If $T^{-1}(D^{\sigma}_{\gamma}){\subset\\!\\!\\!\subset}D^{\sigma}_{\gamma}$
for $\sigma\in\Sigma^{-1}$, then
$\hat{T}_{\sigma}(D^{--}_{\gamma}){\subset\\!\\!\\!\subset}D^{--}_{\gamma}$.
We can then apply the same argument as in the above case (1), using that
$\hat{w}_{\sigma}(\zeta^{*})=\omega_{T}\mu_{1}\mu_{2}$ for $\mu_{1},\mu_{2}$
the multipliers of the unique fixed point $\zeta^{*}\in D^{--}_{\gamma}$ of
$\hat{T}_{\sigma}$, which corresponds to the unique fixed point $z^{\sigma}\in
D^{\sigma}_{\gamma}$ of $T^{-1}$. By the same argument as before, the
multipliers of the respective fixed points in $D^{\sigma}_{\gamma}$ and
$D^{-\sigma}_{\gamma}$ are complex conjugates of each other, and we obtain

$S_{-1}(k)=\frac{(\omega_{T}\mu_{1}\mu_{2})^{k}}{(1-\mu_{1}^{k})(1-\mu_{2}^{k})}+\frac{(\omega_{T}\overline{\mu_{1}}\overline{\mu_{2}})^{k}}{(1-\overline{\mu_{1}}^{k})(1-\overline{\mu_{2}}^{k})}=(\omega_{T})^{k}\sum_{n\in\mathcal{N}^{-1}}((\mu_{1}^{n_{1}}\mu_{2}^{n_{1}})^{k}+(\overline{\mu_{1}}^{n_{1}}\overline{\mu_{2}}^{n_{2}})^{k}),$

where $\mathcal{N}^{-1}=\mathbb{N}^{2}$. A similar calculation to above yields

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

  2. (2’)

Finally, we consider the case
$T^{-1}(D^{\sigma}_{\gamma}){\subset\\!\\!\\!\subset}D^{-\sigma}_{\gamma}$ for
$\sigma\in\Sigma^{-1}$, which implies
$\hat{T}_{\sigma}(D^{\sigma}_{\gamma}){\subset\\!\\!\\!\subset}D^{-\sigma}_{\gamma}$
for $\sigma\in\Sigma^{1}$. If $k\in\mathbb{N}$ is odd, we can apply Lemma 4.8
to $\hat{T}_{\sigma}^{k}$ and the weight function $\hat{w}_{\sigma}$. The
trace of $(M_{\hat{w}_{\sigma}}C_{\hat{T}_{\sigma}})^{k}$ on
$\mathcal{H}^{1}_{\gamma}$ in the lemma exactly coincides with $S_{-1}(k)$,
yielding $S_{-1}(k)=0$. For $k$ even, we can apply case (1’) to $T^{-2}$
instead of $T^{-1}$, again using the fact that the fixed point multipliers
$\mu_{1},\mu_{2}$ are either both real or complex conjugates of each other,
which yields

$S_{-1}(k)=\begin{cases}0\quad\qquad\qquad\qquad\;\text{for $k$ odd},\\\
\frac{2(\mu_{1}\mu_{2})^{k/2}}{(1-\mu^{k/2}_{1})(1-\mu^{k/2}_{2})}\quad\text{for
$k$ even},\end{cases}$

and again using $\lambda_{\sigma}=\lambda_{-\sigma}$ we obtain

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

Claims $(i)$ and $(ii)$ of the theorem now follow by combining the cases
$(1)$-$(2)$ and $(1^{\prime})$-$(2^{\prime})$ for $\ell=1$ and $\ell=-1$,
respectively. ∎

Using the explicit form of the zeros of $\det(\operatorname{Id}-zC_{T})$
obtained in Theorem 1.2, we can calculate the decay rate of their reciprocals,
the eigenvalues of $C_{T}$. For a map $T$ satisfying the assumptions of
Theorem 1.2, for any $\ell\in\\{\pm 1\\}$ and $\sigma\in\Sigma^{\ell}$, we
denote by $\lambda_{\sigma}=(\lambda_{\sigma,1},\lambda_{\sigma,2})$ the
multipliers of the unique attracting fixed point in $D^{\sigma}$ of $T^{\ell}$
if $T^{\ell}(D^{\sigma})\subseteq D^{\sigma}$, and of $T^{2\ell}$ otherwise.

######  Corollary 4.11.

Let $T$ satisfy the assumptions of Theorem 1.2, and let
$\lambda=\lambda_{(-1,-1)}$ and $\mu=\lambda_{(-1,+1)}$. Let $\omega_{T}=1$ if
$T$ is orientation-preserving121212$T$ is orientation-preserving exactly if
either both, or neither of $T$ and $T^{-1}$ satisfy the case $(i)$ in Theorem
1.2., and $\omega_{T}=-1$ otherwise. Then the nonzero eigenvalues of $C_{T}$
on $H_{\alpha,-\gamma}$ are $\\{1\\}\cup\mathcal{E}_{1}\cup\mathcal{E}_{-1}$,
where

$\displaystyle\mathcal{E}_{1}$
$\displaystyle=\begin{cases}\\{\lambda^{n},\overline{\lambda}^{n}:n\in\mathcal{N}^{1}\\},\qquad\qquad\;\;&\text{
if }T(D^{\sigma})\subseteq D^{\sigma},\sigma\in\Sigma^{1},\\\
\\{\pm\lambda^{n/2}:n\in\mathcal{N}^{1}\\},&\text{ if }T(D^{\sigma})\subseteq
D^{-\sigma},\sigma\in\Sigma^{1},\end{cases}$ $\displaystyle\mathcal{E}_{-1}$
$\displaystyle=\begin{cases}\\{\omega_{T}\cdot\mu^{n},\omega_{T}\cdot\overline{\mu}^{n}:n\in\mathcal{N}^{-1}\\},&\text{
if }T^{-1}(D^{\sigma})\subseteq D^{\sigma},\sigma\in\Sigma^{-1},\\\
\\{\pm\mu^{n/2}:n\in\mathcal{N}^{-1}\\},&\text{ if
}T^{-1}(D^{\sigma})\subseteq D^{-\sigma},\sigma\in\Sigma^{-1}.\end{cases}$

Moreover, the algebraic multiplicity of each nonzero eigenvalue is exactly the
number of its occurrences in the above sets.

######  Corollary 4.12.

Let $T$ satisfy the assumptions of Theorem 1.2,
$(\lambda_{n})_{n\in\mathbb{N}}$ be the sequence of eigenvalues of $C_{T}$
sorted in order of decreasing modulus, and
$N_{T}(r)=\\#\\{n\in\mathbb{N}:|\lambda_{n}|\geq r\\}$. Then

$\lim_{r\to 0}\frac{\log N_{T}(r)}{\log|\log r|}=d,$

where

  1. (i)

if $\lambda_{\sigma,1}\cdot\lambda_{\sigma,2}\neq 0$ for some
$\sigma\in\Sigma$, then $d=2$ (stretched-exponential decay), and

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

with
$\eta_{2}=\left(1/2\sum_{\sigma\in\Sigma:\lambda_{\sigma,1}\cdot\lambda_{\sigma,2}\neq
0}(\log|\lambda_{\sigma,1}|\cdot\log|\lambda_{\sigma,2}|)^{-1}\right)^{-1/2}$.

  2. (ii)

if $\lambda_{\sigma,1}\cdot\lambda_{\sigma,2}=0$ for all $\sigma\in\Sigma$,
and $\lambda_{\sigma,k}\neq 0$ for some $\sigma\in\Sigma^{1}$ and
$k\in\\{1,2\\}$, then $d=1$ (exponential decay), and

$\lim_{n\to\infty}\frac{-\log|\lambda_{n}|}{n}=\eta_{1},$

with $\eta_{1}=\left(\sum_{\sigma\in\Sigma^{1}}\sum_{k:\lambda_{\sigma,k}\neq
0}(\log|\lambda_{\sigma,k}|)^{-1}\right)^{-1}$.

  3. (iii)

if $\lambda_{\sigma}=0$ for all $\sigma\in\Sigma^{1}$, and
$\lambda_{\sigma,1}\cdot\lambda_{\sigma,2}=0$ for all $\sigma\in\Sigma^{-1}$,
then $d=0$ (super-exponential decay). In this case the set of eigenvalues of
$C_{T}$ is trivial, and $\operatorname{spec}(C_{T})=\\{0,1\\}$.

