# Resonances for rational Anosov maps on the torus

Julia Slipantschuk111Department of Mathematics, University of Warwick,
Coventry, UK. Email<EMAIL_ADDRESS>Oscar F.
Bandtlow222School of Mathematical Sciences, Queen Mary University of London,
UK. Email<EMAIL_ADDRESS>and Wolfram Just333Institut für Mathematik,
Universität Rostock, Rostock, Germany. Email<EMAIL_ADDRESS>
###### Abstract

A complete description of resonances for rational toral Anosov diffeomorphisms
preserving certain Reinhardt domains is presented. As a consequence it is
shown that every homotopy class of two-dimensional Anosov diffeomorphisms
contains maps with the sequence of resonances decaying stretched-
exponentially. This is achieved by introducing a certain group of rational
toral diffeomorphisms and computing the resonances of the respective
composition operator considered on suitable anisotropic spaces of
hyperfunctions. The class of examples is sufficiently rich to also include
non-linear Anosov maps with trivial resonances, or resonances decaying
exponentially, as well as with or without area-preservation or reversing
symmetries.

###### Contents

  1. 1 Introduction
    1. 1.1 Statement of results
  2. 2 Cone conditions for Anosov diffeomorphisms on ${\mathbb{T}^{2}}$
    1. 2.1 A pedestrian approach
    2. 2.2 Anosov diffeomorphisms with strong mapping conditions
  3. 3 Composition operator for Anosov maps with constant invariant cone fields
    1. 3.1 Hardy-Hilbert spaces on Reinhardt domains
    2. 3.2 Hardy-Hilbert spaces for toral diffeomorphisms
    3. 3.3 Relation to transfer operator
  4. 4 Resonances for certain rational Anosov maps
  5. 5 Anosov maps with different decay rates for resonances
    1. 5.1 A special group of toral diffemorphisms
    2. 5.2 Explicit homotopies of Anosov diffeomorphisms
  6. A Auxiliary results
  7. B Notes on the special group of toral diffeomorphisms

##  1 Introduction

Anosov diffeomorphisms are the simplest truly hyperbolic dynamical systems,
whose long term asymptotic behaviour characterized by correlation decay or
rates of mixing are classical topics in smooth ergodic theory and statistical
physics. The main tool for studying statistical properties for Anosov maps $T$
acting on a compact manifold $M$ is the weighted composition operator444This
operator is also referred to as the transfer operator or Koopman operator,
depending on the weight. defined by

$C_{T,w}\colon f\mapsto w\cdot f\circ T,$

where $w$ is a smooth function on $M$ and $C_{T,w}$ acts on a suitable space
of distributions. There is a considerable amount of recent literature devoted
to the construction of so-called anisotropic spaces for hyperbolic dynamical
systems, on which the above operator is quasicompact, implying a spectral gap
and exponential decay of correlations. The general idea behind the
construction of these spaces is to create sufficient smoothness in the
expanding direction and dual smoothness in the contracting direction, see
[Rug, BKL, GL, BT, FRS, B], to name but a few.

If the Anosov map is real-analytic, it is sometimes possible to prove
compactess and even nuclearity of the above operator, see [Rug, J, FR],
implying that its non-zero spectrum is a sequence of isolated eigenvalues
known as Pollicott-Ruelle resonances, which determine all intrinsic
exponential mixing rates of the given system. However, quantitative results
such as location, or the very existance, of non-trivial resonances are rare, a
few instances in different hyperbolic settings being [FGL, DFG]. Even for
Anosov diffeomorphisms on the torus $\mathbb{T}^{2}$, arguably the simplest
setting of uniformly hyperbolic dynamical systems, it was established only
recently in [A] (after an idea of F. Naud) that non-trivial resonances exist
generically. In [SBJ] the authors presented a one-parameter family of Anosov
maps, proving the presence of infinitely many distinct resonances by
calculating their location explicitly, and also conjectured locations of
resonances for another family of Anosov maps, which was recently proven in
[PoS]. In both cases, after establishing compactness of the transfer operator
on a suitable anisotropic Hilbert space (originally introduced in [FR]), the
eigenvalues are read off from an upper triangular matrix representation of the
operator with respect to a weighted Fourier basis. These results, though
valuable for rigorously establishing the location of resonances for these
particular families, are rather ad hoc as they do not reveal the underlying
spectral structure of the associated operator. In this context, a key
contribution of this article is to explain the underlying structure of
resonances for a class of rational Anosov diffeomorphisms. For this it will be
helpful to work with simple anisotropic spaces, which, albeit less general,
interact well with the analytic structure of the underlying map.

For analytic Anosov maps on $\mathbb{T}^{2}$ with constant invariant stable
and unstable cone fields we define anisotropic Hilbert spaces of
hyperfunctions as a closure of Laurent polynomials under a weighted $L^{2}$
norm with the weight function adapted to the invariant cones, and show that
the respective weighted composition operator is well defined and trace-class.
These spaces are isometrically isomorphic to a direct sum of Hardy-Hilbert
spaces on log-conical Reinhardt domains, with the logarithmic base induced by
the stable and unstable cones. Using this viewpoint and assuming additionally
that these Anosov maps extend holomorphically to certain domains in
${\hat{\mathbb{C}}^{2}}$, we are able to explicitly compute all the
eigenvalues of the respective composition operator. In addition we present a
group of (non-linear) toral diffeomorphisms which satisfy these assumptions
and provide examples in each homotopy class of toral Anosov diffeomorhisms for
which the composition operator has infinitely many distinct non-trivial
resonances $(\lambda_{n})_{n\in\mathbb{N}}$ whose rate of decay is stretched-
exponential, that is, the upper bound of $\exp(-an^{1/2})$ for some $a>0$ is
tight.

Results in [BJS] on resonances for analytic expanding circle maps of degree
$d$, with $|d|>2$, arising from finite Blaschke and anti-Blaschke products,
made it possible to establish exponential lower bounds for the decay rate on a
dense set of analytic expanding circle maps [BN]. In the same vein, the class
of non-linear toral diffeomorphisms presented in this paper will be an
essential ingredient in proving that, generically, the stretched-exponential
decay rate of resonances is optimal within the class of analytic Anosov
diffeomorphisms on $\mathbb{T}^{2}$. Moreover, as all constructions are very
explicit most of our results should extend to Anosov maps in higher
dimensions. These are, however, beyond the current scope and will be pursued
in subsequent works.

###  1.1 Statement of results

We will only consider toral Anosov diffeomorphisms with constant expanding and
contracting cone fields. Restricting to constant invariant cone fields enables
us to work with simple anisotropic Hilbert spaces $H_{\nu}$, which can be seen
as the completion of Laurent polynomials under the $\nu$-weighted
$L^{2}_{\nu}$ inner product for a cone-wise exponential weight function
$\nu\colon{\mathbb{Z}^{2}}\to\mathbb{R}_{\geq 0}$, that is $\nu(n)=e^{f(n)}$
where $f\colon{\mathbb{Z}^{2}}\to\mathbb{R}$ is piecewise linear, with the
pieces being cones in $\mathbb{R}^{2}$. We will make an additional assumption
on the invariant cone fields, termed strongly expanding constant invariant
cone field condition or short-hand (sec), see Definition 2.12 for the precise
definition. If the (constant) unstable invariant cone field can be chosen to
correspond to the positive/negative quadrants $\pm\mathbb{R}_{>0}^{2}$, we
refer to these as positive, and call the respective condition (p-sec).

######  Theorem 1.1.

Let $T$ be an analytic Anosov diffeomorphism of ${\mathbb{T}^{2}}$ satisfying
the (sec) assumption and $w\colon{\mathbb{T}^{2}}\to{\mathbb{C}}$ an analytic
function. Then there exists a cone-wise exponential
$\nu\colon{\mathbb{Z}^{2}}\to\mathbb{R}_{\geq 0}$ such that

$f\mapsto w\cdot f\circ T$

is a well-defined trace-class operator on $H_{\nu}$.

For a special subclass of analytic Anosov diffeomorphisms where cone fields
can be chosen to correspond to quadrants of $\mathbb{R}^{2}$ and the
diffeomorphisms extend holomorphically to certain domains of
${\hat{\mathbb{C}}^{2}}$, we are able to compute all eigenvalues of the above
operator explicitly in terms of multipliers of fixed points on these domains.
To state the results we introduce the notation
$\Sigma=\\{\sigma=(\sigma_{1},\sigma_{2})\colon\sigma_{1},\sigma_{2}\in\\{\pm
1\\}\\}$, and
$D^{\sigma}=\\{z\in{\hat{\mathbb{C}}^{2}}\colon|z_{1}|^{\sigma_{1}}>1,|z_{2}|^{\sigma_{2}}>1\\}$
with $\sigma\in\Sigma$ for the four bidisks in ${\hat{\mathbb{C}}^{2}}$. It
will further be convenient to write
$\mathcal{N}^{1}=\mathbb{N}^{2}_{0}\setminus\\{(0,0)\\}$,
$\mathcal{N}^{-1}=\mathbb{N}^{2}$, and decompose $\Sigma$ as
$\Sigma=\Sigma^{1}\cup\Sigma^{-1}$ with
$\Sigma^{\ell}=\\{\sigma\in\Sigma\colon\sigma_{1}\cdot\sigma_{2}=\ell\\},\ell\in\\{\pm
1\\}$. The notation $T^{\ell}$ for $\ell=-1$ stands for the inverse of $T$. We
write $C_{T}=C_{T,1}$ for the unweighted composition operator.

######  Theorem 1.2.

Let $T$ be an analytic Anosov diffeomorphism on ${\mathbb{T}^{2}}$ satisfying
the (p-sec) assumption. Then there exists a cone-wise exponential
$\nu\colon{\mathbb{Z}^{2}}\to\mathbb{R}_{\geq 0}$ such that $C_{T}$ is a well-
defined trace-class operator on $H_{\nu}$, and its spectral determinant is an
entire function of the form

$\det(\operatorname{Id}-zC_{T})=(1-z)\chi_{T}(z),$

where $\chi_{T}$ is an entire function with zeros outside of
$\operatorname{cl}({\mathbb{D}})$. Moreover, if $T^{\ell}$ holomorphically
extends555Here we view $\mathbb{T}^{2}$ as a subset of
$\hat{}{\mathbb{C}}\times\hat{}{\mathbb{C}}$ and consider extensions of $T$
and $T^{-1}$ to subsets of $\hat{}{\mathbb{C}}\times\hat{}{\mathbb{C}}$ as
holomorphic in the same sense as for mappings of the Riemann sphere. to
$D^{\sigma}$ for all $\sigma\in\Sigma^{\ell}$ and $\ell\in\\{\pm 1\\}$ then
$\chi_{T}$ is the product of two entire functions $\chi^{+1}_{T}$ and
$\chi^{-1}_{T}$, whose zeros are given explicitly. Specifically, for every
$\ell\in\\{\pm 1\\}$ exactly one of the following cases holds:

  1. (i)

if $T^{\ell}(D^{\sigma})\subseteq D^{\sigma}$ for all
$\sigma\in\Sigma^{\ell}$, then

$\chi_{T}^{\ell}(z)=\prod_{n\in\mathcal{N}^{\ell}}\prod_{\sigma\in\Sigma^{\ell}}(1-z\ell^{s}\lambda^{n}_{\sigma})$

where $\lambda_{\sigma}=(\lambda_{\sigma,1},\lambda_{\sigma,2})$ are the
multipliers666The multipliers of $T$ at a point $z$ are the eigenvalues
$\lambda=(\lambda_{1},\lambda_{2})$ of $D_{z}T=\left(\frac{\partial
T_{k}}{\partial z_{l}}\right)_{k,l}$ at $z$. We write
$\lambda^{n}=\lambda_{1}^{n_{1}}\cdot\lambda_{2}^{n_{2}}$ for
$n\in\mathbb{Z}^{2}$. of the unique attracting fixed point $z^{*}_{\sigma}\in
D^{\sigma}$ of $T^{\ell}$, and $s=0$ if $T$ is orientation-preserving and
$s=1$ if it is orientation-reversing. Additionally, it holds that
$z_{\sigma}^{*}=\overline{1/z^{*}_{-\sigma}}$ and
$\lambda_{\sigma}=\overline{\lambda_{-\sigma}}$ for $\sigma\in\Sigma^{\ell}$.

  2. (ii)

if $T^{\ell}(D^{\sigma})\subseteq D^{-\sigma}$ for all
$\sigma\in\Sigma^{\ell}$, then

$\chi^{\ell}_{T}(z)=\prod_{n\in\mathcal{N}^{\ell}}\prod_{\sigma\in\Sigma^{\ell}}(1-z^{2}\lambda^{n}_{\sigma})^{1/2},$

where $\lambda_{\sigma}=\lambda_{-\sigma}$ are the multipliers of the unique
attracting fixed point $z^{*}_{\sigma}\in D^{\sigma}$ of $T^{2\ell}$.
Additionally, $\lambda_{\sigma,1}$ and $\lambda_{\sigma,2}$ are either real,
or complex conjugates of each other.

Clearly, certain linear toral diffeomorphisms satisfy the assumptions of this
theorem. For example, the well-known cat map
$(z_{1},z_{2})\mapsto(z_{1}^{2}z_{2},z_{1}z_{2})$ satisfies assumption (i) and
hence we can compute all fixed points and their multipliers, which however are
all trivial (that is, zero) in this case. The group
${\operatorname{Aut}}(\mathbb{T}^{2})$ of linear diffeomorphisms
(automorphisms) of the torus is isomorphic to
${\operatorname{GL}_{2}}(\mathbb{Z})$. In order to construct non-linear maps,
we shall consider a group of diffeomorphisms generated by a finite set
$\Gamma$ of generators of ${\operatorname{Aut}}(\mathbb{T}^{2})$ and a
(uncountably infinite) set $\mathcal{G}$ of certain rational diffeomorphisms
preserving ${\mathbb{T}^{2}}$. One particular choice for this set is

$\mathcal{G}=\\{(z_{1},z_{2})\mapsto(b_{a}(z_{1}),z_{2})\colon
a\in\mathbb{D}\\}$

with $b_{a}(z)=(z-a)/(1-\overline{a}z)$. We call $\mathcal{F}$ the group of
diffeomorphisms generated by $\Gamma\cup\mathcal{G}$. A certain subset of
$\mathcal{F}$ comprises of hyperbolic diffeomorphisms satisfying the
assumptions of Theorem 1.2. The explicit construction of elements of this set
(see Section 5.1) allows us to compute the sequence of eigenvalues of the
corresponding composition operators and present examples with qualitatively
different decay rates of this sequence (stretched-exponential, exponential,
and trivially super-exponential/all-zero). Using the structure of conjugacy
classes of ${\operatorname{GL}_{2}}(\mathbb{Z})$ we obtain the following
theorem.

######  Theorem 1.3.

Every homotopy class of analytic Anosov diffeomorphisms on ${\mathbb{T}^{2}}$
contains (non-linear) Anosov diffeormorphisms $T\in\mathcal{F}$, such that for
suitable $H_{\nu}$ the corresponding operator $C_{T}$ is well defined and
trace-class, with the entire function
$z\mapsto\det(\operatorname{Id}-zC_{T})=(1-z)\chi_{T}(z)$ as its spectral
determinant. In particular, denoting by $(\lambda_{n})_{n\in\mathbb{N}}$ the
sequence of eigenvalues of $C_{T}$ ordered by modulus in decreasing order, and
counted with multiplicities, we obtain the following.

  1. (i)

For every homotopy class $\mathcal{H}$ and $\eta>0$, there exists
$T\in\mathcal{H}\cap\mathcal{F}$ such that the eigenvalue sequence of $C_{T}$
satisfies

$\lim_{n\to\infty}\frac{-\log|\lambda_{n}|}{n^{1/2}}=\eta.$

  2. (ii)

For every homotopy class $\mathcal{H}$ of orientation-preserving Anosov
diffeomophisms and $\eta>0$, there exists $T\in\mathcal{H}\cap\mathcal{F}$
such that the eigenvalue sequence of $C_{T}$ satisfies

$\lim_{n\to\infty}\frac{-\log|\lambda_{n}|}{n}=\eta.$

  3. (iii)

Every homotopy class $\mathcal{H}$ of Anosov diffeomorphisms not containing a
linear conjugate of one of
$\\{(z_{1},z_{2})\mapsto(z_{1}^{k}z_{2},z_{1}),k\in\mathbb{N}\\}$ has an
element $T\in\mathcal{H}\cap\mathcal{F}$ not smoothly conjugated to a linear
Anosov diffeomorphism, such that

$\chi_{T}(z)=1$

for all $z\in{\mathbb{C}}$.

######  Remark 1.4.

We note that the maps $T\in\mathcal{F}$ above can be written in closed form.
In contrast to the more common setting of analytic perturbations of linear
maps (see, for example, [A, FR]), these maps are not required to be $C^{1}$
close to the respective linear automorphisms, that is, the $C^{1}$ distance
can be arbitrarily large. In fact, as the proof of Theorem 1.3 will show, the
spectral gap for the composition operator associated to the maps in $(i)$ and
$(ii)$ can take an arbitrary value in $(0,1)$. Moreover, under the additional
assumption that $T$ is orientation-preserving, in case $(i)$ the second-
largest eigenvalue $\lambda_{2}\in\mathbb{D}$ and the decay rate $\eta>0$ can
be chosen arbitrarily, independently of each other. The behaviour in $(i)$ is
believed to be generic for two-dimensional Anosov maps, whereas the cases
$(ii)$ and $(iii)$ are exceptional. We also note that under the assumptions of
Theorem 1.2 (as well as of Theorem 1.3), the spectral determinant can be shown
to coincide with the usual dynamical determinant, as the usual trace formula
(see, for example, [FR, Proposition 5]) can be established in this setting.

######  Remark 1.5.

As shown in [FG], any homotopy class $\mathcal{H}$ of analytic Anosov
diffeomorphisms on $\mathbb{T}^{2}$ is path-connected, so that in fact, in the
cases $(i)$-$(iii)$ in Theorem 1.3, any $T^{\prime}\in\mathcal{H}$ is
homotopic via a continuous path of toral diffeomorphisms to a
$T\in\mathcal{H}$ with the respective spectral property. Moreover, by the
Franks-Newhouse classification theorem, every Anosov diffeomorphism of
codimension $1$ on a closed Riemannian manifold777An Anosov diffeomorphism is
said to be of codimension $1$ if the codimension of either its stable or
unstable foliations has dimension $1$. is topologically conjugated to a
hyperbolic toral automorphism, see for example [Hi]. Thus, since any two
smooth surfaces that are homeomorphic are also diffeomorphic, in the above
theorem ${\mathbb{T}^{2}}$ can be replaced with any two-dimensional closed
Riemannian manifold.

This paper is organised as follows. Starting with a motivational example of
the composition operator for the cat map in Section 2.1, we discuss its
boundedness and compactness on an anisotropic space of hyperfunctions relevant
to the current work. In Section 2.2 we consider a class of analytic toral
Anosov diffeomorphisms with strongly expanding constant invariant cone (sec)
fields and show how properties on the tangent bundle are translated into
properties of the map in a small complex neighbourhood of the torus. We start
Section 3 by summarising properties of Hardy-Hilbert spaces on log-conical
Reinhardt domains and realizing the anisotropic Hilbert spaces of
hyperfunctions from Section 2.1 as direct sums of these Hardy-Hilbert spaces.
The main theorem of this section is Theorem 3.24, which together with a
standard factorisation argument yields a trace-class weighted composition
operator for analytic Anosov maps with the (sec) property, thus proving
Theorem 1.1. Section 4 is devoted to the proof of Theorem 1.2, that is, the
computation of resonances of the composition operator associated to rational
Anosov diffeomorphisms satisfying the (p-sec) condition and preserving certain
polydisks. In Section 5 we prove Theorem 1.3. For this, we first introduce in
Section 5.1 the group of diffeomorphisms $\mathcal{F}$, discuss their
properties and present examples of Anosov maps with and without additional
properties such as area-preservation and symmetry reversal. In Section 5.2,
using conjugacy classes of toral Anosov automorphisms, we construct non-linear
Anosov maps in $\mathcal{F}$ satisfying the assertions of Theorem 1.3.

##  2 Cone conditions for Anosov diffeomorphisms on ${\mathbb{T}^{2}}$

###  2.1 A pedestrian approach

In this section we want to explore functional-analytic properties of
composition operators associated to Anosov automorphisms on the torus on
Hilbert spaces that can be defined as the completion of the space of Laurent
polynomials with a norm depending on a certain weight function. We take the
well-known cat map as an example of a toral automorphism, and discuss
boundedness and compactness of the associated composition operator depending
on the weight function. The main result of this section is to show that for a
suitable weight function the composition operator is Hilbert-Schmidt. Since
most of our work in this paper focuses on the two-dimensional case, for
convenience we introduce the following shorthands.

######  Notation 2.1.

For $\alpha,\beta\in{\mathbb{R}^{2}}$ (or $\mathbb{Z}^{2}$) we write
$\alpha>\beta$ if $\alpha_{1}>\beta_{1}$ and $\alpha_{2}>\beta_{2}$, and
moreover for $c\in\mathbb{R}$ the notation $\alpha>c$ will be used as a
shorthand for $\alpha_{1}>c$ and $\alpha_{2}>c$ (and analogously for other
comparison operators). For $z\in{\mathbb{C}}^{2}$ (or $\mathbb{R}^{2}$,
$\mathbb{Z}^{2}$) and $n\in\mathbb{Z}^{2}$ we use the multiindex notation
$z^{n}=z_{1}^{n_{1}}z_{2}^{n_{2}}$, and we write $|z|=(|z_{1}|,|z_{2}|)$.

####  2.1.1 Weighted Hilbert spaces

Let $\mathbb{T}^{2}=\\{z\in{\mathbb{C}}^{2}:|z_{1}|=|z_{2}|=1\\}$, and let
$\mathcal{P}$ denote the space of Laurent polynomials on $\mathbb{T}^{2}$,

$\mathcal{P}=\\{f:\mathbb{T}^{2}\to\mathbb{C}:f(z)=\sum_{n\in\mathbb{Z}^{2},|n|\leq
N}f_{n}z^{n},\text{ with $f_{n}\in\mathbb{C},N\in\mathbb{N}$}\\}\,.$

For any $\nu:\mathbb{Z}^{2}\to\mathbb{R}_{>0}$, we define an inner product on
$\mathcal{P}$ by

$\langle
f,g\rangle_{\nu}=\sum_{n\in\mathbb{Z}^{2}}f_{n}\bar{g}_{n}\nu(n)^{2},$

where $(f_{n})_{n\in\mathbb{Z}^{2}}$ and $(g_{n})_{n\in\mathbb{Z}^{2}}$ are
the Fourier coefficients of $f$ and $g$, and we denote by $\|\cdot\|_{\nu}$
the corresponding norm. Note that $\nu(n)=\|p_{n}\|_{\nu}$ for
$n\in\mathbb{Z}^{2}$, where $p_{n}$ is the monomial $z\mapsto z^{n}$.

######  Definition 2.2.

We write $\mathcal{H}_{\nu}$ for the completion of $\mathcal{P}$ with respect
to the norm $\|\cdot\|_{\nu}$.

