# Anisotropic spaces and nilmanifold automorphisms

Oliver Butterley  (Oliver Butterley) Department of Mathematics, University of
Rome Tor Vergata – Via della Ricerca Scientifica – Roma – 00133 – Italy
<EMAIL_ADDRESS>and  Minsung Kim  (Minsung Kim) Centro di Ricerca
Matematica Ennio De Giorgi, Scuola Normale Superiore, Piazza dei Cavalieri 3,
56126 Pisa, Italy<EMAIL_ADDRESS>
###### Abstract.

We introduce anisotropic Banach spaces on Heisenberg nilmanifolds and study
the resonance spectrum associated to partially hyperbolic automorphisms. In
this work we describe an alternative proof of the spectrum which takes
advantage of geometric-style anisotropic norms.

##  1\. Introduction and results

Let $\mathbb{H}$ be the three-dimensional Heisenberg group, that is the Lie
group equal to $\mathbb{R}^{3}$ with the group law

$(x,y,z)*(x^{\prime},y^{\prime},z^{\prime})=(x+x^{\prime},y+y^{\prime},z+z^{\prime}+\tfrac{1}{2}(xy^{\prime}-x^{\prime}y)).$

The quotient space $M=\raisebox{-1.99997pt}{$\Gamma$}\diagdown\mathbb{H}$,
where $\Gamma$ is a discrete subgroup of $\mathbb{H}$, is a compact manifold
called the Heisenberg nilmanifold. Let $(V,W,Z)$ be elements of the Lie
algebra associated to $\mathbb{H}$ which satisfy the Heisenberg commutation
relations $[V,W]=Z$ and $[V,Z]=[W,Z]=0$. The object of our interest is
$\Phi:M\to M$, an automorphism such that, for some $\lambda>1$,

(1) $\Phi_{*}V=\lambda^{-1}V,\quad\Phi_{*}W=\lambda W,\quad\Phi_{*}Z=Z.$

We say that such an automorphism is _partially hyperbolic_ since it exhibits
contraction, expansion and neutral behaviour. Let $\nu$ denote the probability
measure which is inherited from the Haar measure on $\mathbb{H}$ and which is
preserved by $\Phi$. See Appendix A for the explicit construction of partially
hyperbolic automorphisms. Since $\Phi$ is an isometry in the $Z$ direction it
makes sense to define, for all $N\in\mathbb{Z}$ (i.e., Fourier decomposition
in the $Z$ direction),

(2) $C_{N}^{\infty}(M)=\\{h\in C^{\infty}(M):Zh=2\pi iNh\\}.$

Observe that $h\mapsto h\circ\Phi$ leaves $C_{N}^{\infty}(M)$ invariant.

######  Definition 1.1.

Let $\Xi=\\{\xi_{j}\\}_{j}$ be a set of complex numbers, each with associated
non-negative integer $d_{j}$. We say that $\Phi$ has _resonance spectrum_
$\Xi$ on $\mathcal{C}^{\infty}_{N}(M)$ if, for any
$g,h\in\mathcal{C}^{\infty}_{N}(M)$ and for any $\epsilon>0$,

$\int g\cdot h\cdot\Phi^{n}\
d\nu=\sum_{\left|\lambda_{j}\right|\geq\epsilon}\sum_{k=0}^{d_{j}}\xi_{j}^{n}n^{k}C_{j,k}(g,h)+\operatorname{\mathnormal{o}}(\epsilon^{n})$

where $C_{j,k}(g,h)$ are finite rank, bilinear and non-zero.

This definition matches the one used by Faure, Gouëzel & Lanneau [8,
Definition 1.1]. The resonance spectrum of $\Phi$ on
$\mathcal{C}^{\infty}_{0}(M)$ is equal to $\\{0\\}$ because, in the case
$N=0$, the system reduces to the study of a toral automorphism and the
resonance spectrum can be shown by considering a fourier series decomposition
on the torus.