###### Proof.

This follows directly from Lemma A.3 applied to the eigenvalues of $C_{T}$
written as the values of a cone-wise exponential function
$f\colon\mathbb{Z}^{2}\to{\mathbb{C}}$. In the case when both $T$ and $T^{-1}$
satisfy the case $(i)$ in Theorem 1.2, this function is given by
$f(n)=\lambda_{\sigma}^{|n|}$ with $\sigma=\sigma(n)\in\Sigma$ such that
either $n\in\mathbb{Z}^{2}\cap R^{\sigma}$, $\sigma\in\Sigma^{1}$, or
$n\in(\mathbb{Z}\setminus\\{0\\})^{2}\cap R^{\sigma}$, $\sigma\in\Sigma^{-1}$.
The other cases are similar. ∎

##  5 Anosov maps with different decay rates for resonances

Based on the results of the previous section, in this section we shall prove
our last main result, Theorem 1.3. The proof will use the classical result
that every toral Anosov diffeomorphism is homotopic to a toral automorphism,
and exploit the algebraic structure of ${\operatorname{GL}_{2}}(\mathbb{Z})$,
which is isomorphic to the group of toral automorphisms
${\operatorname{Aut}}(\mathbb{T}^{2})$. To explicitly construct
diffeomorphisms whose corresponding composition operators have resonances
exhibiting a desired decay rate, we shall introduce a special group
$\mathcal{F}$ of toral diffeomorphisms, containing the automorphisms as a
subgroup. The extension will consist of diffeomorphisms each of which is
homotopic to an automorphism in an explicit way.

Beyond its immediate usefulness for our proof, the group $\mathcal{F}$
provides a rich source of explicit examples of toral diffeomorphisms whose
resonances can often be explicitly computed, and which includes both area-
presering and non-area-preserving, orientation-preserving and -reversing
examples, as well as examples satisfying various symmetries. As we believe
this might be of broader interest, we include in Appendix B a more
comprehensive discussion and illustrative set of examples, while constraining
ourselves to a minimal introduction in this section.

###  5.1 A special group of toral diffemorphisms

The group ${\operatorname{Aut}}(\mathbb{T}^{2})$ of linear diffeomorphisms of
$\mathbb{T}^{2}$ is isomorphic to ${\operatorname{GL}_{2}}(\mathbb{Z})$, with
any $A=(a_{ij})\in{\operatorname{GL}_{2}}(\mathbb{Z})$ giving rise to the
toral automorphism
$\tau_{A}(z_{1},z_{2})=(z_{1}^{a_{11}}z_{2}^{a_{12}},z_{1}^{a_{21}}z_{2}^{a_{22}})$.
For our purposes it will be convenient to view
${\operatorname{Aut}}(\mathbb{T}^{2})$ as generated by the following
automorphisms, which can also be viewed as rational maps of
${\hat{\mathbb{C}}}^{2}$:

  1. (i)

a map $F$ given by $F(z_{1},z_{2})=(z_{1}z_{2},z_{2})$, with
$F^{-1}(z_{1},z_{2})=(z_{1}/z_{2},z_{2})$,

  2. (ii)

an involution $R$ given by $R(z_{1},z_{2})=(z_{2},z_{1})$,

  3. (iii)

involutions $I_{kl}$ for $k,l\in\\{0,1\\}$ given by
$I_{kl}(z_{1},z_{2})=(z_{1}^{1-2k},z_{2}^{1-2l})$.

The set $\Gamma=\\{F,R,I_{01}\\}$ generates
${\operatorname{Aut}}(\mathbb{T}^{2})$. To create a richer group of toral
diffeomorphisms, we extend the above by a continuous family of maps. Utilising
automorphisms of $\mathbb{D}$, the so-called Moebius maps
$b_{a}\colon{\hat{\mathbb{C}}}\to{\hat{\mathbb{C}}},a\in\mathbb{D}$, given by

$b_{a}(z)=\frac{z-a}{1-\overline{a}z},$

we define the additional set of generators as

  1. (iv)

a family $\mathcal{G}=\\{G_{a,b}:a,b\in\mathbb{D}\\}$ of maps given by

$G_{a,b}(z_{1},z_{2})=(b_{a}(z_{1}),b_{b}(z_{2})),$

satisfying $G^{-1}_{a,b}=G_{-a,-b}$.

######  Definition 5.1.

Denote by $\mathcal{F}$ the group of diffeomorphisms of $\mathbb{T}^{2}$
generated by the set $\Gamma\cup\mathcal{G}$.

######  Remark 5.2.

The proof of Theorem 1.3 will be based on Theorem 1.2, in particular we will
require all constructed maps $T$ to analytically extend to a neighbourhood of
$\mathbb{T}^{2}$, and to satisfy that

$T^{\ell}\text{ extends holomorpically to }D^{\sigma}\text{ and
}T^{\ell}(D^{\sigma})\subseteq
D^{\pm\sigma}\quad(\sigma\in\Sigma^{\ell},\ell\in\\{\pm 1\\}).$ (16)

A convenient class of maps satisfying (16) is the semigroup of finite
compositions of $\\{F,R,I_{11}\\}\cup\mathcal{G}$.

We remark that the choice of generators $\mathcal{G}$ is not the only
possible, though arguably the simplest choice of non-linear maps satisfying
(16). More generally, this property is satisfied by a certain class of
rational inner skew products, see [ST], taking the form

$(z_{1},z_{2})\mapsto(e^{i\theta}\frac{\tilde{p}(z_{1},z_{2})}{p(z_{1},z_{2})},z_{2}),$

with $\theta\in\mathbb{R}$ and $p$ a polynomial of bidegree $(1,k)$ for
$k\in\mathbb{N}_{0}$, that is, of degree $1$ in $z_{1}$ and degree $k$ in
$z_{2}$. Here, $\tilde{p}$ is the reflection of $p$ defined as
$\tilde{p}(z_{1},z_{2})=z_{1}z_{2}^{k}\overline{p(1/\overline{z_{1}},1/\overline{z_{2}})}$.
The map $G_{a,0}(z)=(b_{a}(z_{1}),z_{2})$ corresponds to the polynomial
$p(z)=1-\overline{a}z_{1}$ of bidegree $(1,0)$, and a general $G_{a,b}$ can be
written as $G_{a,b}=G_{a,0}\circ R\circ G_{b,0}$.

###  5.2 Explicit homotopies of Anosov diffeomorphisms

We proceed by stating an algebraic fact about conjugacy classes of
${\operatorname{GL}_{2}}(\mathbb{Z})$, which will allow us to establish an
analytic conjugacy between an arbitary hyperbolic automorphism of
$\mathbb{T}^{2}$ and an element of $\mathcal{F}$. While our proof is based on
results from [He], for variants of this result see, e.g., [Ka, BR] and
references therein. We defer the proof of the lemma to Appendix A.

######  Lemma 5.3.

Every hyperbolic matrix $M\in{\operatorname{GL}_{2}}(\mathbb{Z})$ is similar
(in ${\operatorname{GL}_{2}}(\mathbb{Z})$) to a matrix of the form

$\pm\begin{pmatrix}k_{1}&1\\\ 1&0\end{pmatrix}\begin{pmatrix}k_{2}&1\\\
1&0\end{pmatrix}\cdots\begin{pmatrix}k_{n}&1\\\ 1&0\end{pmatrix},\qquad
k_{1},\ldots,k_{n}\in\mathbb{N},n\geq 1.$ (17)

######  Corollary 5.4.

Every hyperbolic automorphism of $\mathbb{T}^{2}$ is conjugated via an
(analytic) toral automorphism to an automorphism of the form

$I_{11}^{s}\circ(F^{k_{1}}\circ R)\circ(F^{k_{2}}\circ
R)\circ\cdots\circ(F^{k_{n}}\circ R),\qquad
k_{1},\ldots,k_{n}\in\mathbb{N},n\geq 1,s\in\\{0,1\\}.$ (18)

###### Proof.

Let $\tau_{A}$ be a hyperbolic automorphism of $\mathbb{T}^{2}$ associated to
a hyperbolic matrix $A\in{\operatorname{GL}_{2}}(\mathbb{Z})$. By the previous
lemma, $A$ is similar to a matrix $B$ of the form (17), that is, there exists
$Q\in{\operatorname{GL}_{2}}(\mathbb{Z})$, such that $A=Q^{-1}BQ$ with $B$
decomposing into a product of matrices

$-\mathbb{I}=\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix}\quad\text{ and
}\quad\begin{pmatrix}k&1\\\ 1&0\end{pmatrix}=\begin{pmatrix}1&1\\\
0&1\end{pmatrix}^{k}\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},k\in\mathbb{N}.$

We note that for every $k\in\mathbb{N}$, the latter is equal to
$(M_{F})^{k}M_{R}$, where $M_{F}$, $M_{R}$ correspond to the toral
automorphisms $F$ and $R$ (i.e., $\tau_{M_{F}}=F$ and $\tau_{M_{R}}=R$), and
$\tau_{-\mathbb{I}}=I_{11}$. It follows that $\tau_{B}$ has the desired form,
and $\tau_{A}=\tau_{Q}^{-1}\circ\tau_{B}\circ\tau_{Q}$. ∎

Next, for any hyperbolic automorphism of the form (18) we construct non-linear
area-preserving Anosov diffeomorphisms from $\mathcal{F}$ in the same homotopy
class with easily computable resonances. For this, we derive from the linear
map $F^{k}\circ R,k\in\mathbb{N},$ the one-parameter family of non-linear
toral diffeomorphisms

$U_{k,a}(z)=(G_{0,a}\circ F^{k}\circ R\circ
G_{-a,0})(z)=(b_{-a}(z_{1})^{k}z_{2},z_{1}),\qquad a\in\mathbb{D}.$

Applying this to (18), we define

$\Psi_{K,A}=I_{11}^{s}\circ U_{k_{1},a_{1}}\circ\dots\circ U_{k_{n},a_{n}},$
(19)

where
$s\in\\{0,1\\},n\in\mathbb{N},A=(a_{1},\ldots,a_{n})\in\mathbb{D}^{n},K=(k_{1},\ldots,k_{n})\in\mathbb{N}^{n}$.
We write $A_{o}=(a_{1},a_{3},\ldots)$ and $A_{e}=(a_{2},a_{4},\ldots)$ for the
respective tuples only involving odd or even indices (analogously for $K_{o}$,
$K_{e}$). For convenience, we shall use the multiindex notation
$A^{K}:=a_{1}^{k_{1}}\cdots a_{n}^{k_{n}}$ for arbitrary $n$-tuples $A$ and
$K$, $n\in\mathbb{N}_{0}$, with the convention $A^{K}=1$ when $A$ and $K$ are
of length $0$.