It turns out that $\mathcal{H}_{\nu}$ is a separable Hilbert space, which
contains the Laurent polynomials as a dense subset, with an orthonormal basis
given by the normalised monomials

$e_{n}(z)=\frac{z^{n}}{\nu(n)},\qquad n\in\mathbb{Z}^{2}.$

####  2.1.2 Composition operator for the cat map

The hyperbolic matrix

$A=\begin{pmatrix}2&1\\\ 1&1\end{pmatrix}$

induces the well-known toral automorphism
$T\colon\mathbb{T}^{2}\to\mathbb{T}^{2}$ given by
$(z_{1},z_{2})\mapsto(z_{1}^{2}z_{2},z_{1},z_{2})$, known as the cat map. We
shall study the properties of the composition operator $C_{T}$ associated to
$T$ when considered on different Hilbert spaces $H_{\nu}$. In particular, we
will see that for certain choices of $\nu$ the operator $C_{T}$ is Hilbert-
Schmidt. A bounded operator $L$ on a Hilbert space $H$ is Hilbert-Schmidt if
$\sum_{n}\|Le_{n}\|^{2}<\infty$ where $\\{e_{n}\\}$ is an orthonormal basis of
$H$. A sufficient condition for this is that there are $\delta,c>0$ such that
for all $n$ it holds that

$\displaystyle\|Le_{n}\|\leq\delta\exp(-c\|n\|).$ (1)

We can compute $\|C_{T}e_{n}\|_{\nu}$ explicity, as

$\displaystyle\|C_{T}e_{n}\|^{2}_{\nu}=\sum_{m\in\mathbb{Z}^{2}}\left(\frac{\nu(m)}{\nu(n)}\right)^{2}\delta_{2n_{1}+n_{2},m_{1}}\delta_{n_{1}+n_{2},m2}=\left(\frac{\nu(An)}{\nu(n)}\right)^{2}.$
(2)

We first consider weight functions $\nu$ adapted to the dynamics of $T$
introduced in [FR]. For this, we denote the unstable/stable eigenvalues of $A$
by $\lambda_{u/s}=\varphi^{\pm 2}$, where $\varphi=(1+\sqrt{5})/2$ is the
golden mean, and write $V=(v_{u},v_{s})$ for the matrix with the corresponding
(normalised) unstable/stable eigenvectors as its column vectors. We write
$\langle\cdot,\cdot\rangle$ for the standard inner product on
$\mathbb{R}^{2}$.

######  Lemma 2.3.

For any $a=(a_{1},a_{2})>0$ and $\nu(n)=\nu_{a}(n)=\exp(-a_{1}|\langle
n,v_{u}\rangle|+a_{2}|\langle n,v_{s}\rangle|)$, the composition operator
$C_{T}$ is a well-defined Hilbert-Schmidt operator on
$(H_{\nu},\|\cdot\|_{\nu})$.

###### Proof.

By using $\langle An,v_{u/s}\rangle=\langle
n,A^{T}v_{u/s}\rangle=\lambda_{u/s}\langle n,v_{u/s}\rangle$ and (2), we
obtain

$\displaystyle\|C_{T}e_{n}\|_{\nu}$
$\displaystyle=\exp(-a_{1}\lambda_{u}|\langle
n,v_{u}\rangle|+a_{2}\lambda_{s}|\langle
n,v_{s}\rangle|)\cdot\exp(a_{1}|\langle n,v_{u}\rangle|-a_{2}|\langle
n,v_{s}\rangle|)$ $\displaystyle=\exp(-a_{1}(\lambda_{u}-1)|\langle
n,v_{u}\rangle|-a_{2}(1-\lambda_{s})|\langle n,v_{s}\rangle|).$

As $(\lambda_{u}-1)>0$, $(1-\lambda_{s})>0$, and all norms on $\mathbb{R}^{2}$
are equivalent, we obtain inequality (1). ∎

It turns out that replacing $V$ by the identity matrix yields an operator that
is not even compact. We omit a proof, as this follows by a direct calculation.

######  Lemma 2.4.

Let $a=(a_{1},a_{2})\in\mathbb{R}^{2}_{>0}$ and
$\nu(n)=\nu_{a}(n)=\exp(-a_{1}|n_{1}|+a_{2}|n_{2}|)$. The operator $C_{T}$ is
bounded on $(H_{\nu},\|\cdot\|_{\nu})$ if and only if $a_{1}=a_{2}$. It is
never compact on $(H_{\nu},\|\cdot\|_{\nu})$.

On the other hand, as will become apparent in the next section, working with a
diagonal matrix $V$ yields more convenient function spaces (specifically,
Hilbert spaces of holomorphic functions on polydisks). We can restore the nice
properties of the compositon operator in this setting, by allowing $a$ to be a
function of $n\in\mathbb{Z}^{2}$.

######  Definition 2.5 (Quadrant-wise exponential weight).

Let $\alpha,\gamma\in\mathbb{R}^{2}$ and define
$a\colon\mathbb{Z}^{2}\to\mathbb{R}^{2}$ as

$a_{\alpha,\gamma}(n)=\begin{cases}\alpha,&\text{if }n_{1}\cdot n_{2}\geq 0\\\
\gamma,&\text{if }n_{1}\cdot n_{2}<0,\end{cases}$

and $\nu(n)=\nu_{\alpha,\gamma}(n)=\exp(-\langle
a_{\alpha,\gamma}(n),|n|\rangle)$. We call such weight function quadrant-wise
exponential.

######  Lemma 2.6.

Let $\nu=\nu_{\alpha,\gamma}$ be a quadrant-wise exponential weight. If
$\alpha\in\mathbb{R}^{2}_{>0}$ and $\gamma\in\mathbb{R}^{2}_{<0}$, then the
operator $C_{T}$ is a well-defined Hilbert-Schmidt operator on $H_{\nu}$.

###### Proof.

For $n\in\mathbb{Z}^{2}$ we denote

$\varphi(n)=\langle a_{\alpha,\gamma}(n),|n|\rangle-\langle
a_{\alpha,\gamma}(An),|An|\rangle,$

so that

$\|C_{T}e_{n}\|_{\nu}=\frac{\nu(An)}{\nu(n)}=\exp(\varphi(n)).$

To prove the lemma, it suffices to show that there exists $c>0$ such that

$\varphi(n)<-c(|n_{1}|+|n_{2}|)\quad(\forall n\in\mathbb{Z}^{2}).$ (3)

Set $m=An$. We note that since $\|n\|_{2}\leq\|A^{-1}\|\|m\|_{2}$ and by the
equivalence of norms in $\mathbb{R}^{2}$, there exists $\tilde{c}>0$ such that
$|m_{1}|+|m_{2}|=\|m\|_{1}\geq\tilde{c}\|n\|_{1}=\tilde{c}(|n_{1}|+|n_{2}|)$,
for all $n\in\mathbb{Z}^{2}$ and $m=An$. There are three cases we need to take
care of.

  1. (i)

$n_{1}\cdot n_{2}\geq 0$. Note that in this case $|m|=|An|=A|n|$ and
$m_{1}\cdot m_{2}\geq 0$. It follows that

$\varphi(n)=\langle\alpha,|n|-A|n|\rangle=\langle\alpha-A^{T}\alpha,|n|\rangle=-(\alpha_{1}+\alpha_{2})|n_{1}|-\alpha_{1}|n_{2}|<-c_{1}(|n_{1}|+|n_{2}|),$

for any $0<c_{1}<\alpha_{1}$.

  2. (ii)

$m_{1}\cdot m_{2}<0$, which implies $n_{1}\cdot n_{2}<0$ as $n=A^{-1}m$ and
$n_{1}\cdot
n_{2}=(m_{1}-m_{2})(2m_{2}-m_{1})=3m_{1}m_{2}-2m_{2}^{2}-m_{1}^{2}<0$. Noting
that $|n_{1}|=|m_{1}-m_{2}|=|m_{1}|+|m_{2}|$ and
$|n_{2}|=|2m_{2}-m_{1}|=2|m_{2}|+|m_{1}|$ we obtain

$\varphi(n)=\langle\gamma,|n|-|m|\rangle=\gamma_{1}|m_{2}|+\gamma_{2}(|m_{1}|+|m_{2}|)<-c_{2}(|n_{1}|+|n_{2}|),$

for any $0<c_{2}<-\gamma_{2}\tilde{c}$.

  3. (iii)

$n_{1}\cdot n_{2}<0$ and $m_{1}\cdot m_{2}\geq 0$. In this case

$\varphi(n)=\langle\gamma,|n|\rangle-\langle\alpha,|m|\rangle\leq\max(\gamma_{1},\gamma_{2})(|n_{1}|+|n_{2}|)-\min(\alpha_{1},\alpha_{2})(|m_{1}|+|m_{2}|)<-c_{3}(|n_{1}|+|n_{2}|),$

for any
$0<c_{3}<2\min(-\gamma_{1},-\gamma_{2},\alpha_{1}\tilde{c},\alpha_{2}\tilde{c})$.

Combining the above, we have that (3) holds for all $n\in\mathbb{Z}^{2}$ with
$c=\min(c_{1},c_{2},c_{3})$. ∎

######  Remark 2.7.

It turns out that the use of quadrant-wise exponential weight functions is not
restricted to linear toral Anosov diffeomorphisms, but can also be applied to
certain non-linear maps. It is possible to show that for any map in the family
of non-linear maps studied in [SBJ], one can find suitable $\alpha$ and
$\gamma$, such that the associated composition operator considered on the
Hilbert space $H_{\nu}$ with $\nu=\nu_{\alpha,\gamma}$ is Hilbert-Schmidt, and
even trace-class. The proof is a straightforward but lengthy calculation
involving properties of the underlying map summarized in [SBJ, Lemma 2.3]. See
also [PoS] for this and related results.

###  2.2 Anosov diffeomorphisms with strong mapping conditions

In this section, we establish more general conditions on the toral
diffeomorphisms which will be sufficient to prove that the associated
composition operator is trace-class on a suitable weighted Hilbert space. In
particular, we characterise maps that satisfy the conditions of the main
result of the next section (Theorem 3.24). We start by recalling some well-
known facts about cones and Reinhardt domains.

####  2.2.1 Convex cones in $\mathbb{R}^{2}$

A _cone_ $\Lambda\subset{\mathbb{R}^{2}}$ is a set such that if $v\in\Lambda$,
then $\lambda v\in\Lambda$ for all $\lambda>0$. A cone shifted by a vector,
that is a set of the form $x+\Lambda$, where $x\in\mathbb{R}^{2}$ and
$\Lambda\subset{\mathbb{R}^{2}}$ is a cone, is called an _affine cone_. For
$p_{u},p_{s}\in\mathbb{R}^{2}$ denote by $P=(p_{u},p_{s})$ the matrix having
$p_{u}$ and $p_{s}$ as its column vectors. For an invertible matrix $P$ denote
by $\Lambda_{P}$ the convex open polyhedral cone in $\mathbb{R}^{2}$
(positively) spanned by $p_{u}$ and $p_{s}$, that is,

$\Lambda_{P}=\\{Px=x_{1}p_{u}+x_{2}p_{s}\colon x>0\\}=P(\mathbb{R}^{2}_{>0}).$

Writing $W=(w_{u},w_{s})=(P^{T})^{-1}$, this can equivalently be expressed as

$\Lambda_{P}=\\{x\in{\mathbb{R}^{2}}\colon\langle w_{u},x\rangle>0,\langle
w_{s},x\rangle>0\\}.$

Its polar cone is given by

$(\Lambda_{P})^{o}=\\{x\in\mathbb{R}^{2}\colon\langle x,p\rangle\leq 0\text{
for all }p\in\Lambda_{P}\\}=\\{x\in\mathbb{R}^{2}\colon P^{T}x\leq
0\\}=W(\mathbb{R}^{2}_{\leq 0}).$

While $\Lambda_{P}$ is the image of the positive quadrant of
${\mathbb{R}^{2}}$ under $P$, it will later be useful to consider the images
of all quadrants. For this, we write
$I^{\sigma}=\begin{pmatrix}\sigma_{1}&0\\\ 0&\sigma_{2}\end{pmatrix}$ for
$\sigma\in\Sigma,$ where888For brevity, when using
$\sigma=(\sigma_{1},\sigma_{2})\in\Sigma$ as an index, we will often just
write out its signs, e.g. writing $R^{++}$ for $R^{(+1,+1)}$, $R^{+-}$ for
$R^{(+1,-1)}$, etc.

$\Sigma:=\\{\sigma=(\sigma_{1},\sigma_{2})\colon\sigma_{1},\sigma_{2}\in\\{\pm
1\\}\\},$

and denote by $\Lambda^{\sigma}_{P}$ the image of the quadrant
$R^{\sigma}=I^{\sigma}(\mathbb{R}^{2}_{>0})$ under $P$, that is
$\Lambda^{\sigma}_{P}=P(R^{\sigma})$ for $\sigma\in\Sigma$. A short
calculation reveals that the cone $\Lambda^{\sigma}_{P}$ can be written as

$\Lambda^{\sigma}_{P}=\\{x\in{\mathbb{R}^{2}}\colon\sigma_{1}\langle
w_{u},x\rangle>0,\sigma_{2}\langle w_{s},x\rangle>0\\}.$

All the above cones have $(0,0)$ as their apex. As we shall see shortly we
will need to work with cones translated by a vector. For this, let us denote
by $R^{\sigma}_{\delta}$ the image under $I^{\sigma}$ of the first quadrant
translated by $\delta\in{\mathbb{R}^{2}}$, that is
$R^{\sigma}_{\delta}=I^{\sigma}(\mathbb{R}^{2}_{>0}+\delta)=R^{\sigma}+v_{\delta}^{\sigma}$
with apex $v_{\delta}^{\sigma}=I^{\sigma}\delta$.

######  Definition 2.8.

For $\delta\in\mathbb{R}^{2}$, $\sigma\in\Sigma$ and
$P\in{\operatorname{GL}_{2}}(\mathbb{R})$ we denote
$\Lambda^{\sigma}_{P,\delta}$ the convex affine cone

$\displaystyle\Lambda^{\sigma}_{P,\delta}=P(R_{\delta}^{\sigma})=\Lambda_{P}^{\sigma}+v^{\sigma}_{P,\delta},$
(4)

with apex $v^{\sigma}_{P,\delta}=Pv^{\sigma}_{\delta}$.

We note that $\sigma_{1}\langle
w_{u},v^{\sigma}_{P,\delta}\rangle=\sigma_{1}\langle
P^{T}w_{u},I^{\sigma}\delta\rangle=\sigma_{1}\langle(1,0)^{T},I^{\sigma}\delta\rangle=\delta_{1}$
and $\sigma_{2}\langle w_{s},v^{\sigma}_{P,\delta}\rangle=\delta_{2}$. Using
these equalities, $\Lambda^{\sigma}_{P,\delta}$ can be rewritten as

$\displaystyle\begin{split}\Lambda^{\sigma}_{P,\delta}&=\\{x\in{\mathbb{R}^{2}}\colon\sigma_{1}\langle
w_{u},x-v^{\sigma}_{P,\delta}\rangle>0,\sigma_{2}\langle
w_{s},x-v^{\sigma}_{P,\delta}\rangle>0\\}\\\
&=\\{x\in{\mathbb{R}^{2}}\colon\sigma_{1}\langle
w_{u},x\rangle>\delta_{1},\sigma_{2}\langle
w_{s},x\rangle>\delta_{2}\\}.\end{split}$ (5)

####  2.2.2 Log-conical Reinhardt domains of ${\hat{\mathbb{C}}^{2}}$

The translated convex cones in (4) will be useful for defining certain two-
dimensional complex domains. For this we first require some definitions.

######  Notation 2.9.

We denote the Riemann sphere by
${\hat{\mathbb{C}}}={\mathbb{C}}\cup\\{\infty\\}$, and write
${\hat{\mathbb{C}}}^{2}={\hat{\mathbb{C}}}\times{\hat{\mathbb{C}}}$. For
$z\in{\hat{\mathbb{C}}^{2}}$, $v\in{\mathbb{R}^{2}}$ and $a\in\mathbb{Z}^{2}$
we write $|z|=(|z_{1}|,|z_{2}|)$, $z^{a}=z_{1}^{a_{1}}z_{2}^{a_{2}}$,
$e^{v}=(e^{v_{1}},e^{v_{2}})$, and for $z\in(\mathbb{C}\setminus\\{0\\})^{2}$
we write $\log|z|=(\log|z_{1}|,\log|z_{2}|)$. For any domain
$D\subseteq{\hat{\mathbb{C}}^{2}}$, we write
$D^{+}=D\cap(\mathbb{C}\setminus\\{0\\})^{2}$.

######  Definition 2.10.

A domain $D\subset{\hat{\mathbb{C}}^{2}}$ is called polycircular or a
Reinhardt domain if it is invariant under polyrotations, that is, if $z\in D$
implies $\omega z=(\omega_{1}z_{1},\omega_{2}z_{2})\in D$ for all
$\omega\in{\mathbb{T}^{2}}$. The set $|D|:=\\{|z|\colon z\in
D\\}\subset(\mathbb{R}_{\geq 0}\cup\\{\infty\\})^{2}$ is called absolute
domain of $D$, and $\Lambda=\log|D^{+}|:=\\{\log|z|:z\in
D^{+}\\}\in{\mathbb{R}^{2}}$ the logarithmic base of $D^{+}$. A Reinhardt
domain $D$ is called log-conical if the logarithmic base of $D^{+}$ is a
convex open affine cone.

We define
$\mathbb{T}^{2}_{\rho}=\\{z\in{\mathbb{C}}^{2}:|z_{1}|=\rho_{1},|z_{2}|=\rho_{2}\\}$
for $\rho\in\mathbb{R}^{2}_{>0}$, and write
$e^{\Lambda}{\mathbb{T}^{2}}:=\bigcup_{r\in\Lambda}\mathbb{T}^{2}_{e^{r}}$.
Further let $\mathcal{E}(\Lambda)\subset{\mathbb{R}^{2}}$ denote the union of
the set of _faces_ of $\Lambda$, that is
$\mathcal{E}(\Lambda)=\partial\Lambda\setminus\\{p\\}$ with $p$ the apex of
$\Lambda$. Then, every convex open affine cone
$\Lambda\subset{\mathbb{R}^{2}}$ induces a log-conical Reinhardt domain in
${\hat{\mathbb{C}}}^{2}$ via

$D=\operatorname{cl}({e^{\Lambda}{\mathbb{T}^{2}}})\setminus
e^{\mathcal{E}(\Lambda)}{\mathbb{T}^{2}},$

where the closure is taken in ${\hat{\mathbb{C}}}^{2}$ (that is, it may
contain points of the form $(z_{1},z_{2})$, with either $z_{1}$, or $z_{2}$,
or both, taking the values $0$ or $\infty$). We shall denote by
$D^{\sigma}_{P,\delta}$ the log-conical Reinhardt domain induced by the convex
affine cone $\Lambda^{\sigma}_{P,\delta}$ in (4). We can calculate

$\displaystyle(D^{\sigma}_{P,\delta})^{+}$
$\displaystyle=e^{\Lambda^{\sigma}_{P,\delta}}\mathbb{T}^{2}=\bigcup_{r\in\Lambda^{\sigma}_{P,\delta}}\mathbb{T}^{2}_{e^{r}}=\bigcup_{x\in\Lambda^{\sigma}_{P}}\mathbb{T}^{2}_{\exp(v^{\sigma}_{P,\delta}+x)}$
$\displaystyle=\\{z\in({\mathbb{C}}\setminus\\{0\\})^{2}\colon|z_{1}|=e^{(v^{\sigma}_{P,\delta})_{1}+x_{1}},|z_{2}|=e^{(v^{\sigma}_{P,\delta})_{2}+x_{2}},\,x\in\Lambda^{\sigma}_{P}\\}$
$\displaystyle=\\{z\in({\mathbb{C}}\setminus\\{0\\})^{2}\colon|z|^{\sigma_{1}w_{u}}>e^{\delta_{1}},|z|^{\sigma_{2}w_{s}}>e^{\delta_{2}}\\},$

which yields

$D^{\sigma}_{P,\delta}=\\{z\in{\hat{\mathbb{C}}}^{2}\colon|z|^{\sigma_{1}w_{u}}>e^{\delta_{1}},|z|^{\sigma_{2}w_{s}}>e^{\delta_{2}}\\}.$

The distinguished boundary or Shilov boundary of $D^{\sigma}_{P,\delta}$ is a
torus in ${\mathbb{C}^{2}}$ given by

$\mathbb{T}_{P,\delta}^{\sigma}:=\partial^{*}D^{\sigma}_{P,\delta}:=\\{z\in\mathbb{C}^{2}\colon|z_{1}|=e^{(v^{\sigma}_{P,\delta})_{1}},|z_{2}|=e^{(v^{\sigma}_{P,\delta})_{2}}\\}.$

####  2.2.3 Anosov toral diffeomorphisms with constant invariant cone fields

######  Definition 2.11 (Toral Anosov diffeomorphisms).

Let $M=([0,2\pi]/\sim)^{2}$. A smooth diffeomorphism $\tilde{T}\colon M\to M$
is called Anosov if there exist two uniformly transversal open continuous cone
fields $\mathcal{K}^{u}=\\{K^{u}(x)\\}$, $\mathcal{K}^{s}=\\{K^{s}(x)\\}$ with
cones $K^{u}(x),K^{s}(x)\subset T_{x}M$, a norm $\|\cdot\|$ on $T_{x}M$ and
$\lambda>1$ such that, for all $x\in M$,

  1. (i)

$D_{x}\tilde{T}(\operatorname{cl}({K^{u}(x)}))\subset
K^{u}(\tilde{T}(x))\cup\\{0\\},D_{x}\tilde{T}^{-1}(\operatorname{cl}({K^{s}(x)}))\subset
K^{s}(\tilde{T}^{-1}(x))\cup\\{0\\}$ and

  2. (ii)

$\|D_{x}\tilde{T}(v)\|>\lambda\|v\|\;\forall
v\in\operatorname{cl}({K^{u}(x)})\text{ and
}\|D_{x}\tilde{T}^{-1}(v)\|>\lambda\|v\|\;\forall
v\in\operatorname{cl}({K^{s}(x)})$.

Without loss of generality, the cone fields can be chosen to be complementary,
that is, $K^{s}(x)=T_{x}M\setminus\operatorname{cl}({K^{u}(x)})$ for all $x\in
M$.

If the expanding and contracting cones $K^{u}(x)$ and $K^{s}(x)$ can be chosen
independently of $x$, that is999Here we slightly abuse notation, using the
canonical identification $T_{x}M\cong\mathbb{R}^{2}$ for all $x\in M$.
$K^{u},K^{s}\subset T_{x}M$ such that $K^{u}(x)=K^{u}$ and $K^{s}(x)=K^{s}$
for all $x\in M$, then we say $\tilde{T}$ is an Anosov diffeomorphism with
constant invariant cone fields.