It is known [11, Proposition 2.1] that, up to an automorphism of $\mathbb{H}$,
every discrete subgroup $\Gamma$ is of the form
$\Gamma_{K}=\left\\{(x,y,z)\in\mathbb{H}:x,y\in\mathbb{Z},K\mid y\right\\}$
for some $K\in\mathbb{N}$. Let $K$ be thus fixed.

######  Theorem 1.2.

Let, as above, $\Phi:M\to M$ be a partially hyperbolic automorphism on a
Heisenberg nilmanifold. For each $N\neq 0$ there exists a set of unit complex
numbers ${\\{\mu_{j}\\}}_{j=1}^{K\left|N\right|}$ such that the resonance
spectrum of $\Phi$ on $\mathcal{C}^{\infty}_{N}(M)$ is equal to

$\Xi_{N}=\big{\\{}\lambda^{-(\frac{1}{2}+n)}\mu_{j}:1\leq j\leq
K\left|N\right|,n\in\mathbb{N}_{0}\big{\\}}.$

This result isn’t new, it is contained in the work of Flaminio & Forni [10] on
nilflows since the automorphisms considered here correspond to periodic
renormalization of nilflows and in the work of Faure & Tsujii [9] (under the
name of “prequantum transfer operator”, see also [7]). The first of these
references has the advantage that the full flow is studied, not only the
periodic case. The second of these references has the advantage that, to some
degree, also the non-affine case is permitted and results obtained.

Nonetheless various related problems remain to be properly understood (see,
e.g., [5, 4, 14, 3, 13, 1, 19]). In particular the possibility to extend the
techniques of anisotropic spaces to the cocycle setting and to fully extend
results to the non-algebraic systems. A particular advantage of the
anisotropic Banach spaces is utilized in Section 5 where the full spectrum is
obtained from the peripheral spectrum. This takes the place of an argument
concerning the formal inverse of a given operator in the work of Flaminio &
Forni (see [10, A.3]). Potentially the exploration of anisotropic Banach
spaces in this setting will be an aid for further pushing the ideas to more
settings.

### Overview

In order to prove the theorem we will study the transfer operator on a family
of anisotropic Banach spaces denoted $\mathcal{B}^{p,q}_{N}(M)$. Determining
the resonance spectrum is equivalent to determining the spectrum of the
transfer operator on $\mathcal{B}^{p,q}_{N}(M)$ for $p,q$ sufficiently large.
In Section 2 the anisotropic norms are introduced and various basic properties
are proven. In particular it is shown that the spaces are embedded into the
space of distributions (Lemma 2.2), the operators $V$ and $W$ are continuous
on these spaces (Lemmas 2.5 & 2.6) and that the kernel of $W$ is null (Lemma
2.7). The last fact will be useful later because we see that the operators $V$
and $W$ map from one eigenspace to another. This is arguably the main theme of
the present strategy. In Section 3 the essential spectral radius of the
transfer operator is estimated (Lemma 3.7). That the essential spectral radius
can be made as small as desired means that we can then work just with isolated
spectrum which has many advantages. Section 4 is devoted to studying the
spectrum of the transfer operator restricted to $\ker V$ and it is shown that
the spectrum is a finite set of complex numbers, all with absolute value equal
to $\lambda^{-\frac{1}{2}}$ (Lemma 4.4). Using all the previously established
details, in Section 5 we then show that the rest of the spectrum is a scaled
version of the peripheral spectrum (Lemma 5.3).

##  2\. Anisotropic spaces

In this section we define norms on $\mathcal{C}_{N}^{\infty}(M)$ and then
define Banach spaces by completion with respect to the norms. We also explore
various basic properties of these norms. The norms used are similar to the
“geometric style” anisotropic norms often used for hyperbolic systems [15, 16,
17] and those used by Faure, Gouëzel & Lanneau [8] for pseudo Anosov maps.