######  Proposition 5.5 (Area-preserving maps homotopic to (18)).

For $s\in\\{0,1\\}$, $A=(a_{1},\ldots,a_{n})\in\mathbb{D}^{n}$ and
$K=(k_{1},\ldots,k_{n})\in\mathbb{N}^{n},n\in\mathbb{N}$, the map $\Psi_{K,A}$
is an area-preserving hyperbolic toral diffeomorphism satisfying the
conclusions of Theorem 1.2. In particular, $\Psi_{K,A}$ satisfies Theorem
1.2$(i)$ if $s=0$, and (ii) if $s=1$, and $\Psi_{K,A}^{-1}$ satisfies (i) if
$n+s$ is even, and (ii) if $n+s$ is odd. Moreover, denoting $\lambda_{\sigma}$
the multipliers of the unique attracting fixed point of $\Psi_{K,A}^{\ell}$ in
$D^{\sigma}$ for $\sigma\in\Sigma^{\ell},\ell\in\\{\pm 1\\}$ if
$\Psi_{K,A}^{\ell}(D^{\sigma})\subset D^{\sigma}$ and of $\Psi_{K,A}^{2\ell}$
otherwise, we have:

  1. (i)

If $s=0$ and $n$ is odd, then

$\lambda_{--}=\overline{\lambda_{++}}=((A^{K})^{1/2},-(A^{K})^{1/2})\quad\text{
and
}\quad\lambda_{-+}=\lambda_{+-}=(A_{o}^{K_{o}}\overline{A_{e}^{K_{e}}},\overline{A_{o}^{K_{o}}}A_{e}^{K_{e}}).$

  2. (ii)

If $s=0$ and $n$ is even, then

$\lambda_{--}=\overline{\lambda_{++}}=(A^{K_{o}}_{o},A^{K_{e}}_{e})\quad\text{
and
}\quad\lambda_{-+}=\overline{\lambda_{+-}}=(\overline{A^{K_{o}}_{o}},A^{K_{e}}_{e}).$

  3. (iii)

If $s=1$ and $n$ is odd, then

$\lambda_{--}=\lambda_{++}=(\overline{A^{K_{o}}_{o}}A^{K_{e}}_{e},A^{K_{o}}_{o}\overline{A_{e}^{K_{e}}})\quad\text{
and
}\quad\lambda_{-+}=\overline{\lambda_{+-}}=((A^{K_{o}}_{o}\overline{A_{e}^{K_{e}}})^{1/2},-(A^{K_{o}}_{o}\overline{A_{e}^{K_{e}}})^{1/2}).$

  4. (iv)

If $s=1$ and $n$ is even, then

$\lambda_{\sigma}=(|A^{K_{o}}_{o}|^{2},|A^{K_{e}}_{e}|^{2})\text{ for all
}\sigma\in\Sigma.$

###### Proof.

We begin by showing that $C_{\Psi_{K,A}}$ is trace-class on a suitable Hilbert
space $H_{\alpha,-\gamma}$. In the case $n>1$ this will follow from Theorem
1.2 by proving that $\Psi_{K,A}$ satisfies the (p-sec) condition, while the
case $n=1$ will be handled separately.

For any map $T\colon\mathbb{T}^{2}\to\mathbb{T}^{2}$ and
$M=([0,2\pi]/\sim)^{2}$, we denote by $\tilde{T}\colon M\to M$ the map
determined by $\pi\circ\tilde{T}=T\circ\pi$ with $\pi(x)=e^{ix}$ for $x\in M$,
and analogously for maps on $\mathbb{T}$ and $([0,2\pi]/\sim)$. By Lemma B.1
in the appendix, we have $\tilde{b}_{-a}(e^{i\theta})=\theta+g_{-a}(\theta)$
with $g_{-a}^{\prime}(\theta)>-1$ for all $a\in\mathbb{D}$, and
$\tilde{U}_{k,a}(x_{1},x_{2})=(k(x_{1}+g_{-a}(x_{1}))+x_{2},x_{1})$. Thus we
obtain

$D\tilde{U}_{k,a}(x)=\begin{pmatrix}s_{k,a}(x)&1\\\ 1&0\end{pmatrix},$ (20)

where $s_{k,a}(x)=k(1+g_{-a}^{\prime}(x_{1}))>0$ for all $x=(x_{1},x_{2})\in
M$.

Consider first the case $\Psi_{K,A}=I_{11}^{s}\circ
U_{k_{1},a_{1}}\circ\dots\circ U_{k_{n},a_{n}}$ with $s=0$ and $n>1$. Then
$D\tilde{\Psi}_{K,A}$ is positive and hence
$(D\tilde{\Psi}_{K,A}(x))(\mathbb{R}^{2}_{\geq
0})\subset\mathbb{R}^{2}_{>0}\cup\\{0\\}$ for every $x\in M$. Using (20) we
obtain

$D\tilde{\Psi}_{K,A}(x)=M_{x}+\begin{cases}\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\qquad\text{ if $n$ is even,}\\\ \begin{pmatrix}0&1\\\
1&0\end{pmatrix},\qquad\text{ if $n$ is odd,}\end{cases}$ (21)

with $M_{x}\geq 0$. By the criterion in Remark 2.13, this implies that the
first half of the (p-sec) condition (6) is satisfied. Since $\det
D\tilde{\Psi}_{K,A}(x)=(-1)^{n}$, we also have that
$D\tilde{\Psi}^{-1}_{K,A}(x)=\begin{pmatrix}1+a_{x}&-b_{x}\\\
-c_{x}&1+d_{x}\end{pmatrix}$ if $n$ is even, and
$D\tilde{\Psi}^{-1}_{K,A}(x)=\begin{pmatrix}-a_{x}&1+b_{x}\\\
1+c_{x}&-d_{x}\end{pmatrix}$ for $n$ odd, with $a_{x},b_{x},c_{x},d_{x}>0$. It
is easy to see that these are conjugated to matrices of the form (21) via the
matrix $\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ or $\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}$, which via the criterion in Remark 2.13 implies the second
half of the (p-sec) condition (6). The case of $s=1$ follows immediately,
since condition (6) holds for a map $\tilde{T}$ if and only if it holds for
$-\tilde{T}$, finishing the proof of the (p-sec) condition for $\Psi_{K,A}$
with $n>1$.

In the case $n=1$ the (p-sec) condition does not hold, however one can verify
that the assumptions of Theorem 3.24 still apply (similar to the case in
Remark 3.26), so that by Corollary 3.28, $C_{\Psi_{K,A}}$ is trace-class in
this case also.

Next, we observe that for any $k\in\mathbb{N}$, $a\in\mathbb{D}$ and
$\sigma\in\Sigma^{1}$, the map $U_{k,a}$ extends holomorphically to
$D^{\sigma}$ with $U_{k,a}(D^{\sigma})\subseteq D^{\sigma}$, and fixes
$z^{*}=(0,0)$ with

$DU_{k,a}(z^{*})=\begin{pmatrix}0&a^{k}\\\ 1&0\end{pmatrix},$

while for $\sigma\in\Sigma^{-1}$ the map $U_{k,a}^{-1}$ extends
holomorphically to $D^{\sigma}$ with $U_{k,a}^{-1}(D^{\sigma})\subseteq
D^{-\sigma}$, which implies the claim about how the cases Theorem
1.2$(i)$-$(ii)$ apply to $\Psi_{K,A}$ and $\Psi_{K,A}^{-1}$.

We now proceed to show the assertions $(i)$-$(iv)$. Denoting
$U_{K,A}=U_{k_{1},a_{1}}\circ\cdots\circ U_{k_{n},a_{n}}$ with
$A=(a_{1},\ldots,a_{n})$ and $K=(k_{1},\ldots,k_{n})$, we note that
$DU_{K,A}(z^{*})=DU_{k_{1},a_{1}}(z^{*})\cdots DU_{k_{n},a_{n}}(z^{*})$,
allowing us to compute the relevant multipliers for
$\Psi_{K,A}=I_{11}^{s}\circ U_{K,A}$, starting with the case of $s=0$.