Let $\tilde{T}$ be a toral Anosov diffeomorphism with constant complementary
cone fields $\mathcal{K}^{u}=\\{K^{u}\\}$ and $\mathcal{K}^{s}=\\{K^{s}\\}$.
Then $K^{u}$ can be decomposed as $K^{u}=K^{u}_{+}\cup-K^{u}_{+}$, where
$K^{u}_{+}$ is a convex cone, so there exists a matrix
$P\in{\operatorname{GL}_{2}}(\mathbb{R})$ such that
$K^{u}_{+}=\Lambda^{\sigma}_{P}=PI^{\sigma}(\mathbb{R}^{2}_{>0})$ with
$\sigma=(+1,+1)$. Adopting the notation

$\Sigma^{1}=\\{\sigma\in\Sigma:\sigma_{1}=\sigma_{2}\\}\quad\text{ and
}\quad\Sigma^{-1}=\\{\sigma\in\Sigma:\sigma_{1}=-\sigma_{2}\\}$

we can write $K^{u}=\bigcup_{\sigma\in\Sigma^{1}}\Lambda^{\sigma}_{P}$ and
$K^{s}=\bigcup_{\tilde{\sigma}\in\Sigma^{-1}}\Lambda^{\tilde{\sigma}}_{P}$.

######  Definition 2.12.

Let $\tilde{T}:M\to M$ be a smooth Anosov diffeomorphism.

  1. (i)

We say that $\tilde{T}$ has $P$-induced constant invariant cone fields if it
has constant invariant cone fields $\mathcal{K}^{u}=\\{K^{u}_{0}\\}$ and
$\mathcal{K}^{s}=\\{K^{s}_{0}\\}$ given by
$K^{u}_{\delta}=\bigcup_{\sigma\in\Sigma^{1}}\Lambda^{\sigma}_{P,\delta}$ and
$K^{s}_{\delta}=\bigcup_{\tilde{\sigma}\in\Sigma^{-1}}\Lambda^{\tilde{\sigma}}_{P,\delta}$
with $P\in{\operatorname{GL}_{2}}(\mathbb{R})$ and
$\delta\in{\mathbb{R}^{2}}$.

  2. (ii)

We say that $\tilde{T}$ has $P$-induced strongly expanding constant invariant
cone fields (sec) if it has $P$-induced constant invariant cone fields and if
there are $\delta,\tilde{\delta}\in\mathbb{R}^{2}_{>0}$ and
$\sigma\in\Sigma^{1}$, $\tilde{\sigma}\in\Sigma^{-1}$ such that, for all $x\in
M$,

$\displaystyle(D_{x}\tilde{T})v\in K^{u}_{\delta}\text{ and
}(D_{x}\tilde{T}^{-1})\tilde{v}\in K^{s}_{\tilde{\delta}},$ (6)

where $v=v^{\sigma}_{P,\delta}$,
$\tilde{v}=v^{\tilde{\sigma}}_{P,\tilde{\delta}}$. In the special case
$P=\mathbb{I}$, we refer to
$K_{0}^{u}=\mathbb{R}^{2}_{>0}\cup\mathbb{R}^{2}_{<0}$ as positive, and call
(6) the (p-sec) condition.

Figure 1 illustrates the (p-sec) condition. Note that (i) is satisfied by all
smooth toral Anosov diffeomorphisms with constant invariant cone fields,
whereas (ii) is a stronger condition, which eventually will be required to
prove compactness of the associated weighted composition operator (see Section
3).

Figure 1: Illustration of the (sec) condition in Definition 2.12(ii) for
$P=\mathbb{I}$ and $K^{u}_{\delta}=K^{u}_{\pm}\pm\delta$.

######  Remark 2.13.

The cone conditions in (6) are often easy to check. For example, the (p-sec)
condition requires $K^{u}_{0}=\mathbb{R}^{2}_{>0}\cup\mathbb{R}^{2}_{<0}$ and
$K^{s}_{0}={\mathbb{R}^{2}}\setminus\operatorname{cl}({K^{u}_{0}})$. Assuming
$D_{x}\tilde{T}=\begin{pmatrix}a_{x}&b_{x}\\\ c_{x}&d_{x}\end{pmatrix}$
preserves $\mathbb{R}^{2}_{\geq 0}$ for all $x\in M$, the first part of (6) is
automatically satisfied if either $\inf_{x\in M}{a_{x}}\geq 1$, or $\inf_{x\in
M}{d_{x}}\geq 1$, or if

$\sup_{x\in M}\left(\frac{1-a_{x}}{b_{x}}\right)\cdot\sup_{x\in
M}\left(\frac{1-d_{x}}{c_{x}}\right)<1.$

A similar condition for the first part of (6) in the case when
$D_{x}\tilde{T}$ maps $\mathbb{R}^{2}_{\geq 0}$ to $\mathbb{R}^{2}_{\leq 0}$
can be deduced easily, as can be analogous conditions for the second part of
(6).

We shall next state conditions that can be easily deduced from the fact that
the map $\tilde{T}$ possesses constant invariant expanding and (co)-expanding
cone fields, that is, from (i) in Definition 2.12.

######  Lemma 2.14.

Let $\tilde{T}\colon M\to M$ be a smooth Anosov diffeomorphism with
$P$-induced constant invariant cone fields for some
$P\in{\operatorname{GL}_{2}}(\mathbb{R})$. Then, for any
$\delta,\tilde{\delta}\in\mathbb{R}^{2}_{>0}$, $\sigma\in\Sigma^{1}$ and
$\tilde{\sigma}\in\Sigma^{-1}$, there exists
$q\in\Lambda^{\tilde{\sigma}}_{P,\tilde{\delta}}$ with
$(D_{x}\tilde{T})q\in\Lambda^{\sigma}_{P,\delta}$ for all $x\in M$.

###### Proof.

Fix $\delta,\tilde{\delta}\in\mathbb{R}^{2}_{>0}$, $\sigma\in\Sigma^{1}$ and
$\tilde{\sigma}\in\Sigma^{-1}$, and recall that the (constant)
$(D_{x}\tilde{T})$-invariant cone given by the matrix $P$ is
$K^{u}=\Lambda_{P}\cup-\Lambda_{P}$, with
$\Lambda_{P}=P(\mathbb{R}^{2}_{>0})$, and either
$(D_{x}\tilde{T})(\Lambda_{P})\subset\Lambda_{P}$ or
$(D_{x}\tilde{T})(\Lambda_{P})\subset-\Lambda_{P}$ (for all $x\in M$). For
$x\in M$, we define $N_{x}=P^{-1}(D_{x}\tilde{T})P$, and observe that either
$N_{x}(\mathbb{R}^{2}_{\geq 0})\subset\mathbb{R}^{2}_{>0}\cup\\{0\\}$ or
$N_{x}(\mathbb{R}^{2}_{\geq 0})\subset\mathbb{R}^{2}_{<0}\cup\\{0\\}$, for all
$x\in M$.

Applying Lemma A.2 to $\\{N_{x}:x\in M\\}$, there exists $q^{\prime}\in
R^{\tilde{\sigma}}_{\tilde{\delta}}$ such that $N_{x}(q^{\prime})\in
R^{\sigma}_{\delta}$ for all $x\in M$. Using (4), we obtain that
$Pq^{\prime}\in
P(R^{\tilde{\sigma}}_{\tilde{\delta}})=\Lambda^{\tilde{\sigma}}_{P,\tilde{\delta}}$,
and $(D_{x}\tilde{T})(Pq^{\prime})\in
P(R^{\sigma}_{\delta})=\Lambda^{\sigma}_{P,\delta}$ for all $x\in M$,
finishing the proof with $q=Pq^{\prime}$. ∎

Let $\pi(x_{1},x_{2})=(e^{ix_{1}},e^{ix_{2}})$ be the canonical diffeomorphism
from $M$ to ${\mathbb{T}^{2}}$. We denote by $T$ the diffeomorphism on
${\mathbb{T}^{2}}$ uniquely determined by the equation
$T\circ\pi=\pi\circ\tilde{T}$. We will say that
$T\colon\mathbb{T}^{2}\to\mathbb{T}^{2}$ satisfies the (sec) (or (p-sec))
condition, if the corresponding $\tilde{T}\colon M\to M$ does. As the next
(key) lemma will show, for analytic toral Anosov diffeomorphisms with constant
invariant cone fields, properties of the derivative $D\tilde{T}$ on the
tangent bundle can be translated into properties of the map $T$ in a small
neighbourhood of ${\mathbb{T}^{2}}$. With slight abuse of notation, we
continue writing $T$ for its analytic extension to such small neighbourhood of
${\mathbb{T}^{2}}$.

######  Definition 2.15.

We say that a diffeomorphism $T\colon{\mathbb{T}^{2}}\to{\mathbb{T}^{2}}$ is
_orientation-preserving_ (_orientation-reversing_) if the determinant of
$D_{x}\tilde{T}$ of the conjugated map $\tilde{T}\colon M\to M$ is positive
(negative) for all $x\in M$. We say $T$ is _area-preserving_ if $|\det
D_{x}\tilde{T}|=1$ for all $x\in M$.

######  Lemma 2.16.

Let $\tilde{T}\colon M\to M$ be an analytic Anosov diffeomorphism with
$P$-induced constant invariant cone fields for some
$P\in{\operatorname{GL}_{2}}(\mathbb{R})$, and $T$ the conjugated
diffeomorphism on ${\mathbb{T}^{2}}$. If there are
$\delta\in\mathbb{R}^{2}_{>0}$, $\sigma\in\Sigma$, and $q\in{\mathbb{R}^{2}}$
such that $(D_{x}\tilde{T})q\in\Lambda^{\sigma}_{P,\delta}$ for all $x\in M$,
then there exist $\epsilon>0$ and $\delta^{\prime}>\delta$ such that for all
$t\in(0,\epsilon)$ we have

$T(\mathbb{T}^{2}_{e^{tq}})\subset D^{\sigma}_{P,t\delta^{\prime}},$

where $D^{\sigma}_{P,t\delta^{\prime}}$ is the Reinhardt domain induced by the
logarithmic base $\Lambda^{\sigma}_{P,t\delta^{\prime}}.$

###### Proof.

We denote $w^{(1)}=\sigma_{1}w_{u}$ and $w^{(2)}=\sigma_{2}w_{s}$ (recalling
$W=(w_{u},w_{s})=(P^{T})^{-1}$). Using (5), we observe that
$(D_{x}\tilde{T})q\in\Lambda^{\sigma}_{P,\delta}$ translates to the
inequalities

$\sum_{l=1,2}\sum_{k=1,2}\frac{\partial\tilde{T}_{l}}{\partial
x_{k}}q_{k}w^{(j)}_{l}>\delta_{j},\quad\text{ for }j=1,2\text{ and all }x\in
M.$

Next, for $x\in M$, $s\in\mathbb{R}$ and $a\in\mathbb{R}^{2}$, we define

$f_{x}^{a}(s)=\sum_{l=1,2}a_{l}\log|T_{l}(\xi_{x}(s))|$

with
$\xi_{x}(s)=(e^{sq_{1}+ix_{1}},e^{sq_{2}+ix_{2}})\in\mathbb{T}^{2}_{e^{sq}}$.
We note that $f^{a}_{x}$ is continuously differentiable on
$(0,\epsilon^{\prime})$ for sufficiently small $\epsilon^{\prime}$, with

$\frac{\partial}{\partial
s}f^{a}_{x}(s)=\Re\left(\sum_{l=1,2}\sum_{k=1,2}\frac{e^{ix_{k}}{\partial_{k}}{T_{l}(\xi_{x}(s))}}{T_{l}(\xi_{x}(s))}q_{k}e^{sq_{k}}a_{l}\right),$

where $\partial_{k}T_{l}$ denotes the (complex) derivative of the $l$-th
component of $T$ with respect to the $k$-th variable. It follows that

$\frac{\partial}{\partial
s}f^{w^{(j)}}_{x}(s)|_{s=0}=\Re\left(\sum_{l=1,2}\sum_{k=1,2}\frac{e^{ix_{k}}{\partial_{k}}{T_{l}(e^{ix_{1}},e^{ix_{2}})}}{T_{l}(e^{ix_{1}},e^{ix_{2}})}q_{k}w^{(j)}_{l}\right)=\Re\left(\sum_{l=1,2}\sum_{k=1,2}\frac{\partial\tilde{T}_{l}}{\partial
x_{k}}q_{k}w^{(j)}_{l}\right)>\delta_{j}$

for $j=1,2$ and all $x\in M$. By compactness of $M$ and continuity, we can fix
$\delta^{\prime}>\delta$, so that $\frac{\partial}{\partial
s}f^{w^{(j)}}_{x}(s)>\delta^{\prime}_{j}$, $j=1,2$, holds for all $x\in M$ and
all $s\in(0,\epsilon^{\prime\prime})$ with a sufficiently small
$\epsilon^{\prime\prime}\in(0,\epsilon^{\prime})$. Since $f^{a}_{x}(0)=0$, we
obtain that

$\sum_{l=1,2}w_{l}^{(j)}\log|T_{l}(\xi_{x}(t))|=f_{x}^{w^{(j)}}(t)-f_{x}^{w^{(j)}}(0)=\int_{0}^{t}\frac{\partial}{\partial
s}f_{x}^{w^{(j)}}(s)\,ds>\delta^{\prime}_{j}t,$

for $j=1,2$ and $t\in(0,\epsilon^{\prime\prime})$. Exponentiating both sides,
it follows that $T(\xi_{x}(t))\in D^{\sigma}_{P,t\delta^{\prime}}$ for all
$x\in M$. Since $\\{\xi_{x}(t):x\in M\\}=\mathbb{T}^{2}_{e^{tq}}$, this
completes the proof. ∎

If in addition $\tilde{T}$ satisfies the (sec) condition, then one can
establish all the required properties (for Theorem 1.1) for $T$ in a
neighbourhood of ${\mathbb{T}^{2}}$. First, we need to introduce corresponding
notions on Reinhardt domains induced by convex cones. For
$\delta\in\mathbb{R}^{2}$ and $P\in{\operatorname{GL}_{2}}(\mathbb{R})$, let
$\mathcal{A}_{P,\delta}$ denote a two-dimensional ‘annulus’ containing
${\mathbb{T}^{2}}$, given by

$\mathcal{A}_{P,\delta}=\\{z\in{\mathbb{C}^{2}}\colon-|\delta|<P^{-1}\log|z|<|\delta|\\}.$

For brevity we also write
$\mathcal{A}_{\delta}=\mathcal{A}_{\mathbb{I},\delta}=\\{z\in{\mathbb{C}^{2}}\colon-|\delta|<\log|z|<|\delta|\\}$.

######  Definition 2.17.

Let $T\colon{\mathbb{T}^{2}}\to{\mathbb{T}^{2}}$ be an analytic map with
holomorphic extension to a neighbourhood of
$\operatorname{cl}({\mathcal{A}_{P,\delta}})$ for some
$\delta\in\mathbb{R}^{2}_{>0}$ and $P\in{\operatorname{GL}_{2}}(\mathbb{R})$.
For $\ell\in\\{1,-1\\}$ and $\Delta\in\mathbb{R}^{2}_{>0}$ the map $T$ is said
to have the ($\ell,\delta,\Delta,P$)-strongly expanding mapping property if
one of the following two alternatives holds:

  1. (EP)

$T(\mathbb{T}^{\sigma}_{P,\delta})\subseteq
D^{\sigma}_{P,\Delta}{\subset\\!\\!\\!\subset}D^{\sigma}_{P,\delta}$ for all
$\sigma\in\Sigma^{\ell}$ (“$T$ preserves expanding direction”),

  2. (ER)

$T(\mathbb{T}^{\sigma}_{P,\delta})\subseteq
D^{-\sigma}_{P,\Delta}{\subset\\!\\!\\!\subset}D^{-\sigma}_{P,\delta}$ for all
$\sigma\in\Sigma^{\ell}$ (“$T$ reverses expanding direction”).

######  Proposition 2.18.

Let $\tilde{T}\colon M\to M$ be an analytic Anosov diffeomorphism with
constant $P$-induced strongly expanding invariant cone fields for some
$\delta,\tilde{\delta}\in\mathbb{R}^{2}_{>0}$ in (6). Then there exists
$\eta\in\mathbb{R}^{2}_{>0}$ such that the corresponding diffeomorphisms $T$
and $T^{-1}$ on $\mathbb{T}^{2}$ can be analytically extended to
$\mathcal{A}_{P,\eta}$, and there are
$\Delta,\tilde{\Delta}\in\mathbb{R}^{2}_{>0}$ with $\Delta>\delta$ and
$\tilde{\Delta}<\tilde{\delta}$, such that

  1. (i)

$T$ is $(1,t\delta,t\Delta,P)$-strongly expanding for all sufficiently small
$t>0$,

  2. (ii)

$T^{-1}$ is $(-1,t\tilde{\Delta},t\tilde{\delta},P)$-strongly expanding for
all sufficiently small $t>0$.

Moreover, for any $\sigma\in\Sigma^{1},\tilde{\sigma}\in\Sigma^{-1}$ and any
$\tilde{\delta},\Delta\in\mathbb{R}^{2}_{>0}$,

  1. (iii)

there exists $q\in\Lambda^{\tilde{\sigma}}_{P,\tilde{\delta}}$ such that
$\mathbb{T}^{2}_{e^{tq}}\subset D^{\tilde{\sigma}}_{P,t\tilde{\delta}}$ and
$T(\mathbb{T}^{2}_{e^{tq}})\subset D^{\sigma}_{P,t\Delta}$ for all
sufficiently small $t$.

###### Proof.

To prove $(i)$, we apply Lemma 2.16 with $q=v^{\sigma}_{P,\delta}$,
$\sigma\in\Sigma^{1}$. Since
$tq=tv^{\sigma}_{P,\delta}=v^{\sigma}_{P,t\delta}$ and therefore
$\mathbb{T}^{2}_{e^{tq}}=\mathbb{T}^{\sigma}_{P,t\delta}$, there exist
$\epsilon>0$ and $\Delta>\delta$, such that for all $t\in(0,\epsilon)$,
$T(\mathbb{T}^{\sigma}_{P,t\delta})\subset
D^{\sigma}_{P,t\Delta}{\subset\\!\\!\\!\subset}D^{\sigma}_{P,t\delta}$. Item
$(ii)$ follows analogously by applying Lemma 2.16 with
$\tilde{q}=v^{\tilde{\sigma}}_{P,\tilde{\delta}}$,
$\tilde{\sigma}\in\Sigma^{-1}$, to the map $T^{-1}$. Finally, for $(iii)$, by
Lemma 2.14 there exists $q\in\mathbb{R}^{2}$ satisfying
$tq\in\Lambda^{\tilde{\sigma}}_{P,t\tilde{\delta}}$ (i.e.
$\mathbb{T}^{2}_{e^{tq}}\subset D^{\tilde{\sigma}}_{P,t\tilde{\delta}}$) and
$D_{x}\tilde{T}(tq)\in\Lambda^{\sigma}_{P,t\Delta}$ for all $t>0$ and $x\in
M$. Then, by Lemma 2.16, there exists $\Delta^{\prime}>\Delta$ and
$\epsilon^{\prime}>0$ such that $T(\mathbb{T}^{2}_{e^{tq}})\subset
D^{\sigma}_{P,t\Delta^{\prime}}\subset D^{\sigma}_{P,t\Delta}$ for all
$t\in(0,\epsilon^{\prime})$, as required. ∎

######  Corollary 2.19.

If an analytic Anosov diffeomorphism $\tilde{T}$ on $M$ satisfies the (sec)
condition for some $P\in{\operatorname{GL}_{2}}(\mathbb{R})$, then there exist
$\alpha,A,\gamma,\Gamma,\eta\in\mathbb{R}^{2}_{>0}$ with $\alpha<A<\eta$ and
$\Gamma<\gamma<\eta$, such that the corresponding diffeomorphisms $T$ and
$T^{-1}$ on ${\mathbb{T}^{2}}$ can be analytically extended to
$\mathcal{A}_{P,\eta}$ and the following mapping properties hold:

  1. (i)

$T$ is $(1,\alpha,A,P)$-strongly expanding;

  2. (ii)

$T^{-1}$ is $(-1,\Gamma,\gamma,P)$-strongly expanding;

  3. (iii)

For any $\sigma\in\Sigma^{1},\tilde{\sigma}\in\Sigma^{-1}$, there exists $q$
such that $\mathbb{T}^{2}_{e^{q}}\subset
D^{\tilde{\sigma}}_{P,\gamma}\cap\mathcal{A}_{P,\eta}$ and
$T(\mathbb{T}^{2}_{e^{q}})\subset D^{\sigma}_{P,A}$.

###### Proof.

Using notation from Proposition 2.18, $(i)$-$(iii)$ are satisfied with
$\alpha=t\delta$, $A=t\Delta$, $\Gamma=t\tilde{\Delta}$,
$\gamma=t\tilde{\delta}$, for sufficiently small $t>0$. ∎

##  3 Composition operator for Anosov maps with constant invariant cone fields

In this section, we shall consider (weighted) composition operators associated
to analytic Anosov diffeomorphisms satisfying the (sec) condition. In this
setting, using Corollary 2.19, we will show that there exist Hardy-Hilbert
spaces induced by a suitable cone-wise exponential weight, such that the
operator is trace-class.

###  3.1 Hardy-Hilbert spaces on Reinhardt domains

Before moving to the more general case of Reinhardt domains, we present a few
simplified examples in which the domains are chosen to be polydisks.

Figure 2: Absolute domains of the four polydisks $D^{\sigma}_{\delta}$ for
$\sigma\in\Sigma$ and $\delta=(0.1,0.2)$ (left) and depiction of the dual
domain $(D^{+-}_{\delta})^{\prime}$ of $D^{+-}_{\delta}$ (right).

######  Example 3.1.

Let $P$ be the identity matrix $\mathbb{I}$. Then
$\Lambda_{P,\delta}^{\delta}=R^{\sigma}_{\delta}$, that is, it is one of the
four quadrants translated by $(\pm\delta_{1},\pm\delta_{2})$. For simplicity
we write $D^{\sigma}_{\delta}=D^{\sigma}_{\mathbb{I},\delta}$ for the disk
induced by the cone $R^{\sigma}_{\delta}$. Its distinguished boundary is given
by

$\partial^{*}D^{\sigma}_{\delta}=\mathbb{T}^{\sigma}_{\delta}=\\{z\in{\mathbb{C}^{2}}\colon|z_{1}|=e^{\sigma_{1}\delta_{1}},|z_{2}|=e^{\sigma_{2}\delta_{2}}\\}.$

  1. (i)

Fix $\sigma=(-1,-1)$. Then, for $\delta=(0,0)$ we obtain a unit bidisk, that
is, $D^{\sigma}_{\delta}=\mathbb{D}^{2}$. If $\delta\neq 0$, then
$D^{\sigma}_{\delta}$ is a bidisk centered at $(0,0)$ with radii
$e^{\sigma_{1}\delta_{1}}$ and $e^{\sigma_{2}\delta_{2}}$.

  2. (ii)

The four domains depicted in the left panel of Figure 2 (reduced
representation in the plane of absolute values $(|z_{1}|,|z_{2}|)$)
corresponds to $\delta=(0.1,0.2)$ for all $\sigma\in\Sigma$.

To proceed to the general case of Reinhardt domains, we introduce some general
notation and list simple facts about toral automorphisms.

######  Definition 3.2.

For

$A=\begin{pmatrix}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{pmatrix}\in{\operatorname{GL}_{2}}(\mathbb{Z})$

we define $\tau_{A}\colon{\mathbb{T}^{2}}\to{\mathbb{T}^{2}}$ to be the toral
automorphism given by
$\tau_{A}(z_{1},z_{2})=(z_{1}^{a_{11}}z_{2}^{a_{12}},z_{1}^{a_{21}}z_{2}^{a_{22}})$.
With slight abuse of notation we also write $\tau_{A}$ for the extension of
the map to ${\hat{\mathbb{C}}}^{2}$.

