) $$\begin{eqnarray}\displaystyle \Vert \hat{U} _{-s}(x)\Vert _{{\mathcal{W}}}\leqslant \frac{ce^{\unicode[STIX]{x1D70E}_{1}r(x)/2}}{\max (1,|t|)}\Vert U\Vert _{{\mathcal{B}}_{0}},\quad \Vert \hat{V}_{s}(x)\Vert \leqslant \frac{ce^{\unicode[STIX]{x1D70E}_{1}r(x)/2}}{\max (1,|t|)}\Vert V\Vert _{{\mathcal{B}}_{1}}. & & \displaystyle\end{eqnarray}$$

We can estimate

$$\begin{eqnarray}\displaystyle \Vert {\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}\hat{U} _{-s}(x)\Vert _{{\mathcal{W}}}\leqslant \mathop{\sum }_{\unicode[STIX]{x1D6FE}}^{\ast }|e^{-sr\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}(x)}J\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}(x)|\Vert \hat{U} _{-s}(\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}x)\Vert _{{\mathcal{W}}}. & & \displaystyle \nonumber\end{eqnarray}$$

We were able to remove the factors here coming from $\unicode[STIX]{x1D70C}$ since the representation is unitary. By the estimate for $\Vert \hat{U} _{-s}(x)\Vert _{{\mathcal{W}}}$ in (23), this is

$$\begin{eqnarray}\displaystyle \leqslant \frac{c\Vert U\Vert _{{\mathcal{B}}_{0}}}{\max (1,|t|)}\mathop{\sum }_{\unicode[STIX]{x1D6FE}}^{\ast }e^{(\unicode[STIX]{x1D70E}_{1}/4)r\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}(x)}J\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}(x)e^{(\unicode[STIX]{x1D70E}_{1}/2)r\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}(x)}. & & \displaystyle \nonumber\end{eqnarray}$$

The sum is bounded by a constant $c^{\prime }>0$ given $|\unicode[STIX]{x1D70E}|\leqslant \unicode[STIX]{x1D70E}_{1}/4$ and Theorem 3.10. Hence by increasing $c$ if necessary

$$\begin{eqnarray}\displaystyle \Vert {\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}\hat{U} _{-s}(x)\Vert _{{\mathcal{W}}}\leqslant \frac{c\Vert U\Vert _{{\mathcal{B}}_{0}}}{\max (1,|t|)} & & \displaystyle \nonumber\end{eqnarray}$$

for all $x\in \unicode[STIX]{x1D6EF}$ . We have (recalling footnote 3)

$$\begin{eqnarray}\displaystyle \Vert D({\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}\hat{U} _{-s})(x)\Vert =\biggl\|\mathop{\sum }_{\unicode[STIX]{x1D6FE}}^{\ast }D[e^{-sr\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}}J\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}.\hat{U} _{-s}\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}](x)\biggr\|. & & \displaystyle \nonumber\end{eqnarray}$$

We have to differentiate $e^{-sr\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}}$ , $J\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}$ , the limits of the integral defining $\hat{U} _{-s}$ , or $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}.\hat{U} _{-s}\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}$ . The latter is the only deviation from [Reference Avila, Gouëzel and YoccozAGY06, Lemma 7.18]. Since $\unicode[STIX]{x1D70C}$ is locally constant, we have

$$\begin{eqnarray}\displaystyle \Vert D[\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}.\hat{U} _{-s}\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}](x)\Vert =\Vert \unicode[STIX]{x1D70C}(\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FE}}^{\ast })^{-1}D[\hat{U} _{-s}\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}](x)\Vert =\Vert D[\hat{U} _{-s}\circ \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6FE}}](x)\Vert . & & \displaystyle \nonumber\end{eqnarray}$$

Therefore, since this is the same estimate [Reference Avila, Gouëzel and YoccozAGY06] obtain for the analogous term, the same arguments as in [Reference Avila, Gouëzel and YoccozAGY06, Lemma 7.18] imply $\Vert D({\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}\hat{U} _{-s})(x)\Vert \leqslant c$ .