  1. (i)

If $s=0$ and $n$ is odd, then

$D\Psi_{K,A}({z^{*}})=DU_{K,A}({z^{*}})=\begin{pmatrix}0&A^{K_{o}}_{o}\\\
A^{K_{e}}_{e}&0\end{pmatrix}.$

Thus, the multipliers of $\Psi_{K,A}$ at $z^{*}$ are $\lambda=(v,-v)$ with
$v=(A^{K})^{1/2}$.

  2. (ii)

If $s=0$ and $n$ is even, then

$D\Psi_{K,A}({z^{*}})=DU_{K,A}(z^{*})=\begin{pmatrix}A^{K_{o}}_{o}&0\\\
0&A^{K_{e}}_{e}\end{pmatrix},$

and, the multipliers of $\Psi_{K,A}$ at $z^{*}$ are
$\lambda=(A_{o}^{K_{o}},A_{e}^{K_{e}})$.

For $s=1$ we shall use Lemma 4.9 with $\hat{T}=U_{K,A}$ and $T=I_{11}\circ
U_{K,A}=\Psi_{K,A}$, which yields
$D(\Psi_{K,A}\circ\Psi_{K,A})(z^{*})=\overline{DU_{K,A}(z^{*})}DU_{K,A}(z^{*})$.

  1. (iii)

If $s=1$ and $n$ is odd, then the multipliers of $\Psi_{K,A}^{2}$ at $z^{*}$
are $\lambda=(v,\overline{v})$ with $v=\overline{A_{o}^{K_{o}}}A^{K^{e}}_{e}$.

  2. (iv)

If $s=1$ and $n$ is even, then the multipliers of $\Psi_{K,A}^{2}$ at $z^{*}$
are $\lambda=(|A_{o}^{K_{o}}|^{2},|A_{e}^{K_{e}}|^{2})$.

Now, it remains to compute the multipliers of the inverse of $I_{11}^{s}\circ
U_{K,A}$. We set $S=I_{01}\circ R$ (so that $S^{2}=S^{-2}=I_{11}$), and
observe using Lemma B.2 in the appendix that the inverse
$U^{-1}_{k,a}=G_{a,0}\circ R\circ F^{-1}\circ G_{0,-a}$ obeys the relations

$\displaystyle U^{-1}_{k,a}\circ S$ $\displaystyle=S^{-1}\circ
U_{k,\overline{a}},$ $\displaystyle U^{-1}_{k,b}\circ S^{-1}$
$\displaystyle=S\circ U_{k,b},$

and hence we have $S^{-1}\circ U_{k,a}^{-1}\circ S=I_{11}U_{k,\overline{a}}$
and $S^{-1}\circ(U_{k,a}\circ U_{k,b})^{-1}\circ S=U_{k,b}\circ
U_{k,\overline{a}}$. Iterating, we obtain the conjugation

$S^{-1}\circ U_{K,A}^{-1}\circ S=I_{11}^{n}\circ
U_{k_{n},{\tilde{a}_{n}}}\circ\cdots\circ U_{k_{4},{a_{4}}}\circ
U_{k_{3},\overline{a_{3}}}\circ U_{k_{2},{a_{2}}}\circ
U_{k_{1},\overline{a_{1}}}=:\tilde{U}_{K,A},$ (22)

where $\tilde{a}_{n}$ is $a_{n}$ if $n$ is even or $\overline{a}_{n}$ if n is
odd. With the same conjugacy $S$, we obtain a conjugation

$S^{-1}\circ(I_{11}\circ U_{K,A})^{-1}\circ S=I_{11}^{n-1}\circ
U_{k_{n},{\check{a}_{n}}}\circ\cdots\circ U_{k_{4},\overline{a_{4}}}\circ
U_{k_{3},{a_{3}}}\circ U_{k_{2},\overline{a_{2}}}\circ
U_{k_{1},{a_{1}}}=:\check{U}_{K,A},$ (23)

where $\check{a}^{n}$ is $\overline{a_{n}}$ if $n$ is even or ${a}_{n}$ if n
is odd. We can now compute the relevant multipliers for
$\Psi_{K,A}^{-1}=(I_{11}^{s}\circ U_{K,A})^{-1}$. For $s=0$, by (22) these are
given by the fixed-point multipliers of $\tilde{U}_{K,A}$.

  1. (i)

If $s=0$ and $n$ is odd, by Lemma 4.9 the multipliers of $\tilde{U}_{K,A}^{2}$
at $z^{*}$ are
$\lambda=(v,\overline{v}),v=\overline{A_{o}^{K_{o}}}A_{e}^{K_{e}}$.

  2. (ii)

If $s=0$ and $n$ is even, then the multipliers of $\tilde{U}_{K,A}$ at $z^{*}$
are $\lambda=(\overline{A_{o}^{K_{o}}},A_{e}^{K_{e}})$.

For $s=1$, the fixed-point multipliers of $\Psi_{K,A}^{-1}$ can be computed
via those of $\check{U}_{K,A}$ by (23).

  1. (iii)

If $s=1$ and $n$ is odd, the multipliers of $\check{U}_{K,A}$ at $z^{*}$ are
$\lambda=(v,-v)$ with $v=(A_{o}^{K_{o}}\overline{A_{e}^{K_{e}}})^{1/2}$.

  2. (iv)

If $s=1$ and $n$ is even, by Lemma 4.9 the multipliers of
$\check{U}_{K,A}^{2}$ at $z^{*}$ are
$\lambda=(|A_{o}^{K_{o}}|^{2},|A_{e}^{K_{e}}|^{2}).$

Claims $(i)$-$(iv)$ follow by combining the respective cases for the
multipliers of $\Psi_{K,A}$ and $\Psi_{K,A}^{-1}$. ∎

######  Corollary 5.6.

Let $\Psi_{K,A}$ be as in Proposition 5.5. Then:

  1. (i)

If $a_{i}\neq 0$ for all $i$, then the fixed point multipliers
$\lambda_{\sigma}$ in Theorem 1.2 satisfy
$\lambda_{\sigma,1}\cdot\lambda_{\sigma,2}\neq 0$ for all $\sigma\in\Sigma$.
Moreover, $|\lambda_{\sigma,1}|$ and $|\lambda_{\sigma,2}|$ can
(independently) be chosen to take any value in $(0,1)$ via suitable choice of
$A\in\mathbb{D}^{n}$.

  2. (ii)

If $n$ is even and exactly one of the $a_{i}$ is zero, then either case $(i)$
or $(ii)$ of Theorem 1.2 applies to both $\Psi_{K,A}$ and $\Psi^{-1}_{K,A}$,
and all fixed point multipliers are of the form $(\lambda_{1},0)$ with
$\lambda_{1}\neq 0$. Moreover, $|\lambda_{1}|$ can be chosen to take any value
in $(0,1)$ via suitable choice of $A\in\mathbb{D}^{n}$, and in the case $(ii)$
of Theorem 1.2, $\lambda_{1}\in\mathbb{R}$.

  3. (iii)

If $n>2$ and at most one of the $a_{i}$ is nonzero, then all multipliers are
$(0,0)$.

We shall next construct non-linear non-area-preserving maps in $\mathcal{F}$
homotopic to maps of the form (18) with $n>1$, yielding trivial resonances.
For this we define the toral diffeomorphisms

$W_{k,a}(z)=(F^{k}\circ R\circ
G_{0,a})(z)=(z_{1}^{k}b_{a}(z_{2}),z_{1})\quad(k\in\mathbb{N},a\in\mathbb{D},z\in\mathbb{T})$

and

$\Xi_{K,a}=I_{11}^{s}\circ W_{k_{1},0}\circ W_{k_{2},0}\circ\cdots\circ
W_{k_{n-1,0}}\circ W_{k_{n},a},$

where $s\in\\{0,1\\}$, $n>1$, $a\in\mathbb{D}$ and
$K=(k_{1},\ldots,k_{n})\in\mathbb{N}^{n}$.

######  Lemma 5.7 (Non-area-preserving maps homotopic to (18)).

For any $s\in\\{0,1\\}$, $a\in\mathbb{D}$, $K\in\mathbb{N}^{n}$ with $n>1$,
the map $\Xi_{K,a}$ satisfies the assumptions of Theorem 1.2. Moreover
$\chi_{\Xi_{K,a}}=1.$

###### Proof.

Following similar calculations and notations as in the proof of Proposition
5.5 we first show that $\Xi_{K,a}$ satisfies the (p-sec) condition. We have
that $D\tilde{W}_{k,a}(x)=\begin{pmatrix}k&s_{a}(x)\\\ 1&0\end{pmatrix}$ with
$s_{a}(x)>0$ for all $x\in M$. Thus, for $s=0$ and $n>1$ it follows that
$D\tilde{\Xi}_{K,a}(x)$ can be written as $M_{x}+\begin{pmatrix}1&0\\\
0&0\end{pmatrix}$ with $M_{x}\geq 0$ and thus satisfies the first part of the
(p-sec) condition (6) by Remark 2.13. The case $s=1$ and the second part of
the (p-sec) condition follow similarly.