######  Lemma 3.3.

Let $A,B\in{\operatorname{GL}_{2}}(\mathbb{Z})$ and $f\in
L^{2}(\mathbb{T}^{2})$. Then:

  1. (i)

$\tau_{A}\circ\tau_{B}=\tau_{A\cdot B}$, and in particular
$\tau_{A}^{-1}=\tau_{A^{-1}}$.

  2. (ii)

The map $\tau_{A}$ preserves $\mathbb{T}^{2}$ and is holomorphic in a
neighbourhood.

  3. (iii)

$f\circ\tau_{A}\in L^{2}(\mathbb{T}^{2})$, and
$\|f\circ\tau_{A}\|_{L^{2}(\mathbb{T}^{2})}=\|f\|_{L^{2}(\mathbb{T}^{2})}$.

We note that by Lemma A.1 in the appendix, every Reinhardt domain
$D^{\sigma}_{P,\delta}\subset\hat{{\mathbb{C}}}^{2}$ is the image of a bounded
Reinhardt domain in ${\mathbb{C}}^{2}$ under a holomorphic map of the form
$\tau_{A},A\in{\operatorname{GL}_{2}}(\mathbb{Z})$.

######  Remark 3.4.

In analogy to the case of the one-dimensional Riemann sphere, here we extend
the notion of holomorphic functions on domains in ${\mathbb{C}^{2}}$ to those
on domains in ${\hat{\mathbb{C}}^{2}}$ in the obvious way. For a domain
$\hat{D}\subset{\hat{\mathbb{C}}^{2}}$, a function
$f\colon\hat{D}\to{\mathbb{C}}$ is holomorphic if there exists a domain
$D\subset{\mathbb{C}^{2}}$ and a biholomorphic mapping $\phi\colon
D\to\hat{D}$ such that $f\circ\phi\colon D\to{\mathbb{C}}$ is holomorphic.

######  Definition 3.5.

For $\delta\in\mathbb{R}^{2}$, $\sigma\in\Sigma$ and
$P\in{\operatorname{GL}_{2}}(\mathbb{R})$, the Hardy-Hilbert space
$H^{\sigma}_{P,\delta}:=H^{2}(D_{P,\delta}^{\sigma})$ consists of all
holomorphic functions $f\colon D_{P,\delta}^{\sigma}\to\mathbb{C}$ such that

$\sup_{r\in\Lambda^{\sigma}_{P,\delta}}\frac{1}{(2\pi)^{2}}\int_{[0,2\pi]^{2}}|f(e^{r+it})|^{2}\,dt<\infty.$

In the case $P=\mathbb{I}$ (that is, $D^{\sigma}_{P,\delta}$ a polydisk), we
will write $H^{\sigma}_{\delta}=H^{\sigma}_{P,\delta}$ for brevity.

######  Remark 3.6.

As the definition indicates, the next results (Lemma 3.7 and Proposition 3.9)
will establish that the space $H^{\sigma}_{P,\delta}$ with the inner product
(7) indeed forms a Hilbert space.

We recall that
$\partial^{*}D^{\sigma}_{P,\delta}=\mathbb{T}^{\sigma}_{P,\delta}=\mathbb{T}^{2}_{\exp(v^{\sigma}_{P,\delta})}$,
and proceed with the following generalization of a classical result for the
polydisk $\mathbb{D}^{2}=D^{--}_{\mathbb{I},0}$.

######  Lemma 3.7.

For any $f\in H^{\sigma}_{P,\delta}$ there exists $f^{*}\in
L^{2}(\mathbb{T}_{P,\delta}^{\sigma})$ (the “boundary value function” of $f$),
which satisfies

$\lim_{r\in\Lambda^{\sigma}_{P,\delta},r\to
v^{\sigma}_{P,\delta}}\int_{[0,2\pi]^{2}}|f(e^{r+it})-f^{*}(e^{v^{\sigma}_{P,\delta}+it})|^{2}\,dt=0.$

###### Proof.

Since $D^{\sigma}_{P,\delta}=D^{(-1,-1)}_{P^{\prime},\delta}$ and for
$P^{\prime}=PI^{\sigma}\in{\operatorname{GL}_{2}}(\mathbb{R})$, we can assume
without loss of generality that $\sigma=(-1,-1)$. The case of $P$ being a non-
negative matrix and $\delta=0$ (i.e.,
$D^{\sigma}_{P,\delta}\subseteq\mathbb{D}^{2}$ a bounded Reinhardt domain) is
classical, see [U, Section 2.5] or [L], and the more general case of
$\delta\in\mathbb{R}^{2}$ follows immediately. For general
$P\in{\operatorname{GL}_{2}}(\mathbb{R})$ and $f\in H^{\sigma}_{P,\delta}$, we
write $P=A\tilde{P}$ with $A\in{\operatorname{GL}_{2}}(\mathbb{Z})$ and
$\tilde{P}\in{\operatorname{GL}_{2}}(\mathbb{R})$ non-negative (Lemma A.1). We
use $\tau_{A}\colon D_{\tilde{P},\delta}^{\sigma}\to D_{P,\delta}^{\sigma}$
and consider the function $\tilde{f}=f\circ\tau_{A}\in
H^{\sigma}_{\tilde{P},\delta}$. By the previous case, there exists
$\tilde{f}^{*}\in L^{2}(\mathbb{T}^{\sigma}_{\tilde{P},\delta})$ such that

$\lim_{r\in\Lambda^{\sigma}_{\tilde{P},\delta},r\to
v^{\sigma}_{\tilde{P},\delta}}\int_{[0,2\pi]^{2}}|\tilde{f}(e^{r+it})-\tilde{f}^{*}(e^{v^{\sigma}_{\tilde{P},\delta}+it})|^{2}\,dt=0.$

Writing $f^{*}=\tilde{f}^{*}\circ\tau_{A}^{-1}$, we obtain

$\displaystyle\lim_{r\in\Lambda^{\sigma}_{P,\delta},r\to
v^{\sigma}_{P,\delta}}\int_{[0,2\pi]^{2}}|f(e^{r+it})-f^{*}(e^{v^{\sigma}_{P,\delta}+it})|^{2}\,dt$
$\displaystyle=$
$\displaystyle\lim_{r\in\Lambda^{\sigma}_{\tilde{P},\delta},r\to
v^{\sigma}_{\tilde{P},\delta}}\int_{[0,2\pi]^{2}}|f\circ\tau_{A}(e^{r+it})-f^{*}\circ\tau_{A}(e^{v^{\sigma}_{\tilde{P},\delta}+it})|^{2}\,dt=0.\qed$

We define an inner product on $H^{\sigma}_{P,\delta}$ by setting

$(f,g)_{H^{\sigma}_{P,\delta}}:=\langle
f^{*},\overline{g^{*}}\rangle_{\mathbb{T}^{\sigma}_{P,\delta}}:=(f^{*},g^{*})_{L^{2}(\mathbb{T}^{\sigma}_{P,\delta})},$
(7)

and we will omit the star notation for the boundary value function whenever
there is no ambiguity. Here the $L^{2}$ inner product between $f^{*}$ and
$g^{*}$ is defined as

$(f^{*},g^{*})_{L^{2}(\mathbb{T}^{\sigma}_{P,\delta})}=\int_{[0,2\pi]^{2}}f^{*}(e^{v^{\sigma}_{P,\delta}+it})\overline{g^{*}(e^{v^{\sigma}_{P,\delta}+it})}\,\frac{dt}{(2\pi)^{2}}=\int_{\mathbb{T}^{\sigma}_{P,\delta}}f^{*}(z)\overline{g^{*}(z)}\,dm(z),$

where $dm(z)=\frac{dz_{1}}{2\pi iz_{1}}\frac{dz_{2}}{2\pi iz_{2}}$ is the
normalised Lebesgue measure on $\mathbb{T}^{\sigma}_{P,\delta}$.

######  Notation 3.8.

We denote by $\Lambda^{\sigma,o}_{P}$ the polar cone of
$\Lambda^{\sigma}_{P}=P(R^{\sigma})$, and note that
$\Lambda^{\sigma,o}_{P}=(P^{T})^{-1}(\operatorname{cl}({R^{-\sigma}}))$. We
write $\mathbb{Z}^{\sigma}_{P}={\mathbb{Z}^{2}}\cap\Lambda^{\sigma,o}_{P}$.
For $P=\mathbb{I}$, we will use the shorthand
$\mathbb{Z}^{\sigma}=\mathbb{Z}^{\sigma}_{\mathbb{I}}$.

######  Proposition 3.9 (Alternative charaterisations).

Let $\delta\in{\mathbb{R}^{2}}$, $\sigma\in\Sigma$ and
$P\in{\operatorname{GL}_{2}}(\mathbb{R})$.

  1. (i)

The function $p_{n}(z)=z^{n}$ is in $H^{\sigma}_{P,\delta}$ iff
$n\in\mathbb{Z}^{\sigma}_{P}$.

  2. (ii)

Let $f$ be given by $f(z)=\sum_{n\in{\mathbb{Z}^{2}}}f_{n}z^{n}$. Then, $f\in
H^{\sigma}_{P,\delta}$ iff $f_{n}=0$ for $n\notin\mathbb{Z}^{\sigma}_{P}$ and
$\sum_{n\in{\mathbb{Z}^{2}}}|f_{n}|^{2}e^{2\langle
n,v^{\sigma}_{P,\delta}\rangle}<\infty$.

  3. (iii)

$H^{\sigma}_{P,\delta}$ is a Hilbert space with inner product given by (7).
Moreover, for $f\in H^{\sigma}_{P,\delta}$ with
$f(z)=\sum_{n\in\mathbb{Z}^{\sigma}_{P}}f_{n}z^{n}$, we have

$\|f\|^{2}_{H^{\sigma}_{P,\delta}}=\sup_{r\in\Lambda^{\sigma}_{P,\delta}}\frac{1}{(2\pi)^{2}}\int_{[0,2\pi]^{2}}|f(e^{r+it})|^{2}\,dt=\sum_{n\in\mathbb{Z}^{\sigma}_{P}}|f_{n}|^{2}e^{2\langle
n,v^{\sigma}_{P,\delta}\rangle}.$

  4. (iv)

$H^{\sigma}_{P,\delta}$ is the closure of $\\{p_{n}\colon
n\in\mathbb{Z}^{\sigma}_{P}\\}$ with respect to the norm
$\|\cdot\|_{H^{\sigma}_{P,\delta}}$.

###### Proof.

We begin by recalling that
$\Lambda^{\sigma}_{P,\delta}=\Lambda^{\sigma}_{P}+v^{\sigma}_{P,\delta}$, and
observe that

$\sup_{r\in\Lambda^{\sigma}_{P,\delta}}\frac{1}{(2\pi)^{2}}\int_{[0,2\pi]^{2}}|p_{n}(e^{r+it})|^{2}\,dt=\sup_{r\in\Lambda^{\sigma}_{P,\delta}}e^{2\langle
n,r\rangle}=e^{2\langle
n,v^{\sigma}_{P,\delta}\rangle}\sup_{r\in\Lambda_{P}^{\sigma}}e^{2\langle
n,r\rangle}$

for any $n\in{\mathbb{Z}^{2}}$. Since the supremum is finite if and only if
$\langle n,r\rangle\leq 0$ for all $r\in\Lambda^{\sigma}_{P}$, that is, if and
only if
$n\in\mathbb{Z}^{\sigma}_{P}=\mathbb{Z}^{2}\cap\Lambda_{P}^{\sigma,o}$,
statement $(i)$ follows. Next, we formally calculate for
$f(z)=\sum_{n\in\mathbb{Z}^{2}}f_{n}z^{n}$ that

$\displaystyle\frac{1}{(2\pi)^{2}}\int_{[0,2\pi]^{2}}|f(e^{r+it})|^{2}\,dt$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{[0,2\pi]^{2}}\sum_{n\in\mathbb{Z}^{2}}f_{n}e^{n(r+it)}\overline{\sum_{m\in\mathbb{Z}^{2}}f_{m}e^{m(r+it)}}\,dt$
$\displaystyle=\sum_{n\in\mathbb{Z}^{2}}|f_{n}|^{2}\frac{1}{(2\pi)^{2}}\int_{[0,2\pi]^{2}}|p_{n}(e^{r+it})|^{2}\,dt$
$\displaystyle=\sum_{n\in\mathbb{Z}^{2}}|f_{n}|^{2}e^{2\langle n,r\rangle}.$

If $f_{n}\neq 0$ for some $n\notin\mathbb{Z}^{\sigma}_{P}$, by the above
argument, the supremum over $r\in\Lambda^{\sigma}_{P,\delta}$ is infinite, and
hence $f\notin H^{\sigma}_{P,\delta}$. On the other hand, for
$n\in\mathbb{Z}^{\sigma}_{P}$ and $r\in\Lambda^{\sigma}_{P,\delta}$, we have
$\langle n,r\rangle=\langle P^{T}n,P^{-1}r\rangle=\langle
I^{\sigma}P^{T}n,I^{\sigma}P^{-1}r\rangle$ with
$I^{\sigma}P^{T}n\in\mathbb{R}^{2}_{\leq 0}$ and
$I^{\sigma}P^{-1}r\in\mathbb{R}^{2}_{>0}+\delta$, so that

$\langle n,r\rangle\leq\langle I^{\sigma}P^{T}n,\delta\rangle=\langle
P^{T}n,I^{\sigma}\delta\rangle=\langle n,v^{\sigma}_{P,\delta}\rangle.$

Therefore, if $f_{n}=0$ for all $n\notin\mathbb{Z}^{\sigma}_{P}$ and
$\sum_{n\in{\mathbb{Z}^{2}}}|f_{n}|^{2}e^{2\langle
n,v^{\sigma}_{P,\delta}\rangle}<\infty$, then

$\sup_{r\in\Lambda^{\sigma}_{P,\delta}}\frac{1}{(2\pi)^{2}}\int_{[0,2\pi]^{2}}|f(e^{r+it})|^{2}\,dt=\sup_{r\in\Lambda^{\sigma}_{P,\delta}}\sum_{n\in{\mathbb{Z}^{2}}}|f_{n}|^{2}e^{2\langle
n,r\rangle}=\sum_{n\in{\mathbb{Z}^{2}}}|f_{n}|^{2}e^{2\langle
n,v^{\sigma}_{P,\delta}\rangle}$

implies $f\in H^{\sigma}_{P,\delta}$, proving $(ii)$. For $(iii)$, it remains
to show the last equality, which follows immediately from the last calculation
and (7), as for any $f\in H^{\sigma}_{P,\delta}$ we have that

$\sup_{r\in\Lambda^{\sigma}_{P,\delta}}\frac{1}{(2\pi)^{2}}\int_{[0,2\pi]^{2}}|f(e^{r+it})|^{2}\,dt=\sum_{n\in\mathbb{Z}^{\sigma}_{P}}|f_{n}|^{2}e^{2\langle
n,v^{\sigma}_{P,\delta}\rangle}.$

Finally for $(iv)$, we assume without loss of generality that $\sigma=(-1,-1)$
(since $H^{\sigma}_{P,\delta}=H^{(-1,-1)}_{PI^{\sigma},\delta}$). The case of
$P$ a non-negative matrix is proved in [MX, Proposition 3.6]. For general
$P\in{\operatorname{GL}_{2}}(\mathbb{R})$, we write $P=A\tilde{P}$ with
$A\in{\operatorname{GL}_{2}}(\mathbb{Z})$ and
$\tilde{P}\in{\operatorname{GL}_{2}}(\mathbb{R})$ non-negative, and use the
biholomorphic mapping $\tau_{A}\colon D^{\sigma}_{\tilde{P},\delta}\to
D^{\sigma}_{P,\delta}$, noting that
$\|f\circ\tau_{A}\|_{H^{\sigma}_{\tilde{P},\delta}}=\|f\|_{H^{\sigma}_{P,\delta}}$
for any $f\in H^{\sigma}_{P,\delta}$. Since $A$ maps
$\mathbb{Z}^{\sigma}_{\tilde{P}}$ bijectively onto $\mathbb{Z}^{\sigma}_{P}$,
we have $\\{p_{n}\colon
n\in\mathbb{Z}^{\sigma}_{P}\\}=\\{p_{n}\circ\tau_{A}\colon
n\in\mathbb{Z}^{\sigma}_{\tilde{P}}\\}$, and the general case follows. ∎

We next show shows that two Hardy-Hilbert $H^{\sigma}_{P,\delta}$ and
$H^{\sigma}_{P^{\prime},\delta}$ are isomorphic, whenever
$P^{-1}P^{\prime}\in{\operatorname{GL}_{2}}(\mathbb{Z})$. For any
$\delta\in{\mathbb{R}^{2}}$, $\sigma\in\Sigma$,
$P\in{\operatorname{GL}_{2}}(\mathbb{R})$ and
$A\in{\operatorname{GL}_{2}}(\mathbb{Z})$, we observe that
$A(\Lambda_{P,\delta}^{\sigma})=(AP)(R^{\sigma}_{\delta})=\Lambda_{AP,\delta}^{\sigma}$,
and $AP\in{\operatorname{GL}_{2}}(\mathbb{R})$, from which it follows that

$\tau_{A}(D_{P,\delta}^{\sigma})=\tau_{A}(e^{\Lambda_{P,\delta}^{\sigma}}{\mathbb{T}^{2}})=e^{\Lambda_{AP,\delta}^{\sigma}}{\mathbb{T}^{2}}=D_{AP,\delta}^{\sigma}.$

######  Proposition 3.10.

Let $\delta\in{\mathbb{R}^{2}}$, $\sigma\in\Sigma$,
$P\in{\operatorname{GL}_{2}}(\mathbb{R})$ and
$A\in{\operatorname{GL}_{2}}(\mathbb{Z})$. Then the operator $C_{\tau_{A}}$
given by $C_{\tau_{A}}f=f\circ\tau_{A}$ is an isometric isomorphism from
$H^{\sigma}_{AP,\delta}$ to $H^{\sigma}_{P,\delta}$.

###### Proof.

Let $f=\sum_{n\in{\mathbb{Z}^{2}}}f_{n}p_{n}\in H^{\sigma}_{AP,\delta}$, that
is, by Proposition 3.9, $f_{n}=0$ for all $n\notin\mathbb{Z}^{\sigma}_{AP}$,
and $\sum_{n\in{\mathbb{Z}^{2}}}|f_{n}|^{2}e^{2\langle
n,v^{\sigma}_{AP,\delta}\rangle}<\infty$, and write
$g=C_{\tau_{A}}f=f\circ\tau_{A}$. For $n\in{\mathbb{Z}^{2}}$ we have

$p_{n}\circ\tau_{A}(z)=(z_{1}^{a_{11}}z_{2}^{a_{12}})^{n_{1}}(z_{1}^{a_{21}}z_{2}^{a_{22}})^{n_{2}}=z_{1}^{a_{11}n_{1}+a_{21}n_{2}}z_{2}^{a_{21}n_{1}+a_{22}n_{2}}=p_{A^{T}n}(z).$

Therefore, it holds that

$g=\sum_{n\in{\mathbb{Z}^{2}}}f_{n}p_{n}\circ\tau_{A}=\sum_{n\in{\mathbb{Z}^{2}}}f_{n}p_{A^{T}n}=\sum_{n\in{\mathbb{Z}^{2}}}f_{(A^{T})^{-1}n}p_{n},$

and hence $g=\sum_{n\in{\mathbb{Z}^{2}}}g_{n}p_{n}$ with
$g_{n}=f_{(A^{T})^{-1}n}$. Further, since

$\Lambda^{\sigma,o}_{AP}=((AP)^{T})^{-1}(\operatorname{cl}({R^{-\sigma}}))=(A^{T})^{-1}(P^{T})^{-1}(\operatorname{cl}({R^{-\sigma}}))=(A^{T})^{-1}(\Lambda^{\sigma,o}_{P}),$

it follows that $(A^{T})^{-1}n\in\Lambda^{\sigma,o}_{AP}$ iff
$n\in\Lambda^{\sigma,o}_{P}$. Therefore, $g_{n}=f_{(A^{T})^{-1}n}=0$ whenever
$n\notin\mathbb{Z}^{\sigma}_{P}$. Moreover,

$\displaystyle\|g\|^{2}_{H^{\sigma}_{P,\delta}}$
$\displaystyle=\sum_{n\in{\mathbb{Z}^{2}}}|g_{n}|^{2}e^{2\langle
n,v^{\sigma}_{P,\delta}\rangle}$
$\displaystyle=\sum_{n\in{\mathbb{Z}^{2}}}|f_{n}|^{2}e^{2\langle
A^{T}n,Pv^{\sigma}_{\delta}\rangle}=\sum_{n\in{\mathbb{Z}^{2}}}|f_{n}|^{2}e^{2\langle
n,APv^{\sigma}_{\delta}\rangle}=\sum_{n\in{\mathbb{Z}^{2}}}|f_{n}|^{2}e^{2\langle
n,v^{\sigma}_{AP,\delta}\rangle}=\|f\|^{2}_{H^{\sigma}_{AP,\delta}},$

which by Proposition 3.9$(ii)$ implies that $g=C_{\tau_{A}}f\in
H^{\sigma}_{P,\delta}$, and
$\|C_{\tau_{A}}f\|_{H^{\sigma}_{P,\delta}}=\|f\|_{H^{\sigma}_{AP,\delta}}$ for
all $f\in H^{\sigma}_{AP,\delta}$, finishing the proof. ∎

The next two propositions are important ingredients for proving the main
result of this section.

######  Proposition 3.11.

Let $K$ be a compact subset of
$D_{P,\delta}^{\sigma}\subset{\hat{\mathbb{C}}}^{2}$ and $f\in
H^{\sigma}_{P,\delta}=H^{2}(D_{P,\delta}^{\sigma})$. Then, there is a
$C_{K}>0$ such that

$\sup_{z\in K}|f(z)|\leq C_{K}\|f\|_{H^{\sigma}_{P,\delta}}.$

###### Proof.

For $P=\mathbb{I}$ and $\sigma=(-1,-1)$ (that is, $D_{P,\delta}^{\sigma}$ a
polydisk with radii $(e^{-\delta_{1}},e^{-\delta_{2}})$) the result follows
directly from [BJ1, Lemma 2.9]. In particular, with
$D^{\prime}{\subset\\!\\!\\!\subset}D_{P,\delta}^{\sigma}$ a domain containing
$K$, and $U(D^{\prime})$ the space of functions holomorphic on $D^{\prime}$
and continuous on $\operatorname{cl}({D^{\prime}})$ endowed with the supremum
norm, the natural embedding $H^{--}_{\mathbb{I},\delta}\hookrightarrow
U(D^{\prime})$ is bounded, and its operator norm provides the constant
$C_{K}$.