We consider, as introduced in Section 1, $\Phi:M\to M$ and $N\in\mathbb{N}$
fixed for the remainder of this section. The three elements of the basis of
the Lie algebra $(V,W,Z)$ can each be seen as vector fields on $M$. We will
use the same notation for the vector fields since the meaning is clear from
the context. These three vector fields define a splitting of tangent space
which is invariant (1) under the action of the partially hyperbolic
transformations which we study. The _stable_ and _unstable_ bundles are
respectively the ones corresponding to $V$ and $W$. Each vector field defines
a foliation which is naturally orientated. For each $u\in\mathbb{R}$ let

$\varphi_{u}^{V}:M\to M$

be defined as sliding a distance $u$ along the $V$-foliation. Similarly
$\varphi_{u}^{W}$ and $\varphi_{u}^{Z}$ are defined. In this Lie group context
we could also have written $(x,y,z)\mapsto(x,y,z)*e^{uV}$ for
$\varphi_{u}^{V}$, etc.

We fix $\delta>0$, once and for all, sufficiently small that there exists a
covering of $M$ into sets of diameter not greater than $\delta$ and,
subordinated to which, there exists a smooth partition of unity. For
notational convenience let $I_{\delta}=(-\delta,\delta)$ and let
$\mathcal{C}^{\infty}_{\operatorname{c}}(I_{\delta})$ denote the set of
$\mathcal{C}^{\infty}$ functions with support compactly contained in the
interval $I_{\delta}$. For $h\in\mathcal{C}_{N}^{\infty}(M)$ and
$\eta\in\mathcal{C}^{\infty}_{\operatorname{c}}(I_{\delta})$ and $m\in M$, let

(3) $\ell_{\eta,m}(h)=\int_{-\delta}^{\delta}\eta(u)\cdot
h\circ\varphi_{u}^{W}(m)\ du.$

For $p\in\mathbb{N}_{0}$, $q\in\mathbb{N}$ we define a norm on
$\mathcal{C}_{N}^{\infty}(M)$,

$\left\|h\right\|_{p,q}=\sup\left\\{\left|\ell_{\eta,m}(V^{j}h)\right|:0\leq
j\leq p,m\in
M,\eta\in\mathcal{C}^{\infty}_{\operatorname{c}}(I_{\delta}),\left\|\eta\right\|_{\mathcal{C}^{q}}\leq
1\right\\}.$

We then define the Banach space $\mathcal{B}_{N}^{p,q}(M)$ as the completion
of $\mathcal{C}_{N}^{\infty}(M)$ with respect to this norm. Essentially this
norm requires the function to be rather smooth in the $V$-direction and allows
rather bad behaviour in the $W$-direction.

Suppose that $g\in\mathcal{C}^{\infty}(M)$ with support in a
$\delta$-neighbourhood of a point $m\in M$. Then, for notational convenience,
we understand $\eta_{g,m}(h)$ to be equal to $\ell_{\eta,m}(h)$, as defined
above (3), where $\eta(u)=g\circ\varphi_{u}^{V}(m)$. I.e., we can consider $g$
on $M$ or in local chart along the unstable curve.

These linear functionals behave continuously in the $V$-direction in the
following sense.

######  Lemma 2.1.

Suppose that $m,m^{\prime}\in M$ are $\epsilon$-close and that
$g\in\mathcal{C}_{-N}^{\infty}(M)$ has support within a $\delta$-neighbourhood
of both $m$ and $m^{\prime}$. Then, for all $h\in\mathcal{C}^{\infty}_{N}(M)$,
$q\geq 2$,

$\left|\ell_{g,m}(h)-\ell_{g,m^{\prime}}(h)\right|\leq
2\epsilon\left\|g\right\|_{\mathcal{C}^{q}}\left\|h\right\|_{1,q-1}.$

###### Proof.

Observe that $(g\cdot h)\in\mathcal{C}^{\infty}_{0}(M)$. We have that
$m^{\prime}=\varphi_{a}^{V}\circ\varphi_{c}^{Z}(m)$ and so,