To show that $\Xi_{K,a}$ does not yield any non-trivial resonances, first note
that for $\ell\in\\{1,-1\\}$ the map $W_{k,a}^{\ell}$ extends holomorphically
to $D^{\sigma}$ with $W_{k,a}^{\ell}(D^{\sigma})\subseteq D^{\ell\sigma}$ for
all $\sigma\in\Sigma^{\ell}$. As the forward map $W_{k,a}$ fixes $z^{*}=(0,0)$
and

$DW_{k,a}(z^{*})=\begin{pmatrix}-a\delta_{k,1}&0\\\ 1&0\end{pmatrix},$

we see that the multipliers of $\Xi_{K,a}$ at $z^{*}$ are trivial for any
$n>1$ and $s\in\\{0,1\\}$. For the inverse map we use the conjugation
$S=I_{01}\circ R$, obtaining the relations

$\displaystyle W_{k,a}^{-1}\circ S$ $\displaystyle=S^{-1}\circ G_{-a,0}\circ
W_{k,0},$ $\displaystyle W_{k,a}^{-1}\circ S^{-1}$ $\displaystyle=S\circ
G_{-\overline{a},0}\circ W_{k,0},$

yielding $S^{-1}\circ\Xi_{K,a}^{-1}\circ S=G_{-\overline{a},0}\circ
I_{11}^{n+s}\circ W_{k_{n},0}\circ\cdots\circ W_{k_{1},0}:=E_{K,a}$. One can
calculate that $(\overline{a},0)$ is a fixed point of $E_{K,a}$ for $n+s$ is
even, and of $(E_{K,a})^{2}$ if $n+s$ is odd. In both cases, its multipliers
are $\lambda=(c,0)$ for some $c\in\mathbb{D}$. Thus, by Corollary 4.12$(iii)$
all resonances are trivial. ∎

######  Proposition 5.8.

The homotopy class of every map of the form (18) contains non-linear Anosov
diffeomorphisms $T\in\mathcal{F}$, such that the corresponding operator
$C_{T}$ is well defined and trace-class on $H_{\alpha,-\gamma}$ for some
$\alpha,\gamma\in\mathbb{R}^{2}_{>0}$. Moreover, $T$ can be chosen such that
the eigenvalue sequence of $C_{T}$ satisfies either of the cases (i)-(iii)
from the conclusions of Theorem 1.3.

###### Proof.

To prove cases $(i)$ and $(ii)$ from Theorem 1.3, we choose $T$ to be of the
form $\Psi_{K,A}$ from (19), noting that for any fixed $s\in\\{0,1\\}$ and
$K\in\mathbb{N}^{n}$, $n\in\mathbb{N}$, the map $\Psi_{K,0}$ is an area-
preserving hyperbolic automorphism of the form (18), homotopic to any
$\Psi_{K,A}$, $A\in\mathbb{D}^{n}$. The claim follows directly from
Proposition 5.5 together with Corollaries 5.6 and 4.12.

The case $(iii)$ follows from Lemma 5.7, noting that the maps $\Xi_{K,a}$ are
not area-preserving and hence not $C^{1}$-conjugated to toral automorphisms
for $a\neq 0$, but have trivial resonances. ∎

We will also need the following well-known result (see, e.g., [F, Lemma 1.1]
and [M, Theorem A]):

######  Lemma 5.9.

For any Anosov diffeomorphism $f\colon\mathbb{T}^{n}\to\mathbb{T}^{n}$ there
exists a hyperbolic automorphism $g\colon\mathbb{T}^{n}\to\mathbb{T}^{n}$
which is homotopic to $f$. That is, there exists a continuous one-parameter
family of maps $h\colon[0,1]\times\mathbb{T}^{n}\to\mathbb{T}^{n}$, such that
$h(0,\cdot)=f$ and $h(1,\cdot)=g$.

We are now ready to prove our last main theorem, concluding that every
homotopy class of toral Anosov diffeomorphisms contains elements with
resonances exhibiting any of stretched-exponential, exponential, or trivially
super-exponential decay rate.

###### Proof of Theorem 1.3.

Let $\mathcal{H}$ be any homotopy class of toral Anosov diffeomorphisms. By
Lemma 5.9, there exists a hyperbolic matrix
$B\in{\operatorname{GL}_{2}}(\mathbb{Z})$ with $\tau_{B}\in\mathcal{H}$.
Corollary 5.4 yields that there are
$A,Q\in{\operatorname{GL}_{2}}(\mathbb{Z})$ such that $\tau_{A}$ is of the
form (18), and $\tau_{B}$ is analytically conjugated to $\tau_{A}$, via
$\tau_{A}=\tau_{Q}\circ\tau_{B}\circ\tau_{Q}^{-1}$. We call
$\mathcal{H}^{\prime}$ the homotopy class containing $\tau_{A}$. We note that
because of its special form (18) and using Remark 3.22, the operator
$C_{\tau_{A}}$ given by $f\mapsto f\circ\tau_{A}$ yields an isomorphism from
$H_{\alpha,-\gamma}$ to itself, while $\tau_{Q}$ gives rise to the isometric
isomorphism $C_{\tau_{Q}}\colon H_{Q,\alpha,-\gamma}\to H_{\alpha,-\gamma}$,
conjugating $C_{\tau_{A}}\colon H_{\alpha,-\gamma}\to H_{\alpha,-\gamma}$ and
$C_{\tau_{B}}\colon H_{Q,\alpha,-\gamma}\to H_{Q,\alpha,-\gamma}$, for any
$\alpha,\gamma\in\mathbb{R}^{2}$.

Finally, by Proposition 5.8, there exists an Anosov diffeomorphism
$\tau^{\prime}\in\mathcal{H}^{\prime}$ whose corresponding composition
operator on $H_{\alpha^{\prime},-\gamma^{\prime}}$ for suitable
$\alpha^{\prime},\gamma^{\prime}\in\mathbb{R}^{2}_{>0}$ has an eigenvalue
sequence with any one of the desired decay rates. Writing out the homotopy
explicitly, we have a one-parameter family of maps
$h^{\prime}\colon[0,1]\times\mathbb{T}^{2}\to\mathbb{T}^{2}$,
$h^{\prime}_{t}=h^{\prime}(t,\cdot)\in\mathcal{H}^{\prime}$, such that
$h^{\prime}_{0}=\tau_{A}$ and $h^{\prime}_{1}=\tau^{\prime}$. Conjugating with
$\tau_{Q}$ we obtain a homotopy $h_{t}=\tau_{Q}^{-1}\circ
h^{\prime}_{t}\circ\tau_{Q}\in\mathcal{H}$ with $h_{0}=\tau_{B}$ and
$h_{1}=\tau_{Q}^{-1}\circ\tau^{\prime}\circ\tau_{Q}$. Since
$C_{\tau_{Q}}\colon H_{Q,\alpha^{\prime},-\gamma^{\prime}}\to
H_{\alpha^{\prime},-\gamma^{\prime}}$ is an isometric isomorphism, the spectra
of $C_{\tau^{\prime}}$ and $C_{h_{1}}=C_{\tau^{-1}_{Q}}\circ
C_{\tau^{\prime}}\circ C_{\tau_{Q}}$ coincide, and so the composition operator
associated to $T=h_{1}$ satisfies the assertion of the proposition for
$\nu=\nu_{P,\alpha,-\gamma}$ with $P=Q$, $\alpha=\alpha^{\prime}$ and
$\gamma=\gamma^{\prime}$. ∎

##  Appendix A Auxiliary results

Here we list a number of auxiliary results and proofs omitted but used in the
main text.

######  Lemma A.1.

For every $P\in{\operatorname{GL}_{2}}(\mathbb{R})$, there exist
$A\in{\operatorname{GL}_{2}}(\mathbb{Z})$ and
$\tilde{P}\in{\operatorname{GL}_{2}}(\mathbb{R})$ with $\tilde{P}$ having only
non-negative entries, such that $P=A\tilde{P}$.