For the general case, let
$\psi(z)=z^{-\sigma}=(z_{1}^{-\sigma_{1}},z_{2}^{-\sigma_{2}})$. By
Proposition 3.10, the operator $C_{\psi\circ\tau_{P}}\colon
H^{--}_{\mathbb{I},\delta}\to H^{\sigma}_{P,\delta}$ is an isometric
isomorphism, and the result follows by observing that

$\sup_{z\in K\subset
D_{P,\delta}^{\sigma}}|f(z)|=\sup_{z\in(\psi\circ\tau_{P})^{-1}(K)}|f\circ\psi\circ\tau_{P}(z)|$

with $(\psi\circ\tau_{P})^{-1}(K){\subset\\!\\!\\!\subset}D^{--}_{\delta}$. ∎

######  Definition 3.12 (Dual domains and dual space).

  1. (i)

We denote by $(H^{\sigma}_{P,\delta})^{\prime}$ the dual space of
$H^{\sigma}_{P,\delta}=H^{2}(D_{P,\delta}^{\sigma})$, that is the space of all
continuous linear functionals on $H^{\sigma}_{P,\delta}$.

  2. (ii)

We denote by $(D_{P,\delta}^{\sigma})^{\prime}$ the dual domain of
$D_{P,\delta}^{\sigma}$, which is given by
$(D_{P,\delta}^{\sigma})^{\prime}=D_{P,-\delta}^{-\sigma}$ (see the right
panel of Figure 2 for an illustration of the case $P=\mathbb{I}$).

Clearly, $((D_{P,\delta}^{\sigma})^{\prime})^{\prime}=D_{P,\delta}^{\sigma}$
and $\mathbb{T}_{P,\delta}^{\sigma}=\mathbb{T}^{-\sigma}_{P,-\delta}$.

######  Proposition 3.13.

$(H^{\sigma}_{P,\delta})^{\prime}=(H^{2}(D_{P,\delta}^{\sigma}))^{\prime}$ is
isometrically isomorphic to
$H^{-\sigma}_{P,-\delta}=H^{2}((D_{P,\delta}^{\sigma})^{\prime})$, via the
isomorphism $J\colon
H^{-\sigma}_{P,-\delta}\to(H^{\sigma}_{P,\delta})^{\prime}$ given by

$g\mapsto\langle\cdot,g\rangle_{\mathbb{T}^{\sigma}_{P,\delta}}.$

###### Proof.

We begin by showing that $J$ is injective. Assume $J(g)=0$ for some $g\in
H^{-\sigma}_{P,-\delta}$,
$g(z)=\sum_{n\in\mathbb{Z}_{P}^{-\sigma}}g_{n}z^{n}$, that is, $\langle
f,g\rangle_{\mathbb{T}^{\sigma}_{P,\delta}}=0$ for all $f\in
H^{\sigma}_{P,\delta}$. In particular, for every $m\in\mathbb{Z}_{P}^{\sigma}$
we have

$0=\langle
p_{m},g\rangle_{\mathbb{T}^{\sigma}_{P,\delta}}=\sum_{n\in\mathbb{Z}_{P}^{-\sigma}}g_{n}\langle
p_{m},p_{n}\rangle_{\mathbb{T}^{\sigma}_{P,\delta}}=\sum_{n\in\mathbb{Z}_{P}^{-\sigma}}g_{n}e^{\langle
m+n,v^{\sigma}_{P,\delta}\rangle}\delta_{m,-n}=g_{-m}.$

It follows that $g_{n}=0$ for all $n\in\mathbb{Z}_{P}^{-\sigma}$ and hence
$g=0$, proving injectivity of $J$.

To show that $J$ is surjective, let $\ell\in(H^{\sigma}_{P,\delta})^{\prime}$.
By the Riesz representation theorem, there exists $g\in
H^{\sigma}_{P,\delta}$, so that
$\ell=\ell_{g}=(\cdot,g)_{H^{\sigma}_{P,\delta}}$, and we can write
$g(z)=\sum_{n\in\mathbb{Z}_{P}^{\sigma}}g_{n}z^{n}$. We define
$h(z)=\sum_{n\in\mathbb{Z}_{P}^{-\sigma}}\overline{g_{-n}}e^{-2\langle
n,r\rangle}z^{n}$ with $r=v^{\sigma}_{P,\delta}$, and observe that $h\in
H^{2}((D_{P,\delta}^{\sigma})^{\prime})=H^{-\sigma}_{P,-\delta}$. For
$z=e^{r}e^{it}\in\mathbb{T}^{\sigma}_{P,\delta}$, $t\in[0,2\pi)^{2}$, we have
$e^{-2r}z=e^{-r}e^{it}=1/\overline{z}$, and hence

$\overline{h(z)}=\sum_{n\in\mathbb{Z}_{P}^{-\sigma}}g_{-n}\overline{e^{-2\langle
n,r\rangle}z^{n}}=\sum_{n\in\mathbb{Z}_{P}^{-\sigma}}g_{-n}z^{-n}=\sum_{n\in\mathbb{Z}_{P}^{\sigma}}g_{n}z^{n}=g(z).$

By (7) this implies

$(J(h))(f)=\langle
f,h\rangle_{\mathbb{T}^{\sigma}_{P,\delta}}=(f,g)_{H^{\sigma}_{P,\delta}}=\ell(f),$

for any $f\in H^{\sigma}_{P,\delta}$. Hence $J(h)=\ell$, proving surjectivity
of $J$.

Note that the Riesz representation theorem also yields
$\|\ell_{g}\|_{(H^{\sigma}_{P,\delta})^{\prime}}=\|g\|_{H^{\sigma}_{P,\delta}}$.
On the other hand, for $J(h)=\ell_{g}$ as above, we have

$\|h\|^{2}_{H^{-\sigma}_{P,-\delta}}=\sum_{n\in\mathbb{Z}_{P}^{-\sigma}}|h_{n}|^{2}|e^{2\langle
n,r\rangle}|=\sum_{n\in\mathbb{Z}_{P}^{-\sigma}}|g_{-n}e^{-2\langle
n,r\rangle}|^{2}|e^{2\langle
n,r\rangle}|=\sum_{n\in\mathbb{Z}_{P}^{\sigma}}|g_{n}|^{2}|e^{2\langle
n,r\rangle}|=\|g\|^{2}_{H^{\sigma}_{P,\delta}},$

and so we obtain
$\|J(h)\|_{(H^{\sigma}_{P,\delta})^{\prime}}=\|h\|_{H^{-\sigma}_{P,-\delta}}$,
proving that $J$ is an isometry. ∎

######  Remark 3.14.

Using the above proposition we have the following reformulation: if $f\in
H^{\sigma}_{P,\delta}$ then

$\|f\|_{H^{\sigma}_{P,\delta}}=\sup\left\\{\left|\langle
f,g\rangle_{\mathbb{T}^{\sigma}_{P,\delta}}\right|\colon g\in
H^{2}((D_{P,\delta}^{\sigma})^{\prime})\cap\mathcal{P}\,,\|g\|_{H^{2}((D_{P,\delta}^{\sigma})^{\prime})}\leq
1\right\\},$ (8)

where as before, $\mathcal{P}$ denotes the space of Laurent polynomials.

###  3.2 Hardy-Hilbert spaces for toral diffeomorphisms

For convenience, here we introduce notation to succinctly express monomial
bases for various Hilbert spaces which we will require below.

######  Notation 3.15.

Let $R^{\sigma,o}$ be the (closed) polar cone of $R^{\sigma}$ for any
$\sigma\in\Sigma$. Define

$\hat{R}^{\sigma,o}=\begin{cases}R^{\sigma,o}\text{ if }\sigma=(-1,-1),\\\
R^{\sigma,o}\setminus{\\{(0,0)\\}}\text{ if }\sigma=(+1,+1),\\\
\operatorname{int}{(R^{\sigma,o})}\text{ if }\sigma\in\Sigma^{-1},\end{cases}$

and let $\hat{\mathbb{Z}}^{\sigma}_{P}={\mathbb{Z}^{2}}\cap
P(\hat{R}^{\sigma,o})$, noting that the $\hat{R}^{\sigma,o}$,
$\sigma\in\Sigma$, form a partition of $\mathbb{R}^{2}$. In analogy to the
characterization of $H^{\sigma}_{P,\delta}$ in Proposition 3.9$(ii)$, for
$\sigma\in\Sigma$, $\delta\in\mathbb{R}^{2}$ and
$P\in{\operatorname{GL}_{2}}(\mathbb{R})$ we define

$\hat{H}^{\sigma}_{P,\delta}:=\hat{H}^{2}(D^{\sigma}_{P,\delta}):=\\{f\in
H^{\sigma}_{P,\delta}:(f,p_{n})_{H^{\sigma}_{P,\delta}}=0\text{ for
}n\notin\hat{\mathbb{Z}}^{\sigma}_{P}\\}.$

Writing $e_{n}=\frac{p_{n}}{\nu(n)}$ with
$\nu(n)=\|p_{n}\|_{H^{\sigma}_{P,\delta}}=e^{\langle
n,v^{\sigma}_{P,\delta}\rangle}$, we note that
$\\{e_{n}:n\in\hat{\mathbb{Z}}^{\sigma}_{P}\\}$ forms an orthonormal basis for
$\hat{H}^{\sigma}_{P,\delta}$, which is a Hilbert space with the same norm
$\|\cdot\|_{H^{\sigma}_{P,\delta}}$. Furthermore, for conveniently handling
dual spaces, we set
$\check{\mathbb{Z}}^{\sigma}_{P}=-\hat{\mathbb{Z}}^{\sigma}_{P}$, and

$\check{H}^{\sigma}_{P,\delta}=\\{f\in
H^{2}(D^{\sigma}_{P,\delta})\colon\langle
f,p_{n}\rangle=0,n\notin\check{\mathbb{Z}}^{\sigma}_{P}\\}.$

######  Remark 3.16.

With the above notation, the isomorphism defined in Proposition 3.13 also
forms an isometric isomorphism between $\check{H}^{-\sigma}_{P,-\delta}$ and
$(\hat{H}^{\sigma}_{P,\delta})^{\prime}$.

######  Notation 3.17.

For $\ell\in\\{1,-1\\}$, we write

$\mathcal{D}^{\ell}_{P,\delta}:=\bigcup_{\sigma\in\Sigma^{\ell}}D^{\sigma}_{P,\delta},\qquad\text{
and
}\qquad(\mathcal{D}^{\ell}_{P,\delta})^{\prime}:=\bigcup_{\sigma\in\Sigma^{\ell}}(D^{\sigma}_{P,\delta})^{\prime}=\bigcup_{\sigma\in\Sigma^{\ell}}D^{-\sigma}_{P,-\delta},$

and note that the distinguished boundary of $\mathcal{D}^{\ell}_{P,\delta}$ is
$\partial^{*}\mathcal{D}^{\ell}_{P,\delta}=\bigcup_{\sigma\in\Sigma^{\ell}}\mathbb{T}^{\sigma}_{P,\delta}$.

See the left panel of Figure 2 for an illustration in the case $P=\mathbb{I}$:
the green and pink rectangles represent $\mathcal{D}^{1}_{P,\delta}$, whereas
the orange and blue rectangles represent $\mathcal{D}^{-1}_{P,\delta}$, with
correspondingly coloured stars representing the respective distinguished
boundaries.

######  Definition 3.18.

For $\ell\in\\{1,-1\\}$, define
$\mathcal{H}^{\ell}_{P,\delta}=\bigoplus_{\sigma\in\Sigma^{\ell}}\hat{H}^{\sigma}_{P,\delta}$,
which is a Hilbert space with the inner product of
$f=(f^{\sigma})_{\sigma\in\Sigma^{\ell}},g=(g^{\sigma})_{\sigma\in\Sigma^{\ell}}\in\mathcal{H}^{\ell}_{P,\delta}$
given by

$(f,g)_{\mathcal{H}^{\ell}_{P,\delta}}=\sum_{\sigma\in\Sigma^{\ell}}(f^{\sigma},g^{\sigma})_{H^{\sigma}_{P,\delta}}.$

As before, for the case $P=\mathbb{I}$ where the domains
$D^{\sigma}_{P,\delta}$ are polydisks, we will use the shorthands
$\hat{\mathbb{Z}}^{\sigma}=\hat{\mathbb{Z}}^{\sigma}_{\mathbb{I}}$,
$D^{\sigma}_{\delta}=D^{\sigma}_{\mathbb{I},\delta}$,
$\mathcal{D}^{\ell}_{\delta}=\mathcal{D}^{\ell}_{P,\delta}$,
$\hat{H}^{\sigma}_{\delta}=\hat{H}^{\sigma}_{\mathbb{I},\delta}$, and
$\mathcal{H}^{\ell}_{\delta}=\mathcal{H}^{\ell}_{\mathbb{I},\delta}$.

######  Remark 3.19.

Nominally, an $f\in\mathcal{H}^{\ell}_{P,\delta}$ is a tuple
$f=(f^{\sigma})_{\sigma\in\Sigma^{\ell}}$ of two holomorphic functions with
distinct domains,
$f^{\sigma}\in\hat{H}^{\sigma}_{P,\delta}=\hat{H}^{2}(D^{\sigma}_{P,\delta})$.
It will be useful to alternatively consider $\mathcal{H}^{\ell}_{P,\delta}$ as
(isomorphic to) a function space, which requires us to distinguish two cases:

  1. (i)

If $\delta\in\mathbb{R}^{2}_{<0}$, then
$\mathcal{A}_{P,\delta}=D^{\sigma}_{P,\delta}\cap
D^{-\sigma}_{P,\delta}\neq\emptyset$, and for any
$f=(f^{\sigma},f^{-\sigma})\in\mathcal{H}^{\ell}_{P,\delta}$ we can define a
holomorphic function $\tilde{f}$ on $\mathcal{A}_{P,\delta}$ by
$\tilde{f}(z)=f^{\sigma}(z)+f^{-\sigma}(z)$, yielding an isometric isomorphism
between
$\mathcal{H}^{\ell}_{P,\delta}=\hat{H}^{2}(D^{\sigma}_{P,\delta})\oplus\hat{H}^{2}(D^{-\sigma}_{P,\delta})$
and (a subspace of) $H^{2}(\mathcal{A}_{P,\delta})$.

  2. (ii)

If $\delta_{k}\geq 0$ for some $k\in\\{1,2\\}$, then
$D^{\sigma}_{P,\delta}\cap D^{-\sigma}_{P,\delta}=\emptyset$, and for any
$f=(f^{\sigma},f^{-\sigma})\in\mathcal{H}^{\ell}_{P,\delta}$ we can define a
holomorphic function $\tilde{f}$ on
$\mathcal{D}^{\ell}_{P,\delta}=D^{\sigma}_{P,\delta}\cup
D^{-\sigma}_{P,\delta}$ by $\tilde{f}(z)=f^{\sigma}(z)$ for $z\in
D^{\sigma}_{P,\delta}$, $\sigma\in\Sigma^{\ell}$, yielding an isomorphism
between $\mathcal{H}^{\ell}_{P,\delta}$, and (a subspace of) the space of
holomorphic functions on $\mathcal{D}^{\ell}_{P,\delta}$ which extend to
square-integrable functions on
$\partial^{*}\mathcal{D}^{\ell}_{P,\delta}=\partial^{*}D^{\sigma}_{P,\delta}\cup\partial^{*}D^{-\sigma}_{P,\delta}$.

Figure 3: Absolute domains for the Reinhardt domains underlying
$H_{\nu}=H_{P,\alpha,-\gamma}$ with $\alpha=(0.1,0.2)$, $\gamma=(0.4,0.3)$,
for $P=\mathbb{I}$ (left) and the rotation matrix with rotation angle $\pi/10$
(right).

We can now express the weighted function spaces with respect to quadrant-wise
exponential weights considered in Section 2.1 as direct sums of Hardy-Hilbert
spaces on certain disks, see Figure 3 (left). For this, let
$\alpha,\gamma\in\mathbb{R}^{2}$, $\nu=\nu_{\alpha,-\gamma}$ the associated
quadrant-wise exponential weight (see Definition 2.5) and
$H_{\nu}=H_{\alpha,-\gamma}$ the resulting Hilbert space, given as the
completion of $\mathcal{P}$ with respect to the norm $\|\cdot\|_{\nu}$ (see
Definition 2.2). Then

$\displaystyle
H_{\alpha,-\gamma}\cong\mathcal{H}^{1}_{\alpha}\oplus\mathcal{H}^{-1}_{-\gamma}$
$\displaystyle=\hat{H}^{--}_{\alpha}\oplus\hat{H}^{++}_{\alpha}\oplus\hat{H}^{-+}_{-\gamma}\oplus\hat{H}^{+-}_{-\gamma}$
$\displaystyle=\hat{H}^{2}(D^{--}_{\alpha})\oplus\hat{H}^{2}(D^{++}_{\alpha})\oplus\hat{H}^{2}((D^{+-}_{\gamma})^{\prime})\oplus\hat{H}^{2}((D^{-+}_{\gamma})^{\prime})$

with the isometric isomorphism given by

$f=\sum_{n\in\mathbb{Z}^{2}}f_{n}p_{n}\mapsto(f^{\sigma})_{\sigma\in\Sigma},\qquad
f^{\sigma}=\sum_{n\in\hat{\mathbb{Z}}^{\sigma}}f_{n}p_{n}.$

With the obvious generalization of the weight function
$\nu=\nu_{P,\alpha,-\gamma}$ one can also define this for $P\neq\mathbb{I}$,
obtaining the more general space
$H_{\nu}=H_{P,\alpha,-\gamma}\cong\mathcal{H}^{1}_{P,\alpha}\oplus\mathcal{H}^{-1}_{P,-\gamma}$,
see Figure 3 (right).

######  Remark 3.20.

The above isomorphic representation of $H_{\nu}$ reveals an intuitive
structure of this Hilbert space. For $\alpha,\gamma\in\mathbb{R}^{2}_{>0}$, by
Remark 3.19, the first part $\mathcal{H}^{1}_{P,\alpha}$ can be viewed as
(isomorphic to) “$H^{2}(\mathcal{D}^{1}_{P,\alpha})$”, the space of functions
holomorphic on $\mathcal{D}^{1}_{P,\alpha}$ extending to square-integrable
functions on $\partial^{*}\mathcal{D}^{1}_{P,\alpha}$. The second part
$\mathcal{H}^{-1}_{P,-\gamma}$, on the other hand, can be seen to be
isomorphic to the dual $(\mathcal{H}^{-1}_{P,\gamma})^{\prime}$ (see Lemma
3.29), with $\mathcal{H}^{-1}_{P,\gamma}$ isomorphic to a space
“$H^{2}(\mathcal{D}^{-1}_{P,\gamma})$”: the space of holomorphic functions on
$\mathcal{D}^{-1}_{P,\gamma}$ extending to square-integrable functions on
$\partial^{*}\mathcal{D}^{-1}_{P,\gamma}$.

######  Notation 3.21.

For $f\in\mathcal{H}^{\ell}_{P,\delta}$ for some $\ell\in\\{1,-1\\}$, or
$f\in\mathcal{H}^{1}_{P,\alpha}\oplus\mathcal{H}^{-1}_{P,-\gamma}$, we denote
the canonical projection onto one of the components as $\hat{\Pi}^{\sigma}$
(omitting the dependence on $P$), given by

$\hat{\Pi}^{\sigma}p_{n}=\begin{cases}p_{n}&\text{ if
}n\in\hat{\mathbb{Z}}^{\sigma}_{P},\\\ 0&\text{ otherwise,}\end{cases}$

so that if $f=\sum_{n\in\mathbb{Z}^{2}}f_{n}p_{n}$ (in the sense of Remark
3.19), then
$\hat{\Pi}^{\sigma}f=\sum_{n\in\hat{\mathbb{Z}}^{\sigma}_{P}}f_{n}p_{n}$,
where the operator’s domain and range can be inferred from context. Similarly,
we write
$\check{\Pi}^{\sigma}f=\sum_{n\in\check{\mathbb{Z}}^{\sigma}_{P}}f_{n}p_{n}$.

######  Remark 3.22.

With the isometric isomorphism
$\Phi\colon\mathcal{H}^{1}_{P,\alpha}\oplus\mathcal{H}^{-1}_{P,-\gamma}\to
H_{P,\alpha,-\gamma}$, we can now use any well-defined operator $L$ on
$H_{P,\alpha,-\gamma}$ to define a conjugated operator on the respective space
of function tuples. With slight abuse of notation, in such cases we will
continue denoting the respective operator on
$\mathcal{H}^{1}_{P,\alpha}\oplus\mathcal{H}^{-1}_{P,-\gamma}$ by the same
symbol $L$. Furthermore, every toral automorphism $\tau_{Q}$,
$Q\in{\operatorname{GL}_{2}}(\mathbb{Z})$, yields an isometric isomorphism
$C_{\tau_{Q}}\colon H^{\sigma}_{QP,\delta}\to H^{\sigma}_{P,\delta}$ for any
$P\in{\operatorname{GL}_{2}}(\mathbb{R})$, $\delta\in{\mathbb{R}^{2}}$,
$\sigma\in\Sigma$ (Proposition 3.10), which can be used to define an operator
$\mathcal{H}^{1}_{QP,\alpha}\oplus\mathcal{H}^{-1}_{QP,-\gamma}\to\mathcal{H}^{1}_{P,\alpha}\oplus\mathcal{H}^{-1}_{P,-\gamma}$
given by
$(f^{\sigma})_{\sigma\in\Sigma}\mapsto(\hat{\Pi}^{\sigma}(f^{\sigma}\circ\tau_{Q}))_{\sigma\in\Sigma}$,
conjugated to the composition operator $f\mapsto f\circ\tau_{Q}$ viewed as an
operator from $H_{QP,\alpha,-\gamma}$ to $H_{P,\alpha,-\gamma}$. We will refer
to all three of these operators as $C_{\tau_{Q}}$.

The above decomposition of the space $H_{\nu}$ will allow us to prove the main
result of this section (Theorem 3.24) for holomorphic maps on the torus
satisfying the strongly expanding mapping property from Definition 2.17, by
adapting a method previously used in the one-dimensional setting of analytic
expanding circle maps $\tau\colon\mathbb{T}\to\mathbb{T}$, see [BJS]. To
summarize, writing $U_{r}=\\{z\in{\mathbb{C}}\colon|z|<r\\}$ and
$\mathcal{U}_{r}=U_{r}\cup({\hat{\mathbb{C}}}\setminus\operatorname{cl}({U_{1/r}}))$
with $r\in(0,1)$, analyticity and expansivity of $\tau$ imply that there
exists $r\in(0,1)$ such that $\tau$ extends holomorphically to a suitable
neighbourhood of $\mathbb{T}$, and
$\tau(\partial\mathcal{U}_{r}){\subset\\!\\!\\!\subset}\mathcal{U}_{r}$. This
in turn guarantees compactness of $C_{\tau}$ on $H^{2}(U_{r})\oplus
H_{0}^{2}({\hat{\mathbb{C}}}\setminus\operatorname{cl}({U_{1/r}}))$. In the
same spirit, for $T\colon{\mathbb{T}^{2}}\to{\mathbb{T}^{2}}$ an analytic
Anosov map, if $T^{\ell}$ is $(\ell,\delta_{\ell},\Delta_{\ell},P)$-strongly
expanding for $\ell\in\\{1,-1\\}$ and suitable
$\delta_{\ell},\Delta_{\ell}\in\mathbb{R}^{2}_{>0}$, then

$T\left(\partial^{*}\mathcal{D}^{\ell}_{P,\delta_{\ell}}\right)\subset\mathcal{D}^{\ell}_{P,\Delta_{\ell}}{\subset\\!\\!\\!\subset}\mathcal{D}^{\ell}_{P,\delta_{\ell}},$

which will be used to prove compactness of $C_{T}$ on
$\mathcal{H}^{1}_{\delta_{1}}\oplus(\mathcal{H}^{-1}_{\delta_{-1}})^{\prime}$
with similar techniques as in [BJS].