Hence putting the previous estimates together, we have ${\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}\hat{U} _{-s}\in C^{1}(\unicode[STIX]{x1D6EF};\unicode[STIX]{x1D70C})$ with

(24) $$\begin{eqnarray}\displaystyle \Vert {\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}\hat{U} _{-s}\Vert _{C^{1}(\unicode[STIX]{x1D6EF};\unicode[STIX]{x1D70C})}\leqslant c\Vert U\Vert _{{\mathcal{B}}_{0}},\quad \Vert {\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}\hat{U} _{-s}\Vert _{1,t}\leqslant \frac{c\Vert U\Vert _{{\mathcal{B}}_{0}}}{\max (1,|t|)}. & & \displaystyle\end{eqnarray}$$

As a clarifying remark, we would have liked to obtain these bounds for $\hat{U} _{-s}$ , but it was not possible, so we used ${\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}\hat{U} _{-s}$ instead. We also have from the bound for $\Vert \hat{V}_{s}\Vert _{\infty }$ from (23),

(25) $$\begin{eqnarray}\displaystyle \Vert \hat{V}_{s}\Vert _{L^{2}(\unicode[STIX]{x1D6EF})}^{2}\leqslant \int _{x\in \unicode[STIX]{x1D6EF}}\Vert \hat{V}_{s}(x)\Vert ^{2}\,dm(x)\leqslant \frac{c^{2}\Vert V\Vert _{{\mathcal{B}}_{1}}^{2}}{\max (1,|t|)^{2}}\int _{x\in \unicode[STIX]{x1D6EF}}e^{\unicode[STIX]{x1D70E}_{1}r(x)}\,dm(x)\leqslant \frac{c^{\prime }\Vert V\Vert _{{\mathcal{B}}_{1}}^{2}}{\max (1,|t|)^{2}}. & & \displaystyle\end{eqnarray}$$

These are all the functional norm bounds we need for the proof. We now proceed to use the spectral bounds for the transfer operator.

Let $\unicode[STIX]{x1D70E}^{\prime \prime }=\min (\unicode[STIX]{x1D70E}_{1}/4,\unicode[STIX]{x1D70E}_{0}^{\prime },\unicode[STIX]{x1D70E}_{1}^{\prime })$ where $\unicode[STIX]{x1D70E}_{0}^{\prime }$ and $\unicode[STIX]{x1D70E}_{1}^{\prime }$ are the constants from Propositions 3.11 and 3.12. We now specialize $\unicode[STIX]{x1D70C}$ to $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{q}$ . Writing ${\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}^{k}={\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}^{k-1}{\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}$ we obtain from Proposition 3.11 and (24) that for $|\unicode[STIX]{x1D70E}|\leqslant \unicode[STIX]{x1D70E}^{\prime \prime }$ , if $|t|\geqslant T_{0}$ ,

$$\begin{eqnarray}\displaystyle \Vert {\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}^{k}\hat{U} _{-s}\Vert _{L^{2}(\unicode[STIX]{x1D6EF})}\leqslant C\unicode[STIX]{x1D6FD}^{k-1}\Vert {\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}\hat{U} _{-s}\Vert _{1,t}\leqslant \frac{C^{\prime }\unicode[STIX]{x1D6FD}^{k}}{\max (1,|t|)}. & & \displaystyle \nonumber\end{eqnarray}$$

Using (25) and the Cauchy–Schwarz inequality, we can bound the terms defining $\hat{I}$ in (22) by

$$\begin{eqnarray}\displaystyle |\langle {\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}^{k}[\hat{U} _{-s}](y),\hat{V}_{s}(y)\rangle |\leqslant \frac{c\unicode[STIX]{x1D6FD}^{k}\Vert U\Vert _{{\mathcal{B}}_{0}}\Vert V\Vert _{{\mathcal{B}}_{1}}}{t^{2}} & & \displaystyle \nonumber\end{eqnarray}$$

for $|t|\geqslant T_{0}$ and some $c>0$ . Hence for $|t|\geqslant T_{0}$ we have

$$\begin{eqnarray}\displaystyle \hat{I} (s)\leqslant \mathop{\sum }_{k=0}^{\infty }\frac{c\unicode[STIX]{x1D6FD}^{k}\Vert U\Vert _{{\mathcal{B}}_{0}}\Vert V\Vert _{{\mathcal{B}}_{1}}}{t^{2}}\leqslant \frac{c^{\prime }\Vert U\Vert _{{\mathcal{B}}_{0}}\Vert V\Vert _{{\mathcal{B}}_{1}}}{t^{2}}. & & \displaystyle \nonumber\end{eqnarray}$$