###### Proof.

It suffices to show that for $P\in{\operatorname{GL}_{2}}(\mathbb{R})$, there
exists $B\in{\operatorname{GL}_{2}}(\mathbb{Z})$ such that $(BP)_{ij}>0$ for
$i,j=1,2$. Writing the rows of $B$ as $b^{u},b^{s}$, and the columns of $P$ as
$p_{u},p_{s}$ and denoting the cone $\mathcal{C}=\\{v\in\mathbb{R}^{2}:\langle
v,p_{u}\rangle>0,\langle v,p_{s}\rangle>0\\}$, this is equivalent to there
existing $b^{u},b^{s}\in\mathbb{Z}^{2}\cap\mathcal{C}$, such that

$b^{u}_{1}b^{s}_{2}-b^{u}_{2}b^{s}_{1}=1.$ (24)

We fix any $b^{u}\in\mathbb{Z}^{2}\cap\mathcal{C}$ (non-empty since
$\mathcal{C}$ is an open convex cone), without loss of generality satisfying
$\operatorname{gcd}(b^{u}_{1},b^{u}_{2})=1$. By Bezout’s identity, there exist
$\hat{b}^{s}=(\hat{b}^{s}_{1},\hat{b}^{s}_{2})\in\mathbb{Z}^{2}$ such that
$b^{s}_{1}(k)=\hat{b}^{s}_{1}+kb^{u}_{1},b^{s}_{2}(k)=\hat{b}^{s}_{2}+kb^{u}_{2}$
are solutions to (24) for every $k\in\mathbb{Z}$. Moreover, it is easy to see
that for sufficiently large $k\in\mathbb{Z}$,
$b^{s}=(b^{s}_{1}(k),b^{s}_{2}(k))=\hat{b}^{s}+k\cdot b^{u}$ lies in the cone
$\mathcal{C}$, finishing the proof. ∎

######  Lemma A.2.

Let $\hat{\sigma}\in\Sigma^{1}$, and let $\\{D_{c}:c\in
C\\}\subset{\operatorname{GL}_{2}}(\mathbb{R})$ be a continuous family of
matrices indexed by some compact set $C$, satisfying
$D_{c}(\mathbb{R}^{2}_{\geq 0})\subset R^{\hat{\sigma}}\cup\\{0\\}$ for all
$c\in C$. Then, for any $\sigma\in\Sigma^{1}$, $\tilde{\sigma}\in\Sigma^{-1}$,
$\delta,\tilde{\delta}\in\mathbb{R}^{2}_{>0}$, there exists $q\in
R^{\tilde{\sigma}}_{\tilde{\delta}}$ such that $D_{c}(q)\in
R^{\sigma}_{\delta}$ for all $c\in C$.

###### Proof.

Let us first assume that $\hat{\sigma}=(1,1)$, that is,
$D_{c}(\mathbb{R}^{2}_{\geq 0})\subset\mathbb{R}^{2}_{>0}\cup\\{0\\}$. This
implies $(D_{c})_{kl}>0$ for all $k,l\in\\{1,2\\}$ and all $c\in C$. By
compactness of $C$ it follows that there are $\underline{D},\overline{D}>0$
such that $\underline{D}<(D_{c})_{kl}<\overline{D}$ for all $k,l\in\\{1,2\\}$
and all $c\in C$. We fix $\delta,\tilde{\delta}\in\mathbb{R}^{2}_{>0}$, and
set
$\overline{\delta}=\max\\{\delta_{1},\delta_{2},\tilde{\delta}_{1},\tilde{\delta}_{2}\\}$.

We first consider the case $\sigma=(1,1)$, $\tilde{\sigma}=(-1,1)$. Setting
$q_{1}=-\overline{\delta}<-\tilde{\delta}_{1}$ and
$q_{2}=\overline{\delta}\max\\{1,\frac{1+\overline{D}}{\underline{D}}\\}>\tilde{\delta}_{2}$,
we have $q=(q_{1},q_{2})\in R^{\tilde{\sigma}}_{\tilde{\delta}}$, and since
$(D_{c})_{k,1}q_{1}+(D_{c})_{k,2}q_{2}\geq-\overline{D}\overline{\delta}+\underline{D}\frac{1+\overline{D}}{\underline{D}}\overline{\delta}=\overline{\delta}>\delta_{k}$
for $k=1,2$ and all $c\in C$, we obtain $D_{c}(q)\in
R^{2}_{>0}+\delta=R^{\sigma}_{\delta}$, for all $c\in C$, as required.

The case $\sigma=(-1,-1)$, $\tilde{\sigma}=(1,-1)$ is similar, with
$q^{\prime}=-q\in R^{\tilde{\sigma}}_{\tilde{\delta}}$ and
$D_{c}(q^{\prime})=-D_{c}(q)\in R^{2}_{<0}-\delta=R^{\sigma}_{\delta}$ for all
$c\in C$. The other two cases ($\sigma=(1,1)$, $\tilde{\sigma}=(1,-1)$ and
$\sigma=(-1,-1)$, $\tilde{\sigma}=(-1,1)$) can be shown analogously by
swapping the roles of $q_{1}$ and $q_{2}$ in the above construction.

Finally, for $\hat{\sigma}=(-1,-1)$, we note that the claim holds for $-D_{c}$
by the above, and hence it follows for $D_{c}$ by replacing $q$ by $-q$. ∎

######  Lemma A.3.

Let $f\colon\mathbb{Z}^{2}\to{\mathbb{C}}$ be a cone-wise exponential function
with cones being the quadrants $\hat{R}^{\sigma,o}$, $\sigma\in\Sigma$; that
is, for all $\sigma\in\Sigma$ there exist $\lambda_{\sigma}\in\mathbb{D}^{2}$,
such that $f(n)=\lambda_{\sigma}^{n}$ whenever
$n\in\mathbb{Z}^{2}\cap\hat{R}^{\sigma,o}$. Denote by
$(\lambda_{n})_{n\in\mathbb{N}}$ be an enumeration of
$\\{f(n):n\in\mathbb{Z}^{2}\\}$ sorted by decreasing modulus, and
$N(r)=\\#\\{n\in\mathbb{N}:|\lambda_{n}|\geq r\\}$ for $r\in(0,1)$. Then
$(\lambda_{n})_{n\in\mathbb{N}}$ satisfies

$\lim_{r\to 0}\frac{\log N(r)}{\log|\log r|}=d,$

where:

  1. (i)

if $\lambda_{\sigma,1}\cdot\lambda_{\sigma,2}\neq 0$ for some
$\sigma\in\Sigma$, then $d=2$ (stretched-exponential decay) and

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

with
$\eta_{2}=\left(1/2\sum_{\sigma\in\Sigma:\lambda_{\sigma,1}\cdot\lambda_{\sigma,2}\neq
0}(\log|\lambda_{\sigma,1}|\cdot\log|\lambda_{\sigma,2}|)^{-1}\right)^{-1/2}$.

  2. (ii)

if $\lambda_{\sigma,1}\cdot\lambda_{\sigma,2}=0$ for all $\sigma\in\Sigma$,
and $\lambda_{\sigma,k}\neq 0$ for some $\sigma\in\Sigma^{1}$ and
$k\in\\{1,2\\}$, then $d=1$ (exponential decay), and

$\lim_{n\to\infty}\frac{-\log|\lambda_{n}|}{n}=\eta_{1},$

with $\eta_{1}=\left(\sum_{\sigma\in\Sigma^{1}}\sum_{k:\lambda_{\sigma,k}\neq
0}(\log|\lambda_{\sigma,k}|)^{-1}\right)^{-1}$.

  3. (iii)

if $\lambda_{\sigma}=0$ for all $\sigma\in\Sigma^{1}$, and
$\lambda_{\sigma,1}\cdot\lambda_{\sigma,2}=0$ for all $\sigma\in\Sigma^{-1}$,
then $d=0$ (super-exponential decay). In this case
$(\lambda_{n})_{n\in\mathbb{N}}$ is the trivial sequence with
$\lambda_{n}=\delta_{n,1}$.

###### Proof.

We write $\ell_{s}(r)=\log r/\log|s|$ for $s\in\mathbb{D}$ and $r\in(0,1)$.
For $p,q\in\mathbb{D},r\in(0,1)$ we have

$\displaystyle N^{r}_{p}:=$ $\displaystyle\\#\\{n\geq 1:|p|^{n}\geq
r\\}=\lfloor\ell_{p}(r)\rfloor,$ $\displaystyle N^{r}_{p,q}:=$
$\displaystyle\\#\\{n,m\geq 1:|p|^{n}|q|^{m}\geq r\\}$ $\displaystyle=$
$\displaystyle\\#\\{n,m\geq 1:n(-\log|p|)+m(-\log|q|)\leq-\log r\\}$
$\displaystyle=$
$\displaystyle\frac{\lfloor\ell_{p}(r)\rfloor\lfloor\ell_{q}(r)\rfloor}{2}+\delta,\qquad\text{
with }|\delta|\leq 1+\lfloor\ell_{p}(r)\rfloor+\lfloor\ell_{q}(r)\rfloor,$

where the last equality is obtained from counting the number of integer
lattice points in the triangle spanned by $(0,0)$, $(\ell_{p}(r),0)$ and
$(0,\ell_{q}(r))$, and subtracting those lying on one of the axes. From the
definition of $f$ we obtain

$N(r)=1+\sum_{\sigma\in\Sigma}\left(\frac{\lfloor\ell_{\lambda_{\sigma,1}}(r)\rfloor\lfloor\ell_{\lambda_{\sigma,2}}(r)\rfloor}{2}+\delta_{\sigma}\right)+\sum_{\sigma\in\Sigma^{1}}\left(\lfloor\ell_{\lambda_{\sigma,1}}(r)\rfloor+\lfloor\ell_{\lambda_{\sigma,2}}(r)\rfloor\right),$

with $|\delta_{\sigma}|\leq
1+\lfloor\ell_{\lambda_{\sigma,1}}(r)\rfloor+\lfloor\ell_{\lambda_{\sigma,2}}(r)\rfloor$.