######  Lemma 3.23.

Let $T$ be a map with $(\ell,\delta,\Delta,P)$-strongly expanding mapping
property, then for every $\sigma,\tilde{\sigma}\in\Sigma^{\ell}$ there is
$\hat{\sigma}\in\Sigma^{\ell}$ such that

$\mathbb{T}^{\hat{\sigma}}_{P,\delta}\subset\operatorname{cl}({(D^{\sigma}_{P,\delta})^{\prime}})\text{
and }\,T(\mathbb{T}^{\hat{\sigma}}_{P,\delta})\subset
D^{\tilde{\sigma}}_{P,\Delta}{\subset\\!\\!\\!\subset}D^{\tilde{\sigma}}_{P,\delta}.$

###### Proof.

The first property is obvious as
$\mathbb{T}^{\hat{\sigma}}_{P,\delta}\subset\operatorname{cl}({(D^{\sigma}_{P,\delta})^{\prime}})$
for all $\hat{\sigma}\in\Sigma^{\ell}$. Next, if $T$ satisfies the (EP)
property, pick $\hat{\sigma}=\tilde{\sigma}$, whereas if $T$ satisfies (ER)
pick $\hat{\sigma}=-\tilde{\sigma}$. ∎

######  Theorem 3.24.

Let $T$ be an analytic diffeomorphism of ${\mathbb{T}^{2}}$. Further assume
that there are $\alpha,\gamma,A,\Gamma,\eta\in\mathbb{R}^{2}_{>0}$ with
$\alpha,\gamma<\eta$, and $P\in{\operatorname{GL}_{2}}(\mathbb{R})$, such that
$T$ and $T^{-1}$ can be analytically extended to $\mathcal{A}_{P,\eta}$ and
the following mapping properties hold:

  1. (i)

$T$ is $(1,\alpha,A,P)$-strongly expanding;

  2. (ii)

$T^{-1}$ is $(-1,\Gamma,\gamma,P)$-strongly expanding;

  3. (iii)

For any $\tilde{\sigma}\in\Sigma^{1},\sigma\in\Sigma^{-1}$, there exist
$\mathbb{T}^{2}_{q}\subset D^{\sigma}_{P,\gamma}\cap\mathcal{A}_{P,\eta}$ with
$T(\mathbb{T}^{2}_{q}){\subset\\!\\!\\!\subset}D^{\tilde{\sigma}}_{P,A}$.

Then, the composition operator $C_{T}$ given by

$f\mapsto f\circ T$

maps $H_{P,A,-\Gamma}$ continuously to $H_{P,\alpha,-\gamma}$.

Figure 4: Depiction of the $16$ different cases for the absolute domains in
the proof of Theorem 3.24. Each row corresponds to
$\tilde{\sigma}\in\Sigma^{k}$ and $\sigma\in\Sigma^{l}$, for
$k,l\in\\{1,-1\\}$. The blue rectangle represents the domain of holomorphicity
for the function $f$, and the orange rectangle the domain of holomorphicity
for the function $g$. We chose $P=\mathbb{I}$,
$\alpha=(0.05,0.1),A=(0.15,0.2),\Gamma=(0.35,0.4)$ and $\gamma=(0.5,0.5)$.

###### Proof.

Let
$\mathcal{S}_{\delta,\delta^{\prime}}=\\{\hat{H}^{\sigma}_{\delta}:\sigma\in\Sigma^{1}\\}\cup\\{\hat{H}^{\sigma}_{\delta^{\prime}}:\sigma\in\Sigma^{-1}\\}$
be the collection of four spaces such that
$H_{A,-\Gamma}=\bigoplus\mathcal{S}_{A,-\Gamma}$ and
$H_{\alpha,-\gamma}=\bigoplus\mathcal{S}_{\alpha,-\gamma}$. By definition of
the norm on $H_{A,-\Gamma}$ it is enough to prove that there is a constant
$K>0$ such that

$\|\Pi C_{T}\tilde{\Pi}f\|_{H}\leq K\|f\|_{\tilde{H}},\qquad f\in\tilde{H},$
(9)

for any $H\in\mathcal{S}_{\alpha,-\gamma}$ and
$\tilde{H}\in\mathcal{S}_{A,-\Gamma}$, where $\Pi\colon H_{\alpha,-\gamma}\to
H$ and $\tilde{\Pi}\colon H_{A,-\Gamma}\to\tilde{H}$ are the respective
projection operators. We will denote by $D$, $\tilde{D}$ the domains of
holomorphicity of $H$ and $\tilde{H}$, respectively, and we will write
$D^{\prime}$ for the dual domain of $D$. Using (8), for any $f\in\tilde{H}$ we
have

$\|\Pi C_{T}f\|_{H}=\sup\left\\{\left|\langle
C_{T}f,g\rangle_{\partial^{*}D}\right|\colon g\in
H^{2}(D^{\prime})\cap\mathcal{P},\|g\|_{H^{2}(D^{\prime})}\leq 1\right\\}.$

(Note that this holds for all $H\in\mathcal{S}_{A,-\Gamma}$ without the need
to adapt the function space for $g$.) By the density of Laurent polynomials
$\mathcal{P}$ in $H_{A,-\Gamma}$, see Proposition 3.9$(iv)$, it is enough to
establish

$\left|\langle
C_{T}f,g\rangle_{\partial^{*}D}\right|\leq\tilde{C}\|f\|_{\tilde{H}}\|g\|_{H^{2}(D^{\prime})}$

for some $\tilde{C}>0$, for all $f\in\tilde{H}\cap\mathcal{P}$ and $g\in
H^{2}(D^{\prime})\cap\mathcal{P}$. We shall prove this by breaking the 16
possible different configurations (see Figure 4) into several cases.

Fix $H\in\mathcal{S}_{\alpha,-\gamma}$ and
$\tilde{H}\in\mathcal{S}_{A,-\Gamma}$ (with corresponding domains $D$ and
$\tilde{D})$ and let $g\in H^{2}(D^{\prime})$.

  1. (i)

We first consider the case $D=D^{\sigma}_{P,\alpha}$ with
$\sigma\in\Sigma^{1}$ (so that $\partial^{*}D=\mathbb{T}^{\sigma}_{P,\alpha}$)
and $\tilde{D}=D^{\tilde{\sigma}}_{P,A}$ with $\tilde{\sigma}\in\Sigma^{1}$.
Since $T$ is $(1,\alpha,A,P)$-strongly expanding, Lemma 3.23 yields that there
is a $\hat{\sigma}\in\Sigma^{1}$ such that
$\mathbb{T}^{\hat{\sigma}}_{P,\alpha}\subset\operatorname{cl}({D^{\prime}})$
and $T(\mathbb{T}^{\hat{\sigma}}_{P,\alpha})$ is a compact subset of
$\tilde{D}$.

We obtain

$\displaystyle|\langle C_{T}f,g\rangle_{\mathbb{T}^{\sigma}_{P,\alpha}}|$
$\displaystyle=\left|\int_{\mathbb{T}^{\sigma}_{P,\alpha}}(f\circ
T)g\,dm\right|=\left|\int_{\mathbb{T}^{\hat{\sigma}}_{P,\alpha}}(f\circ
T)g\,dm\right|$
$\displaystyle\leq\left(\int_{\mathbb{T}^{\hat{\sigma}}_{P,\alpha}}|f\circ
T|^{2}\,dm\right)^{1/2}\left(\int_{\mathbb{T}^{\hat{\sigma}}_{P,\alpha}}|g|^{2}\,dm\right)^{1/2},$

where the integral equality (in the case $\hat{\sigma}\neq\sigma$) follows by
Cauchy’s Theorem and the holomorphicity of $T$ on
$\operatorname{cl}({\mathcal{A}_{P,\alpha}})$, and the last step is the
Cauchy-Schwarz inequality.

As $T(\mathbb{T}^{\hat{\sigma}}_{P,\alpha})\subset\tilde{D}$ is compact,
Proposition 3.11 yields a $C_{1}>0$ such that $\sup_{z\in
T(\mathbb{T}^{\tilde{\sigma}}_{P,\alpha})}|f(z)|\leq C_{1}\|f\|_{\tilde{H}}$.
Since $\mathbb{T}^{\hat{\sigma}}_{P,\alpha}\in\operatorname{cl}({D^{\prime}})$
and $g\in H^{2}(D^{\prime})$, we obtain

$|\langle C_{T}f,g\rangle_{\mathbb{T}^{\sigma}_{P,\alpha}}|\leq
C_{1}\|f\|_{\tilde{H}}\|g\|_{H^{2}(D^{\prime})}.$

  2. (ii)

The case $D=D^{\sigma}_{P,\alpha}$ with $\sigma\in\Sigma^{1}$ and
$\tilde{D}=(D^{\tilde{\sigma}}_{P,\Gamma})^{\prime}$ with
$\tilde{\sigma}\in\Sigma^{-1}$ is similar to the previous one. In this case,
it holds that
$T(\mathbb{T}^{2})=\mathbb{T}^{2}{\subset\\!\\!\\!\subset}D^{\prime}\cap\tilde{D}$,
and using holomorphicity of $T$ on
$\operatorname{cl}({\mathcal{A}_{P,\alpha}})$ and the Cauchy-Schwarz
inequality, as before we obtain

$|\langle
C_{T}f,g\rangle_{\mathbb{T}^{\sigma}_{P,\alpha}}|=\left|\int_{\mathbb{T}^{2}}(f\circ
T)g\,dm\right|\leq\left(\int_{\mathbb{T}^{2}}|f\circ
T|^{2}\,dm\right)^{1/2}\left(\int_{\mathbb{T}^{2}}|g|^{2}\,dm\right)^{1/2}.$

By Proposition 3.11, we again have that $|\langle
C_{T}f,g\rangle_{\mathbb{T}^{\sigma}_{P,\alpha}}|\leq
C_{2}\|f\|_{\tilde{H}}\|g\|_{H^{2}(D^{\prime})}$ for some $C_{2}>0$.

  3. (iii)

Next we consider the case
$D=(D^{\sigma}_{P,\gamma})^{\prime},\sigma\in\Sigma^{-1}$ and
$\tilde{D}=D^{\tilde{\sigma}}_{P,A},\tilde{\sigma}\in\Sigma^{1}$. From
assumption $(iii)$ we have that there exists a torus
$\mathbb{T}_{q}^{2}\subset D^{\sigma}_{P,\gamma}\cap\mathcal{A}_{P,\eta}$ such
that $T(\mathbb{T}^{2}_{q})$ is a compact subset of $\tilde{D}$. As in case
(i), we obtain

$\displaystyle|\langle C_{T}f,g\rangle_{\mathbb{T}^{\sigma}_{P,\gamma}}|$
$\displaystyle\leq\left(\int_{\mathbb{T}^{2}_{q}}|f\circ
T|^{2}\,dm\right)^{1/2}\left(\int_{\mathbb{T}^{2}_{q}}|g|^{2}\,dm\right)^{1/2}\leq
C_{3}\|f\|_{\tilde{H}}\|g\|_{H^{2}(D^{\prime})}$

for some $C_{3}>0$.

  4. (iv)

Finally we consider the case
$D=(D^{\sigma}_{P,\gamma})^{\prime},\tilde{D}=(D^{\tilde{\sigma}}_{P,\Gamma})^{\prime}$
with $\sigma,\tilde{\sigma}\in\Sigma^{-1}$. As $T$ and $T^{-1}$ are
holomorphic on a neighbourhood of
$\operatorname{cl}({\mathcal{A}_{P,\gamma}})$, by Cauchy’s Theorem we have

$\displaystyle\langle C_{T}f,g\rangle_{\mathbb{T}^{\sigma}_{P,\gamma}}$
$\displaystyle=\int_{\mathbb{T}^{\sigma}_{P,\gamma}}(f\circ
T)g\,dm=\int_{{\mathbb{T}^{2}}}(f\circ T)g\,dm=\int_{{\mathbb{T}^{2}}}f(g\circ
T^{-1})w\,dm$ $\displaystyle=\int_{\mathbb{T}^{\sigma}_{P,\Gamma}}f(g\circ
T^{-1})w\,dm,$

where $w(z)=\omega_{T}\det(DT^{-1}(z))\cdot z/T^{-1}(z)$ with $\omega_{T}=1$
if $T$ is orientation-preserving and $\omega_{T}=-1$ otherwise. As $T^{-1}$ is
$(-1,\Gamma,\gamma,P)$-strongly expanding, by Lemma 3.23, there is
$\hat{\sigma}\in\Sigma^{-1}$ such that
$\mathbb{T}^{\hat{\sigma}}_{\Gamma}\in\operatorname{cl}({(D^{\tilde{\sigma}}_{P,\Gamma})^{\prime}})=\operatorname{cl}({\tilde{D}})$
and $T^{-1}(\mathbb{T}^{\hat{\sigma}}_{P,\Gamma})$ is a compact set in
$D^{\sigma}_{P,\gamma}=D^{\prime}$. By the same argument as before we obtain a
$C_{4}>0$ such that

$\displaystyle|\langle C_{T}f,g\rangle_{\mathbb{T}^{\sigma}_{P,\gamma}}|$
$\displaystyle\leq\sup_{z\in\mathbb{T}^{\sigma}_{P,\Gamma}}|w(z)|\left(\int_{\mathbb{T}^{\sigma}_{P,\Gamma}}|f|^{2}\,dm\right)^{1/2}\left(\int_{\mathbb{T}^{\sigma}_{P,\Gamma}}|g\circ
T^{-1}|^{2}\,dm\right)^{1/2}$ $\displaystyle\leq
C_{4}\|f\|_{\tilde{H}}\|g\|_{H^{2}(D^{\prime})}.$

Setting $\tilde{C}=\max\\{C_{1},C_{2},C_{3},C_{4}\\}$, we obtain the required
inequality. ∎

######  Corollary 3.25.

Let $T$ be an analytic Anosov diffeomorphism of $\mathbb{T}^{2}$ satisfying
the (sec) condition for some $P\in{\operatorname{GL}_{2}}(\mathbb{R})$. Then
there are $\alpha,A,\gamma,\Gamma\in\mathbb{R}^{2}_{>0}$ with $\alpha<A$ and
$\Gamma<\gamma$ such that the associated composition operator $C_{T}$ is a
well-defined and bounded operator from $H_{P,A,-\Gamma}$ to
$H_{P,\alpha,-\gamma}$.

######  Remark 3.26.

Note that the (sec) condition is sufficient, but not necessary for the above
corollary. For example, take $P=\mathbb{I}$ and $\tilde{T}(x,y)=(x+y,x)$, then
the union of the first and third quadrants of ${\mathbb{R}^{2}}$ is not an
expanding invariant cone, as $D_{x}\tilde{T}(0,1)^{T}=(1,0)^{T}$ for all $x\in
M$. However, it is not difficult to find $\alpha,A,\gamma,\Gamma$ satisfying
the assumptions of Theorem 3.24 (for $P=\mathbb{I}$) for the corresponding map
$T\colon(z_{1},z_{2})\mapsto(z_{1}z_{2},z_{1})$ on ${\mathbb{T}^{2}}$.

Using a standard factorisation argument we can now deduce that the composition
operator from the above theorem is trace-class when considered as an operator
on $H_{A,-\Gamma}$. We defer the proof of the following lemma to the appendix.

######  Lemma 3.27.

For $\alpha,A,\gamma,\Gamma\in\mathbb{R}^{2}_{>0}$ with $\alpha<A$ and
$\Gamma<\gamma$ let $J\colon H_{P,\alpha,-\gamma}\to H_{P,A,-\Gamma}$ be the
canonical embedding operator. For $n\in\mathbb{N}$, denote by $s_{n}(J)$ the
$n$-th singular value of $J$. Then

$\lim_{n\to\infty}\frac{-\log s_{n}(J)}{n^{1/2}}=\eta,$

where
$\eta=\left(\frac{1}{\log(A_{1}-\alpha_{1})\log(A_{2}-\alpha_{2})}+\frac{1}{\log(\gamma_{1}-\Gamma_{1})\log(\gamma_{2}-\Gamma_{2})}\right)^{-1/2}$.

######  Corollary 3.28.

Let $T$ be as in Theorem 3.24. Then the are
$\alpha,\gamma\in\mathbb{R}^{2}_{>0}$ such that the composition operator
$C_{T}$ associated to $T$ is well-defined as an operator from
$H_{P,\alpha,-\gamma}$ to $H_{P,\alpha,-\gamma}$. Moreover, there are
constants $\tilde{c}_{1},\tilde{c}_{2},\hat{c}_{1},\hat{c}_{2}>0$ such that

$s_{n}(C_{T})\leq\tilde{c}_{1}e^{-\tilde{c}_{2}n^{1/2}}\quad(n\in\mathbb{N}),$

and

$|\lambda_{n}(C_{T})|\leq\hat{c}_{1}e^{-\hat{c}_{2}n^{1/2}}\quad(n\in\mathbb{N}),$

where $s_{n}(C_{T})$ and $\lambda_{n}(C_{T})$ are the $n-$th largest (counted
with multiplicity) singular values and eigenvalues of $C_{T}$, respectively.
In particular, $C_{T}:H_{P,\alpha,-\gamma}\to H_{P,\alpha,-\gamma}$ is trace-
class.

###### Proof.

By Theorem 3.24 the composition operator lifts to a continuous operator
$\tilde{C}_{T}\colon H_{P,A,-\Gamma}\to H_{P,\alpha,-\gamma}$. Let $J$ be the
embedding operator from Lemma 3.27, then $C_{T}=\tilde{C}_{T}J$ is a well-
defined trace-class operator from $H_{P,\alpha,-\gamma}$ to
$H_{P,\alpha,-\gamma}$, as $\tilde{C}_{T}$ is bounded and $J$ is trace-class.
By [Pi, 2.2] we have $s_{n}(C_{T})\leq Cs_{n}(J)$ with
$C=\|\tilde{C}_{T}\|_{H_{P,A,-\Gamma}\to H_{P,\alpha,-\gamma}}$, and by Lemma
3.27 we have $s_{n}(J)\leq c_{1}e^{-\eta\cdot n^{1/2}}$ for some $c_{1}>0$.
Using the multiplicative Weyl inequality [Pi, 3.5.1] we obtain
$|\lambda_{n}(C_{T})|\leq\hat{c}_{1}e^{-\hat{c}_{2}n^{1/2}}$ with
$\hat{c}_{1}=Cc_{1},\hat{c}_{2}=2/3\eta$ (see, for example, [BJ2, Lemma
5.11]). ∎

We are now ready to precisely state and prove our first main theorem.

######  Theorem 1.1.

Let $T$ be an analytic Anosov diffeomorphism of $\mathbb{T}^{2}$ satisfying
the (sec) condition for some $P\in{\operatorname{GL}_{2}}(\mathbb{R})$, and
let $w\colon\mathbb{T}^{2}\to{\mathbb{C}}$ be analytic. Then there exist
$\alpha,\gamma\in\mathbb{R}^{2}_{>0}$ such that the weighted composition
operator

$f\mapsto w\cdot f\circ T$

is a well-defined trace-class operator on $H_{P,\alpha,-\gamma}$.

###### Proof.

Corollary 3.28 proves the theorem for $w\equiv 1$, and remains valid if the
operator $C_{T}$ is replaced by $M_{w}C_{T}$, with $M_{w}$ the multiplication
operator with a weight function that is holomorphic on $\mathcal{A}_{P,\eta}$
for some $\eta>\alpha,\gamma$. Since $w$ is analytic on $\mathbb{T}^{2}$, for
$\eta$ sufficiently small we can assume without loss of generality that $w$
holomorphically extends to $\mathcal{A}_{P,\eta}$, proving the general case. ∎

###  3.3 Relation to transfer operator

The following lemma is analogous to Proposition 3.13, replacing the Hilbert
space $H^{\sigma}_{P,\delta}$ by $\mathcal{H}^{1}_{P,\delta}$,
$\mathcal{H}^{-1}_{P,\delta}$ or
$\mathcal{H}^{1}_{P,\alpha}\oplus\mathcal{H}^{-1}_{P,-\gamma}$, with the
respective inner products as in Definition 3.18.

######  Lemma 3.29.

  1. (i)

For $\ell\in\\{1,-1\\}$ and $\delta\in{\mathbb{R}^{2}}$, the dual
$(\mathcal{H}^{\ell}_{P,\delta})^{\prime}=(\bigoplus_{\sigma\in\Sigma^{\ell}}\hat{H}^{\sigma}_{P,\delta})^{\prime}$
is isometrically isomorphic to
$\bigoplus_{\sigma\in\Sigma^{\ell}}\check{H}^{-\sigma}_{P,-\delta}$ via the
isomorphism
$\mathcal{J}^{\ell}_{P,\delta}\colon\bigoplus_{\sigma\in\Sigma^{\ell}}\check{H}^{-\sigma}_{P,-\delta}\to(\mathcal{H}^{\ell}_{P,\delta})^{\prime}$,
$g\mapsto l_{g}$, given by

$l_{g}(f)=\sum_{\sigma\in\Sigma^{\ell}}\langle\hat{\Pi}^{\sigma}f,\check{\Pi}^{-\sigma}g\rangle_{\mathbb{T}^{\sigma}_{P,\delta}}.$

  2. (ii)

For $\alpha,\gamma\in{\mathbb{R}^{2}}$, the dual
$(\mathcal{H}^{1}_{P,\alpha}\oplus\mathcal{H}^{-1}_{P,-\gamma})^{\prime}=(\hat{H}^{--}_{P,\alpha}\oplus\hat{H}^{++}_{P,\alpha}\oplus\hat{H}^{-+}_{P,-\gamma}\oplus\hat{H}^{+-}_{P,-\gamma})^{\prime}$
is isometrically isomorphic to
$\check{H}^{++}_{P,-\alpha}\oplus\check{H}^{--}_{P,-\alpha}\oplus\check{H}^{+-}_{P,\gamma}\oplus\check{H}^{-+}_{P,\gamma}$
via the isomorphism $\mathcal{J}_{P,\alpha,\gamma}\colon g\mapsto l_{g}$,
given by

$l_{g}(f)=\sum_{\sigma\in\Sigma^{1}}\langle\hat{\Pi}^{\sigma}f,\check{\Pi}^{-\sigma}g\rangle_{\mathbb{T}^{\sigma}_{P,\alpha}}+\sum_{\sigma\in\Sigma^{-1}}\langle\hat{\Pi}^{\sigma}f,\check{\Pi}^{-\sigma}g\rangle_{\mathbb{T}^{\sigma}_{P,-\gamma}}.$

######  Remark 3.30.

As before we can identify the Hardy-Hilbert space $H_{P,-\alpha,\gamma}$
associated to a cone-wise exponential weight (similar to Definition 2.5), with
a topological direct sum of Hardy-Hilbert spaces on corresponding Reinhardt
domains, that is,

$H_{P,-\alpha,\gamma}\cong\check{H}^{++}_{P,-\alpha}\oplus\check{H}^{--}_{P,-\alpha}\oplus\check{H}^{+-}_{P,\gamma}\oplus\check{H}^{-+}_{P,\gamma}\\\
$

Note that in particular this implies $(H_{P,\alpha,-\gamma})^{\prime}\cong
H_{P,-\alpha,\gamma}$.