$(g\cdot h)\circ\varphi_{u}^{W}(m^{\prime})=(g\cdot
h)\circ\varphi_{u}^{W}\circ\varphi_{a}^{V}\circ\varphi_{c}^{Z}(m)=(g\cdot
h)\circ\varphi_{a}^{V}\circ\varphi_{u}^{W}(m).$

Additionally, by definition of $V$,

$(g\cdot h)\circ\varphi_{a}^{V}(\cdot)=(g\cdot h)(\cdot)+\int_{0}^{a}V(g\cdot
h)\circ\varphi_{s}^{V}(\cdot)\ ds.$

Consequently,

$\displaystyle\int(g\cdot h)\circ\varphi_{u}^{W}(m^{\prime})\ du$
$\displaystyle=\int(g\cdot h)\circ\varphi_{u}^{W}(m)\ du$ $\displaystyle\
+\int_{0}^{a}\left(\int(g\cdot Vh)\circ\varphi_{s}^{V}\circ\varphi_{u}^{W}(m)\
du\right)\ ds$ $\displaystyle\ +\int_{0}^{a}\left(\int(Vg\cdot
h)\circ\varphi_{s}^{V}\circ\varphi_{u}^{W}(m)\ du\right)\ ds.$

The final two terms are bounded from above in absolute value by

$\left|a\right|\left\|g\right\|_{\mathcal{C}^{q}}\left\|h\right\|_{1,q}+\left|a\right|\left\|g\right\|_{\mathcal{C}^{q}}\left\|h\right\|_{0,q-1},$

taking advantage of the definition of the norm. ∎

The following lemma means that one can identify $\mathcal{B}_{N}^{p,q}(M)$
with a space of distributions. For any $r\in\mathbb{N}$ denote by
$\mathcal{D}^{r}_{N}(M)$ the elements of the dual space
$\mathcal{C}^{r}(M)^{\prime}$ which have support in
$\mathcal{C}^{\infty}_{N}(M)$.

######  Lemma 2.2.

Let $p\in\mathbb{N}_{0}$, $q\in\mathbb{N}$. There is a map
$\iota:\mathcal{B}_{N}^{p,q}(M)\to\mathcal{D}_{N}^{q}(M)$ is continuous and
injective.

###### Proof.

The map $\iota$ extends the canonical inclusion
$\mathcal{C}^{\infty}_{N}(M)\rightarrow\mathcal{D}_{N}^{\infty}(M)$ given by
$\langle\iota(h),g\rangle=\int_{M}h\cdot g$. Using the previously mentioned
smooth partition of unity of $M$ and then decomposing the integral into leaves
of the form $\left\\{\varphi_{u}^{W}(m):u\in(-\delta,\delta)\right\\}$ we show
that there exists $C>0$ such that, for all $h\in\mathcal{C}_{N}^{\infty}(M)$,

$\int_{M}h\cdot g\leq
C\left\|h\right\|_{p,q}\left\|g\right\|_{\mathcal{C}^{p}(M)}.$

Considering the completion, this shows that any $h\in\mathcal{B}_{N}^{p,q}(M)$
gives a distribution on $M$ of order at most $p$. That $\iota$ is injective
follows in the same way as the argument of, e.g., Gouëzel & Liverani [16,
Proposition 4.1]. We start by taking $h\in\mathcal{B}_{N}^{p,q}(M)$, $h\neq
0$. Then we deduce that there exists some $m\in M$ such that the distribution
$\eta\mapsto\int_{-\delta}^{\delta}\eta(u)\cdot h\circ\varphi_{u}^{W}(m)\ du$
is non-zero. We finally use this $\eta$ to construct a
$\tilde{\eta}\in\mathcal{C}^{\infty}(M)$ such that $\langle\iota
h,\tilde{\eta}\rangle\neq 0$ and hence $\iota(h)\in\mathcal{D}_{N}^{q}(M)$ is
non-zero as required. ∎

######  Lemma 2.3.

Let $p,p^{\prime}\in\mathbb{N}_{0}$, $q,q^{\prime}\in\mathbb{N}$ such that
$p^{\prime}\leq p$ and $q^{\prime}\geq q$. There is a continuous inclusion
$\mathcal{B}_{N}^{p,q}(M)\subseteq\mathcal{B}_{N}^{p^{\prime},q^{\prime}}(M)$.