We now prove $(i)$. Since $\ell_{s}(r)\to\infty$ as $r\to 0$ for any
$s\in\mathbb{D}$ and $\lambda_{\sigma,1}\cdot\lambda_{\sigma,2}\neq 0$ for
some $\sigma\in\Sigma$, we have that for every $\epsilon>0$ there exists
$r_{\epsilon}>0$ such that for $r\in(0,r_{\epsilon})$ it holds that

$(1-\epsilon)\cdot\left(\frac{\log r}{\eta_{2}}\right)^{2}\leq
N(r)\leq(1+\epsilon)\cdot\left(\frac{\log r}{\eta_{2}}\right)^{2}.$ (25)

The assertion $d=2$ immediately follows. Moreover, since
$n>(1+\epsilon)\cdot(\log r/\eta_{2})^{2}\geq N(r)$ implies $|\lambda_{n}|<r$,
a short calculation yields that $\log|\lambda_{n}|<\log r$ holds for any
sufficiently large $n$ and $\log
r\in\left(-(1+\epsilon)^{-1/2}\eta_{2}\sqrt{n},\log r_{\epsilon}\right)$,
which implies

$\frac{-\log|\lambda_{n}|}{\sqrt{n}}\geq(1+\epsilon)^{-1/2}\eta_{2}.$

Conversely, $n<(1-\epsilon)\cdot(\log r/\eta_{2})^{2}\leq N(r)$ implies
$|\lambda_{n}|\geq r$, and hence for large $n$ we obtain

$\frac{-\log|\lambda_{n}|}{\sqrt{n}}\leq(1-\epsilon)^{-1/2}\eta_{2}.$

Since the choice of $\epsilon>0$ was arbitrary, assertion $(i)$ follows.

The proof of $(ii)$ is very similar; replacing (25) by

$(1-\epsilon)\cdot\frac{|\log r|}{\eta_{1}}\leq
N(r)\leq(1+\epsilon)\cdot\frac{|\log r|}{\eta_{1}}$ (26)

yields $d=1$, as well as
$\eta_{1}/(1+\epsilon)<-\log|\lambda_{n}|/n<\eta_{1}/(1-\epsilon)$ for
sufficiently large $n$.

Finally, $(iii)$ follows directly by observing that in this case
$f(n)=\delta_{n_{1},0}\delta_{n_{2},0}$. ∎

###### Proof of Lemma 3.27.

The proof follows the same steps as [BJ2, Propositions 3.4 & 3.5]. Let
$J^{*}\colon H_{P,A,-\Gamma}\to H_{P,\alpha,-\gamma}$ denote the Hilbert space
adjoint of $J$, then $J^{*}J$ is diagonal in the orthogonal basis of
monomials, as
$(J^{*}Jp_{n},p_{m})_{H_{P,\alpha,-\gamma}}=(Jp_{n},Jp_{m})_{H_{P,A,-\Gamma}}=(p_{n},p_{m})_{H_{P,A,-\Gamma}}=\omega_{n}(p_{n},p_{m})_{H_{P,\alpha,-\gamma}}$
with $\omega_{n}=\nu_{P,A,-\Gamma}(n)/\nu_{P,\alpha,-\gamma}(n)$. Therefore
the eigenvalues of $J^{*}J$, which are square roots of the singular values of
$J$, are given by $\\{\sqrt{\omega_{n}}:n\in\mathbb{Z}^{2}\\}$, which can be
written as the set of all $\lambda^{n},n\in\mathbb{N}_{0}^{2},$ and
$\mu^{n},n\in\mathbb{N}^{2}$, with $\lambda=e^{\frac{A-\alpha}{2}}$ and
$\mu=e^{\frac{\gamma-\Gamma}{2}}$. Since
$A-\alpha,\gamma-\Gamma\in\mathbb{R}^{2}_{>0}$, Lemma A.3$(i)$ applies, with

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

###### Proof of Lemma 5.3.

We first note that it is sufficient to consider the case
$\operatorname{Tr}M\geq 0$, as the general case easily follows by considering
$-M$ in the opposite case. By [He, Theorem 3], every
$M\in{\operatorname{GL}_{2}}(\mathbb{Z})$ with $\operatorname{Tr}M\geq 0$ is
similar to a so-called ‘standard matrix’ (see [He, Definition 1]), which in
the case of a hyperbolic matrix (implying $\operatorname{Tr}M\neq 0$ and real
eigenvalues $\neq-1,1$) reduces to two (alternative) cases:

  1. (i)

$M$ is similar to $\begin{pmatrix}1&1\\\ 1&0\end{pmatrix}$ in the case
$\operatorname{Tr}M=1$,

  2. (ii)

$M$ is similar to $\begin{pmatrix}a&b\\\ c&d\end{pmatrix},0\leq d\leq b,c<a$
in the case $\operatorname{Tr}M\geq 2$.

We are left to show that every matrix in case (ii) is similar to one of (17).
We note that every matrix of this form satisfies $a>0$, since $c<a\leq 0$
would imply $d=\operatorname{Tr}M-a\geq 2$, and hence $1\geq\det M=ad-
bc\geq(a-c)d\geq 2$, a contradiction. We also note that $c=0$ yields a non-
hyperbolic matrix, so we can assume $c\neq 0$.

For $c>0$ the claim follows immediately from [He, Theorem 4]. For $c<0$, we
note that $0\leq d\leq b$ implies $b>0$ (as otherwise $\det M=0$), and hence
$-bc\geq 1$. Since $ad-bc\leq 1$, it follows that $ad=0$ and hence $d=0$, as
well as $b=1$ and $c=-1$. We obtain that $M$ is similar to a matrix of the
form $M_{a}=\begin{pmatrix}a&1\\\ -1&0\end{pmatrix}$, which is only hyperbolic
for $a\geq 3$. The claim follows by observing that $M_{a},a\geq 3$, is similar
to

$N_{a}=\begin{pmatrix}a-1&1\\\ a-2&1\end{pmatrix}=\begin{pmatrix}1&1\\\
1&0\end{pmatrix}\begin{pmatrix}a-2&1\\\
1&0\end{pmatrix}\in{\operatorname{GL}_{2}}(\mathbb{Z})$

via $M_{a}=C^{-1}N_{a}C$ with $C=\begin{pmatrix}1&0\\\ 1&1\end{pmatrix}$. ∎

##  Appendix B Notes on the special group of toral diffeomorphisms

Here we briefly restate the definition of the group of toral diffeomorphisms
$\mathcal{F}$ from Section 5, before expanding on the various types of maps
that can be constructed within this group via a set of examples. For
$a\in\mathbb{D}$, let $b_{a}\colon{\hat{\mathbb{C}}}\to{\hat{\mathbb{C}}}$ be
an automorphism of $\mathbb{D}$ (or Moebius map) given by

$b_{a}(z)=\frac{z-a}{1-\overline{a}z}.$

Every such map satisfies $b_{a}(\mathbb{T})=\mathbb{T}$, and a straightforward
calculation yields the following lemma.

######  Lemma B.1.

Fix $a=|a|e^{i\alpha}\in\mathbb{D}$ and let $b_{a}$ be as above. For
$M=([0,2\pi]/\sim)$ let $\tilde{b}_{a}\colon M\to M$ be the map determined by
$\pi\circ\tilde{b}_{a}=b_{a}\circ\pi$ with $\pi(\theta)=e^{i\theta}$ for
$\theta\in M$. Then

$\tilde{b}_{a}(\theta)=\theta+g_{a}(\theta),$

with
$g_{a}(\theta)=2\arctan\left(\frac{|a|\sin(\theta-\alpha)}{1-|a|\cos(\theta-\alpha)}\right)$
and
$g^{\prime}_{a}(\theta)=2\left(\frac{|a|\cos(\theta-\alpha)-|a|^{2}}{1-2|a|\cos(\theta-\alpha)+|a|^{2}}\right)>-1$.

We recall the definition of the maps
$F\colon(z_{1},z_{2})\mapsto(z_{1}z_{2},z_{2})$,
$R\colon(z_{1},z_{2})\mapsto(z_{2},z_{1})$,
$I_{kl}\colon(z_{1},z_{2})\mapsto(z_{1}^{1-2k},z_{2}^{1-2l})$ for
$k,l\in\\{0,1\\}$, and
$\mathcal{G}=\\{G_{a,b}\colon(z_{1},z_{2})\mapsto(b_{a}(z_{1}),b_{a}(z_{2})):a,b\in\mathbb{D}\\}$
from Section 5, as well as the definition of $\mathcal{F}$ as the group of
toral diffeomorphisms generated by these maps. The group of toral
automorphisms ${\operatorname{Aut}}(\mathbb{T}^{2})$ is generated by the set
$\Gamma=\\{F,R,I_{01}\\}$. Any $T\in\\{F,I_{00},I_{11}\\}\cup\mathcal{G}$
yields an orientation-preserving diffeomorphism of ${\mathbb{T}^{2}}$, while
all of $\\{I_{01},I_{10},R\\}$ are orientation-reversing. The next lemma
summarises some basic properties of these maps.