Let $\tilde{T}\colon M\to M$ be a smooth diffeomorphism of $M$, then the
associated Perron-Frobenius operator $\mathcal{L}_{\tilde{T}}$ given by
$g\mapsto|\det{D\tilde{T}^{-1}}|\cdot(g\circ\tilde{T}^{-1})$ is a well-defined
operator on $L^{2}(M)$. The respective operator on $L^{2}({\mathbb{T}^{2}})$
is given by101010Use the relation $\pi\circ\tilde{T}=T\circ\pi$ with
$\pi(x,y)=(\exp(ix),\exp(iy))$. Also observe that
$\det{D\tilde{T}^{-1}}(x,y)=(\det{DT^{-1}}(z))\cdot\frac{z_{1}z_{2}}{(T^{-1}(z))_{1}(T^{-1}(z))_{2}}$
for $z=(e^{ix},e^{iy})$.

$(\mathcal{L}_{T}g)(z)=w(z)\cdot(g\circ T^{-1})(z),$ (10)

with
$w(z)=\omega_{T}\cdot(\det{DT^{-1}}(z))\cdot\frac{p_{1,1}(z)}{p_{1,1}(T^{-1}(z))}$,
where $p_{1,1}(z)=z_{1}\cdot z_{2}$ and $\omega_{T}=1$ if $T$ is orientation-
preserving and $\omega_{T}=-1$ otherwise.

######  Proposition 3.31.

Let $T$ satisfy the assumptions of Theorem 3.24 and let $C_{T}$ be the
respective operator on a suitable space $H_{P,\alpha,-\gamma}$. Then the
isomorphism $\mathcal{J}=\mathcal{J}_{P,\alpha,\gamma}$ from Lemma 3.29
conjugates the adjoint $(C_{T})^{*}$ of $C_{T}$ to the operator
$\mathcal{L}_{T}$ given by (10), which is well defined and bounded as an
operator on $H_{P,-\alpha,\gamma}$.

###### Proof.

For notational simplicity, we assume $P=\mathbb{I}$, the proof for general
$P\in{\operatorname{GL}_{2}}(\mathbb{R})$ being identical. We want to show
that $C_{T}^{*}\mathcal{J}=\mathcal{J}\mathcal{L}_{T}$. By the density of
Laurent polynomials in $H_{P,-\alpha,\gamma}$ and $H_{P,\alpha,-\gamma}$, it
suffices to show this for monomials, i.e.
$(C_{T}^{*}\mathcal{J}(p_{n}))(p_{m})=(\mathcal{J}\mathcal{L}_{T}(p_{n}))(p_{m})$
for all $n,m\in\mathbb{Z}^{2}$. For any $n,m\in\mathbb{Z}^{2}$ we have that

$(C_{T}^{*}\mathcal{J}(p_{n}))(p_{m})=(\mathcal{J}(p_{n}))(C_{T}p_{m})=\sum_{\sigma\in\Sigma}\langle\hat{\Pi}^{\sigma}(C_{T}p_{m}),\check{\Pi}^{-\sigma}p_{n}\rangle_{\mathbb{T}^{\sigma}_{\delta(\sigma)}}$

with $\delta(\sigma)=\alpha$ for $\sigma\in\Sigma^{1}$, and
$\delta(\sigma)=-\gamma$ otherwise. We note that
$\mathbb{Z}^{2}=\bigcup_{\sigma\in\Sigma}\check{\mathbb{Z}}^{\sigma}$ is a
disjoint union, so that for every $n\in\mathbb{Z}^{2}$ there exists exactly
one $\sigma^{\prime}\in\Sigma$ such that
$\check{\Pi}^{-\sigma^{\prime}}p_{n}=p_{n}$ and $\check{\Pi}^{-\sigma}p_{n}=0$
for all $\sigma\neq\sigma^{\prime}$. Moreover we have
$\langle\hat{\Pi}^{\sigma}f,\check{\Pi}^{\sigma^{\prime}}g\rangle_{\mathbb{T}^{\sigma}_{\delta}}=0$
whenever $\sigma^{\prime}\neq-\sigma$. It follows that

$\sum_{\sigma\in\Sigma}\langle\hat{\Pi}^{\sigma}(C_{T}p_{m}),\check{\Pi}^{-\sigma}p_{n}\rangle_{\mathbb{T}^{\sigma}_{\delta(\sigma)}}=\langle\hat{\Pi}^{\sigma^{\prime}}(C_{T}p_{m}),\check{\Pi}^{-\sigma^{\prime}}p_{n}\rangle_{\mathbb{T}^{\sigma^{\prime}}_{\delta(\sigma^{\prime})}}=\langle
C_{T}p_{m},p_{n}\rangle_{\mathbb{T}^{\sigma^{\prime}}_{\delta(\sigma^{\prime})}},$

where the second step follows from
$\sum_{\sigma\in\Sigma}\hat{\Pi}^{\sigma}(C_{T}p_{m})=C_{T}p_{m}$.
Analogously,

$(\mathcal{J}\mathcal{L}_{T}(p_{n}))(p_{m})=\sum_{\sigma\in\Sigma}\langle\hat{\Pi}^{\sigma}p_{m},\check{\Pi}^{-\sigma}(\mathcal{L}_{T}(p_{n}))\rangle_{\mathbb{T}_{\delta(\sigma)}^{\sigma}}=\langle
p_{m},\mathcal{L}_{T}(p_{n})\rangle_{\mathbb{T}_{\delta(\sigma^{\prime\prime})}^{\sigma^{\prime\prime}}},$

with suitable $\sigma^{\prime\prime}$. Finally, we have that

$\displaystyle\langle
C_{T}p_{m},p_{n}\rangle_{\mathbb{T}_{\delta(\sigma^{\prime})}^{\sigma^{\prime}}}=\int_{\mathbb{T}_{\delta(\sigma^{\prime})}^{\sigma^{\prime}}}(p_{m}\circ
T)p_{n}\,dm=\int_{\mathbb{T}^{2}}(p_{m}\circ T)p_{n}\,dm$ $\displaystyle=$
$\displaystyle\int_{\mathbb{T}^{2}}p_{m}(p_{n}\circ
T^{-1})w\,dm=\int_{\mathbb{T}_{\delta(\sigma^{\prime\prime})}^{\sigma^{\prime\prime}}}p_{m}(p_{n}\circ
T^{-1})w\,dm=\langle
p_{m},\mathcal{L}_{T}(p_{n})\rangle_{\mathbb{T}_{\delta(\sigma^{\prime\prime})}^{\sigma^{\prime\prime}}},$

where $w(z)=\omega_{T}\det(DT^{-1}(z))\cdot z/T^{-1}(z)$ and we have used that
the integrands are holomorphic on a neighbourhood of
$\operatorname{cl}({A_{\gamma}})$. Combining the above, we obtain the claim of
the proposition. ∎

######  Remark 3.32.

Using Theorem 1.1 with the weight function being (the complex version of) the
determinant of $DT$ gives rise to a transfer operator corresponding to the map
$T^{-1}$, which is well defined and trace-class on a suitable
$H_{P,\alpha,-\gamma}$. Now, using the previous proposition, it follows that
the operator $f\mapsto f\circ T^{-1}$ is well-defined and trace-class on
$H_{P,-\alpha,\gamma}$.

##  4 Resonances for certain rational Anosov maps

This section is devoted to proving Theorem 1.2, that is, determining the
explicit form of eigenvalues of composition operators associated to analytic
maps with holomorphic extensions to certain domains of
${\hat{\mathbb{C}}^{2}}$. In order to capture all the intricacies of the
resonances in this result, we will use a fundamental result by Rudin and Stout
[Rud, Theorem 5.2.5] characterising inner functions on polydisks. An inner
function on $\mathbb{D}^{n}$ is a function $f\in H^{\infty}(\mathbb{D}^{n})$
whose radial boundary values satisfy $|f^{*}(z)|=1$ almost everywhere111111
Knese [Kn, Cor. 14.6] proved that the radial limit exists and is unimodular at
every point $z\in\mathbb{T}^{n}$. However, in general $f^{*}$ need not be
continuous on $\mathbb{T}^{n}$. on $\mathbb{T}^{n}=\partial\mathbb{D}^{n}$. We
denote by $U(\mathbb{D}^{n})$ the class of all continuous complex functions on
$\operatorname{cl}({\mathbb{D}^{n}})$ whose restriction to $\mathbb{D}^{n}$ is
holomorphic.

######  Theorem 4.1.

Every rational inner function $f$ on $\mathbb{D}^{n}$, $n\in\mathbb{N}$, has
the form

$f(z)=\frac{M(z)\tilde{Q}(1/z)}{Q(z)},$ (11)

where $M$ is a monomial, $Q$ a polynomial with no zeros in $\mathbb{D}^{n}$,
and $\tilde{Q}$ is the polynomial whose coefficients are the complex
conjugates of the coefficients of $Q$. Moreover, every function $f\in
U(\mathbb{D}^{n})$ which is inner is rational, and in this case $Q$ has no
zeros in $\operatorname{cl}({\mathbb{D}^{n}})$.

A direct consequence of this theorem is a characterization of the form that
analytic Anosov diffeomorphisms on $\mathbb{T}^{2}$ from Theorem 1.2 can take,
namely that each component is necessarily a rational function satisfying a
certain set of properties. For for $k,l\in\\{0,1\\}$, we denote by
$I_{kl}\colon{\mathbb{T}^{2}}\to{\mathbb{T}^{2}}$ the map

$I_{kl}(z_{1},z_{2})=(z_{1}^{1-2k},z_{2}^{1-2l}),$

noting that $I_{kl}=\tau_{I^{\sigma}}$ with $\sigma=(1-2k,1-2l)$.

######  Corollary 4.2.

Let $T$ be an analytic diffeomorphism of $\mathbb{T}^{2}$ with holomorphic
extension to a neighbourhood of $\mathbb{T}^{2}$. Assume there exist
$\sigma,\sigma^{\prime}\in\Sigma$ so that $T$ holomorphically extends to
$D^{\sigma}$ with $T(D^{\sigma})\subset D^{\sigma^{\prime}}$. Then each
component of $T$ can be written as a rational function. Moreover, writing
$T=T_{A}$ with $A=(a_{1},\ldots,a_{n})$ the collection of all coefficients
occurring in $T$ (in any order), and denoting
$\overline{A}=(\overline{a_{1}},\ldots,\overline{a_{n}})$, we have the
following properties:

  1. (i)

$I_{11}\circ T_{A}\circ I_{11}=T_{\overline{A}}$,

  2. (ii)

$T_{\overline{A}}(\overline{z})=\overline{T_{A}(z)}$ for any
$z\in{\hat{\mathbb{C}}^{2}}$.

###### Proof.

We define $\phi^{\sigma}\colon\mathbb{D}^{2}\to D^{\sigma}$ by
$\phi^{\sigma}(z)=z^{-\sigma}=(z_{1}^{-\sigma_{1}},z_{2}^{-\sigma_{2}})$, then
each component of $\hat{T}=(\phi^{\sigma^{\prime}})^{-1}\circ
T\circ\phi^{\sigma}\colon\mathbb{D}^{2}\to\mathbb{D}^{2}$ is a rational inner
function in $\mathbb{D}^{2}$ continuous on all of
$\operatorname{cl}({\mathbb{D}^{2}})$. By Theorem 4.1, each component is a
rational function of the form (11), and this property is preserved under
composition with $\phi^{\sigma}$. Furthermore, property $(i)$ holds for maps
whose components are of the form (11). Since $I_{11}$ and $\phi^{\sigma}$
commute for any $\sigma\in\Sigma$, we have that

$I_{11}\circ T_{A}\circ I_{11}=\phi^{\sigma^{\prime}}\circ
I_{11}\circ\hat{T}_{A}\circ
I_{11}\circ(\phi^{\sigma})^{-1}=\phi^{\sigma^{\prime}}\circ\hat{T}_{\overline{A}}\circ(\phi^{\sigma})^{-1}=T_{\overline{A}},$

proving that property $(i)$ is preserved under composition with
$\phi^{\sigma}$. Lastly, $(ii)$ holds for all maps whose components are
rational functions, and hence for the given map $T=T_{A}$. ∎

We shall require the following lemma, a direct consequence of the maximum
modulus principle.

######  Lemma 4.3.

Fix $\sigma,\sigma^{\prime}\in\Sigma$, $a\in\mathbb{R}^{2}_{>0}$, and let
$T\colon D^{\sigma}\to D^{\sigma^{\prime}}$ be holomorphic with
$T(\mathbb{T}^{\sigma}_{a})\subset D^{\sigma^{\prime}}_{a}$. Then
$T(D^{\sigma}_{a}){\subset\\!\\!\\!\subset}D^{\sigma^{\prime}}_{a}$.

###### Proof.

We write $\hat{\sigma}=(-1,-1)$, and begin with the case
$\sigma=\sigma^{\prime}=\hat{\sigma}$ (i.e.
$D^{\sigma}=D^{\sigma^{\prime}}=\mathbb{D}^{2}$), whose proof is a direct
application of the (multivariate) maximum modulus principle. By compactness,
there exists
$z^{*}=(z_{1}^{*},z_{2}^{*})\in\operatorname{cl}({D^{\sigma}_{a}})$ such that
$|T_{1}(z^{*})|=\max_{z\in\operatorname{cl}({D^{\sigma}_{a}})}|T_{1}(z)|$.
Defining $f\colon\mathbb{D}\to\mathbb{D}$ by
$f(z_{1})=T_{1}(z_{1},z^{*}_{2})$, the maximum modulus principle implies that

$|T_{1}(z^{*})|=\max_{|z_{1}|\leq
e^{-a_{1}}}|f(z_{1})|=\max_{|z_{1}|=e^{-a_{1}}}|f(z_{1})|,$

and hence without loss of generality we can assume $|z_{1}^{*}|=e^{-a_{1}}$.
Analogously, $z_{2}^{*}$ can be assumed to have modulus $e^{-a_{2}}$. It
follows that
$\max_{z\in\operatorname{cl}({D^{\sigma}_{a}})}|T_{1}(z)|=\max_{z\in\mathbb{T}^{\sigma}_{a}}|T_{1}(z)|<e^{-a_{1}}$
(using $T(\mathbb{T}^{\sigma}_{a})\subset D^{\sigma}_{a}$). Repeating the
argument for $T_{2}$ yields
$\max_{z\in\operatorname{cl}({D^{\sigma}_{a}})}|T_{2}(z)|<e^{-a_{2}}$, and
hence $T(D^{\sigma}_{a}){\subset\\!\\!\\!\subset}D^{\sigma^{\prime}}_{a}$.

For general $\sigma,\sigma^{\prime}\in\Sigma$, let
$\phi^{\sigma}\colon\mathbb{D}^{2}\to D^{\sigma}$ be given by
$\phi^{\sigma}(z)=z^{-\sigma}=(z_{1}^{-\sigma_{1}},z_{2}^{-\sigma_{2}})$, so
that $\phi(D^{\hat{\sigma}}_{a})=D^{\sigma}_{a}$ and
$\phi(\mathbb{T}^{\hat{\sigma}}_{a})=\mathbb{T}^{\sigma}_{a}$. Then
$\tilde{T}=(\phi^{\sigma^{\prime}})^{-1}\circ
T\circ\phi^{\sigma}\colon\mathbb{D}^{2}\to\mathbb{D}^{2}$ is holomorphic and
satisfies $\tilde{T}(\mathbb{T}^{\hat{\sigma}}_{a})\subset
D^{\hat{\sigma}}_{a}$. It follows that
$\tilde{T}(D^{\hat{\sigma}}_{a}){\subset\\!\\!\\!\subset}D^{\hat{\sigma}}_{a}$,
and the assertion
$T(D^{\sigma}_{a}){\subset\\!\\!\\!\subset}D^{\sigma^{\prime}}_{a}$ follows. ∎

The explicit form of the spectral determinant in Theorem 1.2 will follow by
computing traces of certain trace-class operators. If $L\colon H\to H$ is a
trace-class operator on a separable Hilbert space $(H,(\cdot,\cdot))$ and
$\\{e_{n}\\}_{n\in\mathcal{I}}$ is an orthonormal basis of $H$ with some index
set $\mathcal{I}$, then the trace of $L$ is

$\operatorname{Tr}(L)=\sum_{n\in\mathcal{I}}(Le_{n},e_{n}),$ (12)

and its determinant is given by

$\det{(\operatorname{Id}-zL)}=\exp\left(-\sum_{k=1}^{\infty}\frac{z^{k}}{k}\operatorname{Tr}(L^{k})\right),$
(13)

for all $z\in{\mathbb{C}}$ in a suitable neighbourhood of $0$. Moreover, both
$\operatorname{Tr}$ and $\det$ are spectral, that is,
$\operatorname{Tr}(L)=\sum_{k=1}^{\infty}\lambda_{k}(L)$, and counting
multiplicities, the zeros of the entire function
$z\mapsto\det{(\operatorname{Id}-zL)}$ are precisely the reciprocals of the
eigenvalues $\lambda_{k}(L)$ of $L$ (see [Pi, 4.62, 4.7.14, 4.7.15]).

######  Notation 4.4.

The multipliers at a fixed point $z^{*}\in{\hat{\mathbb{C}}^{2}}$ of a
rational map $T\colon{\hat{\mathbb{C}}^{2}}\to{\hat{\mathbb{C}}^{2}}$ are
given by the eigenvalues of $DT(z^{*})$ if $z^{*}\in{\mathbb{C}^{2}}$, and by
the eigenvalues of $D\hat{T}(I_{kl}(z^{*}))$ with $(k,l)=(1,1),(1,0)$ or
$(0,1)$ for $z^{*}=(\infty,\infty),(\infty,w)$ or $(w,\infty)$ respectively,
where $w\in{\mathbb{C}}$ and $\hat{T}=I_{kl}\circ T\circ I_{kl}$.

######  Lemma 4.5.

For $\sigma\in\Sigma$ and $\delta\in{\mathbb{R}^{2}}$ let $\varphi\colon
D^{\sigma}_{\delta}\to D^{\sigma}_{\delta}$ be a holomorphic map such that
$\varphi(D^{\sigma}_{\delta}){\subset\\!\\!\\!\subset}D^{\sigma}_{\delta}$.
Let $C_{\varphi}$ denote the corresponding composition operator and $M_{w}$
the multiplication operator with $w$ a holomorphic function on a neighbourhood
of $\operatorname{cl}({D^{\sigma}_{\delta}})$. Then $M_{w}C_{\varphi}$ is
trace-class on $H^{\sigma}_{\delta}$ and

$\operatorname{Tr}((M_{w}C_{\varphi})^{k})=\frac{w(z^{*})^{k}}{\det(\mathbb{I}-D\varphi^{k}(z^{*}))}=\frac{w(z^{*})^{k}}{(1-\mu^{k}_{1})(1-\mu^{k}_{2})},$

with $\mu_{1},\mu_{2}$ the (not necessarily distinct) multipliers at the
unique attracting fixed point $z^{*}\in D^{\sigma}_{\delta}$ of $\varphi$.

###### Proof.

Let $\hat{\sigma}=(-1,-1)$ and consider the case $\sigma=\hat{\sigma}$. As
$H^{\hat{\sigma}}_{\delta}$ is a ‘favourable Hilbert space’ (see [BJ1,
Definition 2.7]), the result follows by [BJ1, Proposition 2.10 and Theorem
4.2].

For general $\sigma\in\Sigma$, $D^{\sigma}_{\delta}$ is biholomorphically
equivalent to $D^{\hat{\sigma}}_{\delta}$ under the map $\phi^{\sigma}\colon
D^{\hat{\sigma}}_{\delta}\to D^{\sigma}_{\delta}$ given by
$\phi^{\sigma}(z)=z^{-\sigma}$, and
$\hat{\varphi}=(\phi^{\sigma})^{-1}\circ\varphi\circ\phi^{\sigma}\colon
D^{\hat{\sigma}}_{\delta}\to D^{\hat{\sigma}}_{\delta}$ satisfies
$\hat{\varphi}(D^{\hat{\sigma}}_{\delta}){\subset\\!\\!\\!\subset}D^{\hat{\sigma}}_{\delta}$.
Defining $\hat{w}=w\circ\phi^{\sigma}$ on a neighbourhood of
$\operatorname{cl}({D^{\hat{\sigma}}_{\delta}})$, the first case implies the
statement of the lemma for the operator $M_{\hat{w}}C_{\hat{\varphi}}$ on
$H^{\hat{\sigma}}_{\delta}$. By Proposition 3.10 the operator
$C_{\phi^{\sigma}}\colon H^{\sigma}_{\delta}\to H^{\hat{\sigma}}_{\delta}$ is
an isometric isomorphism, which conjugates $M_{w}C_{\varphi}$ to
$M_{\hat{w}}C_{\hat{\varphi}}$. The statement for $M_{w}C_{\varphi}$ follows,
as the multipliers of $z^{*}\in D^{\sigma}_{\delta}$ for $\varphi$ coincide
with those of the unique attracting fixed point
$(\phi^{\sigma})^{-1}(z^{*})\in D^{\hat{\sigma}}_{\delta}$ for
$\hat{\varphi}$. ∎

######  Lemma 4.6.

Let $T$ be a smooth diffeomorphism of ${\mathbb{T}^{2}}$, and let
$\omega_{T}=1$ if $T$ is orientation-preserving, and $\omega_{T}=-1$
otherwise. Let $r,s\in\\{0,1\\}$, and write $\hat{T}=I_{rs}\circ T^{-1}\circ
I_{rs}$ and $\hat{w}=\omega_{T}\det D\hat{T}$. Then for any
$n\in\mathbb{Z}^{2}$and $k\in\mathbb{N}$ we have

$(C_{T}^{k}p_{n},p_{n})_{L^{2}({\mathbb{T}^{2}})}=((M_{\hat{w}}C_{\hat{T}})^{k}p_{m},p_{m})_{L^{2}({\mathbb{T}^{2}})}\text{
with }m=((-1)^{1-r}n_{1}-1,(-1)^{1-s}n_{2}-1).$

###### Proof.

All the steps follow by change of variables. For any $n\in\mathbb{Z}^{2}$, we
have

$\displaystyle(C_{T}^{k}p_{n},p_{n})_{L^{2}({\mathbb{T}^{2}})}$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}p_{n}(T^{k}(z))p_{-n}(z)\,\frac{dz_{1}}{z_{1}}\frac{dz_{2}}{z_{2}}$
$\displaystyle=\frac{\omega_{T}^{k}}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}p_{n}(w)p_{-n}(T^{-k}(w))\det{DT^{-k}(w)}\,\frac{dw_{1}}{(T^{-k}(w))_{1}}\frac{dw_{2}}{(T^{-k}(w))_{2}}$
$\displaystyle=\frac{\omega_{T}^{k}}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}p_{n+1}(w)p_{-(n+1)}(T^{-k}(w))\det{DT^{-k}(w)}\,\frac{dw_{1}}{w_{1}}\frac{dw_{2}}{w_{2}},$

using the shorthand $n+1=(n_{1}+1,n_{2}+1)$. Next, observe that

$\det{D(T^{-k}\circ
I_{rs})(z)}=\det{D(I_{rs}\circ\hat{T}^{k})(z)}=(-1)^{r+s}p_{-2r,-2s}(\hat{T}^{k}(z))\cdot\det{D\hat{T}^{k}(z)}.$

Substituting $w=I_{rs}(z)$ and using that $I_{rs}$ is orientation-preserving
iff $r+s$ is even, we obtain

$\displaystyle(C_{T}^{k}p_{n},p_{n})_{L^{2}({\mathbb{T}^{2}})}$
$\displaystyle=\frac{(-1)^{r+s}\omega_{T}^{k}}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}p_{n+1}(I_{rs}(z))p_{-(n+1)}(T^{-k}(I_{rs}(z)))\det{D(T^{-k}\circ
I_{rs})(z)}\,\frac{dz_{1}}{z_{1}^{1-2r}}\frac{dz_{2}}{z_{2}^{1-2s}},$
$\displaystyle=\frac{\omega_{T}^{k}}{(2\pi)^{2}}\int_{\mathbb{T}^{2}}p_{-m}(z)p_{m}(\hat{T}^{k}(z))\det{D\hat{T}^{k}(z)}\,\frac{dz_{1}}{z_{1}}\frac{dz_{2}}{z_{2}}$
$\displaystyle=((\omega_{T}^{k}\det{D\hat{T}^{k}})\cdot
C_{\hat{T}}^{k}p_{m},p_{m})_{L^{2}({\mathbb{T}^{2}})},$

with $m=(-(-1)^{r}n_{1}-1,-(-1)^{s}n_{2}-1)$, as claimed. ∎

######  Remark 4.7.

If $T\colon{\mathbb{T}^{2}}\to{\mathbb{T}^{2}}$ has an analytic extension to a
neighbourhood of $\operatorname{cl}({\mathcal{A}_{\delta}})$ for some
$\delta$, then one can check that
$(C_{T}p_{n},p_{n})_{L^{2}({\mathbb{T}^{2}})}=\langle
C_{T}e_{n},\overline{e_{n}}\rangle_{\mathbb{T}^{\sigma}_{\delta}}=(C_{T}e_{n},e_{n})_{H^{\sigma}_{\delta}}$
for any $n\in{\mathbb{Z}^{2}},\sigma\in\Sigma$.