###### Proof.

Let $h\in\mathcal{C}^{\infty}_{N}(M)$. By definition

$\left\|h\right\|_{p^{\prime},q^{\prime}}=\sup\left\\{\left|\ell_{\eta,m}(V^{j}h)\right|:0\leq
j\leq p^{\prime},m\in M,\left\|\eta\right\|_{\mathcal{C}^{q^{\prime}}}\leq
1\right\\}.$

Observe that $j\leq p^{\prime}$ implies that $j\leq p$ and
$\left\|\eta\right\|_{\mathcal{C}^{q^{\prime}}}\leq 1$ implies that
$\left\|\eta\right\|_{\mathcal{C}^{q}}\leq 1$ because $p^{\prime}\leq p$ and
$q^{\prime}\geq q$. Consequently the above term is bounded by
$\left\|h\right\|_{p,q}$. By density the full result follows. ∎

The following compact embedding result and argument is very similar to the one
appearing in other works using geometric-style anisotropic space (e.g., [16,
§5] and [8, §2.2]).

######  Lemma 2.4.

Let $p,p^{\prime}\in\mathbb{N}_{0}$, $q,q^{\prime}\in\mathbb{N}$,
$p^{\prime}>p$, $q^{\prime}<q$. Then, the inclusion
$\mathcal{B}_{N}^{p^{\prime},q^{\prime}}(M)\subset\mathcal{B}_{N}^{p,q}(M)$ is
compact.

###### Proof.

By Lemma 2.3 it suffices to prove the result for the case $p^{\prime}=p+1$,
$q^{\prime}=q-1$ and by density it suffices to work with
$h\in\mathcal{C}^{\infty}_{N}(M)$. We will show that, for each $\epsilon>0$
sufficiently small and given $N>0$, there exist $N_{\epsilon}>0$ and a finite
set ${\\{\rho_{k}\\}}_{k=1}^{N_{\epsilon}}$ of linear functionals on
$\mathcal{B}_{N}^{p+1,q-1}(M)$ such that, for any
$h\in\mathcal{C}^{\infty}_{N}(M)$,

(4) $\left\|h\right\|_{p,q}\leq\sup_{1\leq k\leq
N_{\epsilon}}\left|\rho_{k}(h)\right|+\epsilon\left\|h\right\|_{p+1,q-1}.$

This implies the claimed compactness (see, e.g., [8, Proof of Prop. 2.8]).

Fix $0<\epsilon<\frac{\delta}{4}$. Let ${\\{m_{k}\\}}_{k=1}^{N_{\epsilon}}$
denote a finite set of points in $M$ such that every point of $M$ is
$\epsilon$-close to at least one of the $m_{k}$. Moreover, using the
compactness of $\mathcal{C}^{q}$ in $\mathcal{C}^{q-1}$, choose a finite set
${\\{\zeta_{k}\\}}_{k=1}^{N_{\epsilon}}$ of functions in
$\mathcal{C}^{\infty}_{\operatorname{c}}(I_{\delta})$ such that, for every
$\eta$ in the unit ball of $\mathcal{C}^{q}_{\operatorname{c}}(I_{\delta})$,
there exists some $\zeta_{k}$ in the set such that
$\left\|\eta-\zeta_{k}\right\|_{\mathcal{C}^{q-1}}\leq\epsilon$.