For $|t|<T_{0}$ we apply Proposition 3.12 with $t_{0}=T_{0}$ and $u={\mathcal{L}}_{s,\unicode[STIX]{x1D70C}_{q}}\hat{U} _{-s}$ to obtain

$$\begin{eqnarray}\displaystyle \Vert {\mathcal{L}}_{s,\unicode[STIX]{x1D70C}_{q}}^{k}\hat{U} _{-s}\Vert _{C^{1}}\leqslant C(1-\unicode[STIX]{x1D716})^{k-1}q^{\unicode[STIX]{x1D702}}\Vert {\mathcal{L}}_{s,\unicode[STIX]{x1D70C}_{q}}\hat{U} _{-s}u\Vert _{C^{1}}\leqslant C^{\prime }(1-\unicode[STIX]{x1D716})^{k}q^{\unicode[STIX]{x1D702}}\Vert U\Vert _{{\mathcal{B}}_{0}}, & & \displaystyle \nonumber\end{eqnarray}$$

where the last inequality used (24). Hence for $|t|<T_{0}$ , using the Cauchy–Schwarz inequality again to bound $|\langle {\mathcal{L}}_{s,\unicode[STIX]{x1D70C}}^{k}[\hat{U} _{-s}](y),\hat{V}_{s}(y)\rangle |$ ,

$$\begin{eqnarray}\displaystyle \hat{I} (s)\leqslant \mathop{\sum }_{k=0}^{\infty }C^{\prime }(1-\unicode[STIX]{x1D716})^{k}q^{\unicode[STIX]{x1D702}}\Vert U\Vert _{{\mathcal{B}}_{0}}\Vert V\Vert _{{\mathcal{B}}_{1}}\leqslant c^{\prime }q^{\unicode[STIX]{x1D702}}\Vert U\Vert _{{\mathcal{B}}_{0}}\Vert V\Vert _{{\mathcal{B}}_{1}}. & & \displaystyle \nonumber\end{eqnarray}$$

These estimates prove that the expression defining $\hat{I} (s)$ is absolutely uniformly convergent on compact sets in $|\unicode[STIX]{x1D70E}|\leqslant \unicode[STIX]{x1D70E}^{\prime \prime }$ . Since each of the terms are analytic, this establishes analytic continuation of $\hat{I} (s)$ to $\Re (s)>-\unicode[STIX]{x1D70E}^{\prime \prime }$ . Since we have established the estimate

$$\begin{eqnarray}\displaystyle \hat{I} (s)\leqslant \frac{c^{\prime }q^{\unicode[STIX]{x1D702}}}{1+|t|^{2}}\Vert U\Vert _{{\mathcal{B}}_{0}}\Vert V\Vert _{{\mathcal{B}}_{1}},\quad |\unicode[STIX]{x1D70E}|\leqslant \unicode[STIX]{x1D70E}^{\prime \prime }, & & \displaystyle \nonumber\end{eqnarray}$$

by inverting the Laplace transform, using a contour integral over the vertical line $\Re (s)=-\unicode[STIX]{x1D70E}^{\prime \prime }/2$ as in [Reference Oh and WinterOW16, Proposition 5.5], we obtain for some $\unicode[STIX]{x1D6FF}>0$

$$\begin{eqnarray}\displaystyle I(t)\leqslant c^{\prime \prime }q^{\unicode[STIX]{x1D702}}e^{-\unicode[STIX]{x1D6FF}t}\Vert U\Vert _{{\mathcal{B}}_{0}}\Vert V\Vert _{{\mathcal{B}}_{1}}. & & \displaystyle \nonumber\end{eqnarray}$$

This completes the proof of Theorem 3.9. ◻

Now, the only outstanding proofs required for Theorem 3.5 are those of Propositions 3.11 and 3.12. These are given in §§ 4 and 5 respectively.