######  Lemma B.2.

Let $a,b\in\mathbb{D}$ and $k,l,m,n\in\\{0,1\\}$. The following commutation
relations hold.

  1. (i)

$F\circ I_{kl}=I_{kl}\circ F^{-1}$ and $F^{-1}\circ I_{kl}=I_{kl}\circ F$ for
$k\neq l$,

  2. (ii)

$F\circ I_{11}=I_{11}\circ F$ and $F^{-1}\circ I_{11}=I_{11}\circ F^{-1}$,

  3. (iii)

$R\circ I_{kl}=I_{lk}\circ R$,

  4. (iv)

$I_{kl}\circ I_{mn}=I_{1-\delta_{km},\,1-\delta_{ln}}$,

  5. (v)

$G_{a,b}\circ I_{kl}=I_{kl}\circ
G_{(1-k)a+k\overline{a},\;(1-l)b+l\overline{b}}$,

  6. (vi)

$G_{a,b}\circ R=R\circ G_{b,a}$.

###### Proof.

All statements follow by direct computation, with part (v) using
$b_{a}(z^{-1})=b_{\overline{a}}(z)^{-1}$. ∎

We now provide examples from various interesting sub-classes of
diffeomorphisms in $\mathcal{F}$.

##### Anosov diffeomorphisms in $\mathcal{F}$.

For $a\in\mathbb{D}$ let $T_{a}\colon{\mathbb{T}^{2}}\to{\mathbb{T}^{2}}$ be
given by

$T_{a}=G_{0,-a}\circ F\circ R\circ G_{a,0}.$

Using Lemma B.1, we can see that the derivative of the respective map
$\tilde{T}_{a}\colon M\to M$ is given by

$D\tilde{T}_{a}(x)=\begin{pmatrix}s_{a}(x)&1\\\ 1&0\end{pmatrix},$

with $s_{a}(x)=(1+g_{a}^{\prime}(x_{1}))>0$ for all $x=(x_{1},x_{2})\in M$.

######  Example B.3.

  1. (i)

For $a=0$ the map $T_{0}(z)=(z_{1}z_{2},z_{1})$ is an Anosov automorphism
induced by $\begin{pmatrix}1&1\\\ 1&0\end{pmatrix}$ with eigenvalues
$\lambda_{u/s}=\varphi^{\pm 1}$ where $\varphi=(1+\sqrt{5})/2$ is the golden
mean.

  2. (ii)

One can check that $T_{a}(z)=(b_{a}(z_{1})z_{2},z_{1})$ is Anosov for all
$a\in\mathbb{D}$ by finding suitable cone fields (see Definition 2.11).

  3. (iii)

The maps $T_{b}\circ T_{a}$ given by

$(T_{b}\circ T_{a})(z)=(b_{b}(b_{a}(z_{1})z_{2})z_{1},b_{a}(z_{1})z_{2})$

are Anosov for all $a,b\in\mathbb{D}$, and in fact
$\mathbb{R}^{2}_{>0}\cup\mathbb{R}^{2}_{<0}$ can be chosen as the invariant
expanding cone, and its complementary cone as the invariant contracting one.
These are the maps considered in [SBJ] and [PoS].

The maps in (iii) are orientation-preserving while the ones in (i) and (ii)
are orientation-reversing.

##### Area-preserving diffeomorphisms in $\mathcal{F}$.

######  Lemma B.4.

Let $F_{a}=G_{0,-a}\circ F\circ G_{0,a}$ for $a\in\mathbb{D}$. Then any finite
composition of the elements of
$\Gamma_{ap}=\Gamma\cup\\{F_{a}:a\in\mathbb{D},k\in\mathbb{N}\\}$ is area-
preserving.

###### Proof.

For $z\in{\mathbb{T}^{2}}$ we have
$F_{a}(z_{1},z_{2})=(z_{1}b_{a}(z_{2}),z_{2})$. As $b_{a}$ preserves
$\mathbb{T}$ we have $|\det{DF_{a}}(z)|=|b_{a}(z_{2})|=1$ for all
$z\in{\mathbb{T}^{2}}$. As all elements in $\Gamma_{ap}$ are area-preserving,
so is their composition. ∎

######  Example B.5.

  1. (i)

The maps in Example B.3 are area-preserving Anosov diffeomorphisms as
$T_{a}=G_{0,-a}\circ F\circ G_{0,a}\circ R=F_{a}\circ R$ with $F_{a}$ as in
Lemma B.4.

  2. (ii)

The map $T=F\circ R\circ G_{0,a}\circ F\circ R$ given by

$T(z_{1},z_{2})=(z_{1}b_{a}(z_{1})z_{2},z_{1}z_{2})$

is not area-preserving for $a\in\mathbb{D}\setminus\\{0\\}$ as
$|\det{DT}(z)|=|z^{2}_{1}b_{a}^{\prime}(z_{1})z_{2}|=|b_{a}^{\prime}(z_{1})|$
for $z\in{\mathbb{T}^{2}}$.

  3. (iii)

The map $T=F\circ R\circ G_{0,-a}\circ F\circ R\circ G_{a,b}$ given by

$T(z_{1},z_{2})=(z_{1}b_{a}(z_{1})b_{b}(z_{2}),b_{a}(z_{1})b_{b}(z_{2})),\quad(a\in\mathbb{D}),$

is area-preserving for $b=0$ (see Example B.3(ii)), but is not area-preserving
for $b\neq 0$.

##### Maps with symmetries.

An automorphism $T$ of some topological space is said to have a symmetry if
there exists an automorphism $H$ so that

$H^{-1}\circ T\circ H=T,$

and to have a reversing symmetry if there exists an automorphism $H$ so that

$H^{-1}\circ T\circ H=T^{-1}.$

Clearly, for any rational map $T$ with only real coefficients, Corollary
4.2$(i)$ implies that $I_{11}$ is a symmetry of $T$, which by Theorem 1.2 and
its proof induces symmetry relations on the resonances.

On the other hand, presence of reversing symmetries is of considerable
interest in classical and quantum mechanics. For systems with time-reversal
symmetry the reverse motion satisfies the same laws of motion as the forward
motion. Usually this time-reversal symmetry corresponds to a particular
involution map $H$. However, the notion of reversible systems was extended to
include all involutions and even non-involutary reversing symmetries, see for
example the survey [LR] adapted to dynamical systems or [BR] specifially for
toral automorphisms. We will next present some Anosov diffeomorphisms in
$\mathcal{F}$ which have reversing symmetries.

######  Lemma B.6.

Let $k\in\mathbb{N}$ and $a\in(0,1)$, and define the maps $T_{k,a}=F^{k}\circ
R\circ G_{a,-a}\circ F^{k}\circ R$ and $U_{k,a}=G_{0,-a}\circ T_{k,a}\circ
G_{a,0}$.

  1. (i)

The map $T_{k,a}$ is a non-area-preserving Anosov diffeomorphism with a
reversing symmetry.

  2. (ii)

The map $U_{k,a}$ is an area-preserving Anosov diffeomorphism with a reversing
symmetry.

###### Proof.

Using Lemma B.1 it is not difficult to see that both $T_{k,a}$ and $U_{k,a}$
are Anosov with the first and third quadrant of $\mathbb{R}^{2}$ forming an
unstable, and the second and forth quadrant forming a stable invariant cone.
Area preservation of $U_{k,a}$ and non-preservation for $T_{k,a}$ can be
computed directly, noting that $U_{k,a}=(G_{0,-a}\circ F^{k}\circ R\circ
G_{a,0})^{2}$. For the symmetries, a calculation with $H=I_{01}\circ R$ using
Lemma B.2 and the fact that $a\in\mathbb{R}$ reveals that $H^{-1}\circ
T_{k,a}^{-1}\circ H=T_{k,a}$. Since $G_{0,a}\circ H=H\circ G_{a,0}$ we also
have $H^{-1}\circ U_{k,a}^{-1}\circ H=U_{k,a}$. ∎

##### Comparison to Blaschke product diffeomorphisms.

In [PS] the authors coined the notion of Blaschke product diffeomorphisms,
which are maps of the form

$T(z_{1},z_{2})=(A(z_{1})B(z_{2}),C(z_{1})D(z_{2})),$

where $A,B,C,D$ are Blaschke products in one variable. They state that these
are precisely the analytic maps on a neighbourhood of the open bidisk
${\mathbb{D}^{2}}$, mapping ${\mathbb{D}^{2}}$ to itself and
${\mathbb{T}^{2}}$ diffeomorphically to itself, and provide an explanation of
this in [PS, Remark 5.2]. Here we observe that this claim is inaccurate: while
Examples B.5(ii)-(iii) are instances of Blaschke product diffeomorphisms,
Example B.3(iii) is a hyperbolic diffeomorphism in $\mathcal{F}$ containing
Blaschke factors of a product of two variables, and cannot be written as a
Blaschke product diffeomorphism.

##### Acknowledgements

All authors gratefully acknowledge the support for the research presented in
this article by the EPSRC grant EP/RO12008/1. JS gratefully acknowledges the
support by the ERC grant 833802-Resonances.

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