We recall that
$\mathcal{H}^{1}_{\delta}=\hat{H}^{--}_{\delta}\oplus\hat{H}^{++}_{\delta}$
and
$\mathcal{H}^{-1}_{\delta}=\hat{H}^{-+}_{\delta}\oplus\hat{H}^{+-}_{\delta}$,
and that $\\{e_{n}:n\in\hat{\mathbb{Z}}^{\sigma}\\}$ forms an orthonormal
basis for $\hat{H}^{\sigma}_{\delta}$ for $\sigma\in\Sigma$.

######  Lemma 4.8.

Let $\delta\in\mathbb{R}^{2}$ and
$T\colon{\hat{\mathbb{C}}^{2}}\to{\hat{\mathbb{C}}^{2}}$ be holomorphic on a
neighbourhood of $\operatorname{cl}({\mathcal{A}_{\delta}})$. Assume
additionally that $T$ is holomorphic on $D^{\sigma}_{\delta}$ and
$T(D^{\sigma}_{\delta}){\subset\\!\\!\\!\subset}D^{-\sigma}_{\delta}$ for
every $\sigma\in\Sigma^{1}$. Let $w$ be a holomorphic function on a
neighbourhood of $\operatorname{cl}({\mathcal{A}_{\delta}})$. Then
$M_{w}C_{T}$ is a well-defined and trace-class operator on
$\mathcal{H}^{1}_{\delta}$ with trace
$\operatorname{Tr}(M_{w}C_{T})=\sum_{\sigma\in\Sigma^{1}}\sum_{n\in\hat{}\mathbb{Z}^{\sigma}}\langle
M_{w}C_{T}e_{n},e_{n}\rangle_{\mathbb{T}^{\sigma}_{\delta}}$. Moreover,

  1. (i)

if $w\equiv 1$, then
$\operatorname{Tr}(M_{w}C_{T})=\operatorname{Tr}(C_{T})=1$,

  2. (ii)

if $w=\det DT$, then $\operatorname{Tr}(M_{w}C_{T})=0$.

###### Proof.

We define $\tilde{T}=I_{11}\circ T$, so that $\tilde{T}$ is holomorphic on a
neighbourhood of $\operatorname{cl}({\mathcal{A}_{\delta}})$, and moreover,
for every $\sigma\in\Sigma^{1}$ it is holomorphic on $D^{\sigma}_{\delta}$
with
$\tilde{T}(D^{\sigma}_{\delta}){\subset\\!\\!\\!\subset}D^{\sigma}_{\delta}$.
By Lemma 4.5 the composition operator $C_{\tilde{T}}$ is trace-class on
$\hat{H}^{\sigma}_{\delta}$, and hence $C_{T}$ is well-defined and trace-class
viewed as an operator on $\mathcal{H}^{1}_{\delta}$. Since $M_{w}$ is well-
defined and bounded as an operator on $\mathcal{H}^{1}_{\delta}$, it follows
that $M_{w}C_{T}$ is also trace-class. Since
$\overline{e}_{n}(z)=p_{-n}(z)\nu(n)$ for $z\in\mathbb{T}^{\sigma}_{\delta}$
and $n\in{\mathbb{Z}^{2}}$, we have that

$(M_{w}C_{T}e_{n},e_{n})_{\hat{H}^{\sigma}_{\delta}}=\langle
M_{w}C_{T}e_{n},\overline{e}_{n}\rangle_{\mathbb{T}^{\sigma}_{\delta}}=\langle
w\cdot p_{n}\circ T,p_{-n}\rangle_{\mathbb{T}^{\sigma}_{\delta}}=\langle
w\cdot p_{-n}\circ\tilde{T},p_{-n}\rangle_{\mathbb{T}^{\sigma}_{\delta}}.$

Moreover, $p_{-n}\circ\tilde{T}\in H^{\sigma}_{\delta}$ for any
$\sigma\in\Sigma^{1}$ and $-n\in\hat{\mathbb{Z}}^{\sigma}$, that is,
$p_{-n}\circ\tilde{T}=\sum_{m\in\mathbb{Z}^{\sigma}}c_{m}p_{m}$. For $w\equiv
1$, recalling that $\langle
p_{k},p_{l}\rangle_{\mathbb{T}^{\sigma}_{\delta}}=0$ whenever $k\neq-l$, and
noting
$(-\hat{\mathbb{Z}}^{\sigma})\cap\mathbb{Z}^{\sigma}\subset\\{(0,0)\\}$, it
follows that

$(M_{w}C_{T}e_{n},e_{n})_{\hat{H}^{\sigma}_{\delta}}=\delta_{n,0}\cdot\langle
p_{0}\circ\tilde{T},p_{0}\rangle_{\mathbb{T}^{\sigma}_{\delta}}=\delta_{n,0},$

which yields $\operatorname{Tr}(C_{T})=1$.

For $w=w_{T}=\det DT$, it is easy to see that
$w=\sum_{m\in\mathbb{Z}^{2}_{<0}}w_{m}p_{m}$ for suitable
$w_{m}\in{\mathbb{C}}$. We consider first the case
$n\in\hat{\mathbb{Z}}^{--}$. In this case we have $p_{n}\circ
T=\sum_{m\in\mathbb{Z}^{2}_{\leq 0}}c_{m}p_{m}$, and hence $w\cdot p_{n}\circ
T=\sum_{m\in\mathbb{Z}^{2}_{<0}}d_{m}p_{m}$, for suitable coefficients
$c_{m},d_{m}\in{\mathbb{C}}$. It follows that $\langle w_{T}\cdot p_{n}\circ
T,p_{-n}\rangle_{\mathbb{T}^{\sigma}_{\delta}}=0$. For
$n\in\hat{\mathbb{Z}}^{++}$, let $\tilde{T}=I_{11}\circ T\circ I_{11}$. Direct
calculation using a change of variables $y=I_{11}(z)$ yields

$\langle w_{T}\cdot p_{n}\circ
T,p_{-n}\rangle_{\mathbb{T}^{\sigma}_{\delta}}=\langle w_{\tilde{T}}\cdot
p_{-(n+2)}\circ\tilde{T},p_{n+2}\rangle_{\mathbb{T}^{\sigma}_{\delta}}.$

Noting that $-(n+2)\in\hat{Z}^{--}$ and that $\tilde{T}$ and $w_{\tilde{T}}$
also satisfy the assumptions of the lemma, we can apply the previous case and
obtain that again $\langle w_{T}\cdot p_{n}\circ
T,p_{-n}\rangle_{\mathbb{T}^{\sigma}_{\delta}}=0$. The conclusion
$\operatorname{Tr}(M_{w}C_{T})=0$ follows. ∎

######  Lemma 4.9.

Let $\ell\in\\{\pm 1\\}$, $\delta\in\mathbb{R}^{2}$, and let $T$ be an
analytic diffeomorphism of $\mathbb{T}^{2}$, holomorphic on a neighbourhood of
$\operatorname{cl}({\mathcal{A}_{\delta}})$. Assume additionally that $T$
extends holomorphically to $D^{\sigma}_{\delta}$ with
$T(D^{\sigma}_{\delta}){\subset\\!\\!\\!\subset}D^{-\sigma}_{\delta}$ for
every $\sigma\in\Sigma^{\ell}$. Then for any $\sigma\in\Sigma^{\ell}$, $T\circ
T$ has a unique fixed point $z^{\sigma}\in D^{\sigma}_{\delta}$, and

$D(T\circ T)(z^{\sigma})=\overline{D\hat{T}(z^{\sigma})}D\hat{T}(z^{\sigma}),$

where $\hat{T}=I_{11}\circ T$.

###### Proof.

We first note that $(T\circ
T)(D^{\sigma}_{\delta}){\subset\\!\\!\\!\subset}D^{\sigma}_{\delta}$ for
$\sigma\in\Sigma^{\ell}$, implying the existence of a unique fixed point
$z^{\sigma}\in D^{\sigma}_{\delta}$ for $\sigma\in\Sigma^{\ell}$. The map
$\hat{T}=I_{11}\circ T$ also satisfies
$\hat{T}(D^{\sigma}_{\delta}){\subset\\!\\!\\!\subset}D^{\sigma}_{\delta}$ for
$\sigma\in\Sigma^{\ell}$, so by Corollary 4.2 each component of $\hat{T}$ is a
rational function, and we write $\hat{T}=\hat{T}_{A}$ with
$A=(a_{1},\ldots,a_{n})$ the collection of all coefficients occurring in $T$
(in any order). Using Corollary 4.2$(i)$ we obtain

$T\circ T=I_{11}\circ\hat{T}_{A}\circ
I_{11}\circ\hat{T}_{A}=\hat{T}_{\overline{A}}\circ\hat{T}_{A}.$

Next, observe that on the one hand, we have

$(\hat{T}_{A}\circ\hat{T}_{\overline{A}})(\hat{T}_{A}(z^{\sigma}))=\hat{T}_{A}(\hat{T}_{\overline{A}}\circ\hat{T}_{A}(z^{\sigma}))=\hat{T}_{A}(z^{\sigma}),$

and on the other hand, using Corollary 4.2$(ii)$, we have

$(\hat{T}_{A}\circ\hat{T}_{\overline{A}})\left(\overline{z^{\sigma}}\right)=\hat{T}_{A}\left(\overline{\hat{T}_{A}(z^{\sigma})}\right)=\overline{(\hat{T}_{\overline{A}}\circ\hat{T}_{A})(z^{\sigma})}=\overline{z^{\sigma}},$

so that $\hat{T}_{A}(z^{\sigma})=\overline{z^{\sigma}}$. It follows that

$D(T\circ
T)(z^{\sigma})=D(\hat{T}_{\overline{A}}\circ\hat{T}_{A})(z^{\sigma})=D\hat{T}_{\overline{A}}\left(\overline{z^{\sigma}}\right)D\hat{T}_{A}(z^{\sigma})=\overline{D\hat{T}_{A}(z^{\sigma})}D\hat{T}_{A}(z^{\sigma}).\qed.$

######  Remark 4.10.

The above lemma implies that the two multipliers of a fixed point $z^{\sigma}$
of $T\circ T$ are either real or complex conjugates of each other, as

$\det(D(T\circ
T)(z^{\sigma}))=\overline{\det(D\hat{T}_{A}(z^{\sigma}))}\cdot\det(D\hat{T}_{A}(z^{\sigma}))=|\det(D\hat{T}_{A}(z^{\sigma}))|^{2}$

and $\operatorname{Tr}(D(T\circ
T)(z^{\sigma}))=\operatorname{Tr}\left(\overline{D\hat{T}_{A}(z^{\sigma})}D\hat{T}_{A}(z^{\sigma})\right)\in\mathbb{R}$.
In contrast to the one-dimensional setting of anti-Blaschke products [BN],
examples of orientation-reversing circle maps allowing for explicit
determination of resonances, the multipliers are no longer necessarily real.
Note also that under the assumptions of the lemma for $\ell=1$ or $\ell=-1$
the two attracting fixed points of $T\circ T$ in $\mathcal{D}^{\ell}_{\delta}$
($z^{\sigma}$, $\sigma\in\Sigma^{\ell}$) have identical sets of multipliers.

We are now ready to prove our second main theorem.

###### Proof of Theorem 1.2.

By Theorem 1.1, $C_{T}$ is trace-class on $H_{\alpha,-\gamma}$ for suitable
$\alpha,\gamma\in\mathbb{R}_{>0}^{2}$, and its trace is given by
$\operatorname{Tr}C_{T}=\sum_{n\in{\mathbb{Z}^{2}}}\langle
C_{T}e_{n},e_{n}\rangle_{\nu_{\alpha,-\gamma}}$. Using the isometric
isomorphism between $H_{\alpha,-\gamma}$ and
$\mathcal{H}^{1}_{\alpha}\oplus\mathcal{H}^{-1}_{-\gamma}$ and the fact that
$C^{k}_{T}=C_{T^{k}}$, for every $k\in\mathbb{N}$ we have

$\operatorname{Tr}(C_{T}^{k})=\sum_{\sigma\in\Sigma^{1}}\sum_{n\in\hat{}\mathbb{Z}^{\sigma}}(C_{T}^{k}e_{n},e_{n})_{\hat{H}^{\sigma}_{\alpha}}+\sum_{\sigma\in\Sigma^{-1}}\sum_{n\in\hat{}\mathbb{Z}^{-\sigma}}(C_{T}^{k}e_{n},e_{n})_{\hat{H}^{-\sigma}_{-\gamma}}=:S_{1}(k)+S_{-1}(k),$
(14)

as well as

$\log\det(\operatorname{Id}-zC_{T})=-\sum_{k=1}^{\infty}\frac{z^{k}}{k}S_{1}(k)-\sum_{k=1}^{\infty}\frac{z^{k}}{k}S_{-1}(k).$
(15)

We note that the assumptions and Lemma 4.3 imply that for every $\ell\in\\{\pm
1\\}$ and $\sigma\in\Sigma^{\ell}$, $T^{\ell}$ is holomorphic in a
neighbourhood of $\operatorname{cl}({D^{\sigma}_{\delta}})$ and
$T^{\ell}(D^{\sigma}_{\delta}){\subset\\!\\!\\!\subset}D^{\pm\sigma}_{\delta}$,
where $\delta=\alpha$ for $\ell=1$ and $\delta=-\gamma$ for $\ell=-1$. We will
calculate (15) by handling the two sums $S_{\ell},\ell\in\\{\pm 1\\}$,
separately, considering for each the two possible cases
$T^{\ell}(D^{\sigma}_{\delta})\subseteq D^{\sigma}_{\delta}$ and
$T^{\ell}(D^{\sigma}_{\delta})\subseteq D^{-\sigma}_{\delta}$ for all
$\sigma\in\Sigma^{\ell}$. The claim of the theorem will follow with
$(1-z)\chi_{T}^{1}(z)=\exp(-\sum_{k=1}^{\infty}\frac{z^{k}}{k}S_{1}(k))$ and
$\chi_{T}^{-1}(z)=\exp(-\sum_{k=1}^{\infty}\frac{z^{k}}{k}S_{-1}(k))$. We
first calculate $S_{1}(k)$:

  1. (1)

Consider first the case
$T(D^{\sigma}_{\alpha}){\subset\\!\\!\\!\subset}D^{\sigma}_{\alpha}$ for
$\sigma\in\Sigma^{1}$. For $\sigma=(-1,-1)\in\Sigma^{1}$, the composition
operator $\tilde{C}_{T}$ associated to $T$ on
$H^{\sigma}_{\alpha}=\hat{H}^{\sigma}_{\alpha}$ is trace-class, and its trace,
computed using (12), coincides with the term in (14) corresponding to $\sigma$
(note that $\tilde{C}^{k}_{T}=\tilde{C}_{T^{k}}$). By Lemma 4.5, we obtain the
value $((1-\lambda_{1}^{k})(1-\lambda_{2}^{k}))^{-1}$, where
$\lambda_{1},\lambda_{2}$ are the multipliers of the unique fixed point
$z^{\sigma}\in D^{\sigma}_{\alpha}$. Similarly, for
$\sigma=(+1,+1)\in\Sigma^{1}$, the associated composition operator
$\tilde{C}_{T}$ is trace-class on $H^{\sigma}_{\alpha}$. Writing $T=T_{A}$ for
some $A\in{\mathbb{C}}^{m}$, $m\in\mathbb{N}_{0}$, by Corollary 4.2$(i)$ we
have $T_{A}=I_{11}\circ T_{\overline{A}}\circ I_{11}$, where
$T_{\overline{A}}$ is a holomorphic map on $D^{-\sigma}_{\alpha}$ with unique
attracting fixed point $\overline{z^{\sigma}}$ (see Corollary 4.2$(ii)$).
Moreover, it follows that
$DT_{\overline{A}}(\overline{z^{\sigma}})=\overline{DT_{A}(z^{\sigma})}$, thus
by Lemma 4.5 we have
$\operatorname{Tr}\tilde{C}^{k}_{T}=((1-\overline{\lambda_{1}}^{k})(1-\overline{\lambda_{2}}^{k}))^{-1}$.
Since

$\operatorname{Tr}\tilde{C}^{k}_{T}={(\tilde{C}^{k}_{T}e_{0},e_{0})_{H^{\sigma}_{\alpha}}+\sum_{n\in\hat{\mathbb{Z}}^{++}}}(\tilde{C}^{k}_{T}e_{n},e_{n})_{H^{\sigma}_{\alpha}}=1+\sum_{n\in\hat{\mathbb{Z}}^{++}}(C^{k}_{T}e_{n},e_{n})_{\hat{H}^{\sigma}_{\alpha}},$

we obtain

$S_{1}(k)=1+D(\lambda_{1}^{k},\lambda_{2}^{k})+D(\overline{\lambda_{1}}^{k},\overline{\lambda_{2}}^{k}),$

where
$D(a,b):=\frac{1}{(1-a)(1-b)}-1=\sum_{n\in\mathcal{N}^{1}}a^{n_{1}}b^{n_{2}}$
for $a,b\in\mathbb{D}$,
$\mathcal{N}^{1}=\mathbb{N}^{2}_{0}\setminus\\{(0,0)\\}$. Calculating

$-\sum_{k=1}^{\infty}\frac{z^{k}}{k}D(a^{k},b^{k})=-\sum_{n\in\mathcal{N}^{1}}\sum_{k=1}^{\infty}\frac{z^{k}a^{kn_{1}}b^{kn_{2}}}{k}=\sum_{n\in\mathcal{N}^{1}}\log(1-za^{n_{1}}b^{n_{2}}),$

we finally obtain

$-\sum_{k=1}^{\infty}\frac{z^{k}}{k}S_{1}(k)=\log(1-z)+\sum_{\sigma\in\Sigma^{1}}\sum_{n\in\mathcal{N}^{1}}\log(1-z\lambda^{n}_{\sigma}).$

  2. (2)

Next we consider the case
$T(D^{\sigma}_{\alpha}){\subset\\!\\!\\!\subset}D^{-\sigma}_{\alpha}$ for
$\sigma\in\Sigma^{1}$. For $k\in\mathbb{N}$ odd, $T^{k}$ satisfies the
assumptions of Lemma 4.8, and the trace of the composition operator associated
to $T^{k}$ on $\mathcal{H}^{1}_{\alpha}$ exactly corresponds to the first sum
in (14), yielding $S_{1}(k)=1$. For $k$ even, $T^{k}$ satisfies
$T^{k}(D^{\sigma}_{\alpha}){\subset\\!\\!\\!\subset}D^{\sigma}_{\alpha}$ for
$\sigma\in\Sigma^{1}$, and so we can apply case $(1)$. Moreover, by Remark
4.10 in this case the fixed point multipliers $\lambda_{1},\lambda_{2}$ of
$T^{2}$ are either real or complex conjugates of each other, and hence

$S_{1}(k)=\begin{cases}1\qquad\qquad\qquad\qquad\;\;\text{for $k$ odd},\\\
1+2D(\lambda_{1}^{k/2},\lambda_{2}^{k/2})\quad\text{for $k$ even}.\end{cases}$

A straightforward calculation using the fact that
$\lambda_{\sigma}=\lambda_{-\sigma}$ now yields

$-\sum_{k=1}^{\infty}\frac{z^{k}}{k}S_{1}(k)=\log(1-z)+\frac{1}{2}\sum_{\sigma\in\Sigma^{1}}\sum_{n\in\mathcal{N}^{1}}\log(1-z^{2}\lambda^{n}_{\sigma}).$

Next, we proceed to calculate $S_{-1}(k)$. The approach to calculating
$S_{1}(k)$ does not immediately translate to this case, as the bidisks
$D^{\sigma}_{\delta},\sigma\in\Sigma^{-1},$ are not invariant under the map
$T$, and so do not directly give rise to trace-class composition operators on
the respective spaces $H^{\sigma}_{\delta}$. Instead, we will show that the
sums in $S_{-1}(k)$ correspond to the traces of certain weighted composition
operators $M_{\hat{w}}C_{\hat{T}}$ on $H^{--}_{\delta}$, where $\hat{T}$ will
be a map conjugated to $T^{-1}$ from Lemma 4.6. Combining Remark 4.7 with
Lemma 4.6, we calculate for $\sigma=(-1,+1)$ that

$\sum_{n\in\hat{\mathbb{Z}}^{-\sigma}}(C^{k}_{T}e_{n},e_{n})_{\hat{H}^{-\sigma}_{-\gamma}}=\sum_{n\in\hat{\mathbb{Z}}^{-\sigma}}(C^{k}_{T}p_{n},p_{n})_{L^{2}(\mathbb{T}^{2})}=\sum_{n\in\hat{\mathbb{Z}}^{--}}((M_{\hat{w}_{\sigma}}C_{\hat{T}_{\sigma}})^{k}p_{n},p_{n})_{L^{2}(\mathbb{T}^{2})},$

with $\hat{w}_{\sigma}=\omega_{T}\det D\hat{T}_{\sigma}$ and
$\hat{T}_{\sigma}=I_{01}\circ T^{-1}\circ I_{01}$. For $\sigma=(+1,-1)$, Lemma
4.6 yields the exact same equality with $\hat{T}_{\sigma}=I_{10}\circ
T^{-1}\circ I_{10}$. We now consider two cases again.

  1. (1’)