We need to estimate integrals of the form,

$\int\eta(u)\cdot V^{j}h\circ\varphi_{u}^{W}(m)\ du,\quad 0\leq j\leq p.$

For any $m\in M$, there exists $m_{k}$ in the chosen finite set such that $m$
and $m_{k}$ are $\epsilon$-close. We can extend $\eta\circ\varphi^{W}_{u}(m)$
to a neighbourhood of $m$ in such a way that we can apply Lemma 2.1.
Consequently, paying a price of
$\epsilon\left\|V^{j}h\right\|_{1,q-1}\leq\epsilon\left\|h\right\|_{p+1,q-1}$,
it suffices to consider only integrals centred at some $m_{k}$.

Summarising what we have shown so far, we know the existence of a constant
$C>0$ such that

$\displaystyle\left|\int\eta(u)\cdot V^{j}h\circ\varphi_{u}^{W}(m)\ du\right|$
$\displaystyle\leq\sup_{k,\widetilde{\eta}}\left|\int\widetilde{\eta}(u)\cdot
V^{j}h\circ\varphi_{u}^{W}(m_{k})\ du\right|$ $\displaystyle\ +\epsilon
C\left\|h\right\|_{p+1,q-1}.$

To finish, we recall that there exists some
$\zeta_{k^{\prime}}\in{\\{\zeta_{k}\\}}_{k=1}^{N_{\epsilon}}$ with the
property that
$\left\|\widetilde{\eta}-\zeta_{k^{\prime}}\right\|_{\mathcal{C}^{q-1}}\leq\epsilon$.
This means that

$\left|\int\widetilde{\eta}(u)\cdot V^{j}h\circ\varphi_{u}^{W}(m_{k})\
du\right|$

is bounded from above by

$\left|\int\zeta_{k^{\prime}}(u)\cdot V^{j}h\circ\varphi_{u}^{W}(m_{k})\
du\right|+\left|\int(\widetilde{\eta}-\zeta_{k^{\prime}})(u)\cdot
V^{j}h\circ\varphi_{u}^{W}(m_{k})\ du\right|\\\
\leq\left|\int\zeta_{k^{\prime}}(u)\cdot V^{j}h\circ\varphi_{u}^{W}(m_{k})\
du\right|+\epsilon C\left\|h\right\|_{p+1,q-1}.$

As required (4), we have proved that the existence of a finite number of
continuous linear functional $(\rho_{k})_{1\leq k\leq N_{\epsilon}}$ on
$\mathcal{B}_{N}^{p,q}$. ∎

######  Lemma 2.5.

Let $p,q\in\mathbb{N}$. The operator
$V:\mathcal{B}_{N}^{p,q}(M)\to\mathcal{B}_{N}^{p-1,q}(M)$ is continuous.111In
the sense that $V$ extends to a continuous operator on these spaces and,
abusing notation, we use the same symbol for the extension.

###### Proof.

Let $h\in\mathcal{C}^{\infty}_{N}(M)$. Since $[V,Z]=0$ we know that
$Vh\in\mathcal{C}^{\infty}_{N}(M)$. Further observe that

$\displaystyle\left\|Vh\right\|_{p-1,q}$
$\displaystyle=\sup\left\\{\left|\ell_{\eta,m}(V^{j}(Vh))\right|:0\leq j\leq
p-1,m\in M,\left\|\eta\right\|_{\mathcal{C}^{q}}\leq 1\right\\}$
$\displaystyle=\sup\left\\{\left|\ell_{\eta,m}(V^{j}h)\right|:1\leq j\leq
p,m\in M,\left\|\eta\right\|_{\mathcal{C}^{q}}\leq 1\right\\}$
$\displaystyle\leq\left\|h\right\|_{p,q}.$

By density the full result follows. ∎

######  Lemma 2.6.

Let $p\in\mathbb{N}_{0}$, $q\in\mathbb{N}$. The operator
$W:\mathcal{B}_{N}^{p,q}(M)\to\mathcal{B}_{N}^{p,q+1}(M)$ is continuous.

###### Proof.

Let $h\in\mathcal{C}_{N}^{\infty}$. Let $\eta\in\mathcal{C}^{\infty}$ and
$\left\|\eta\right\|_{\mathcal{C}^{q+1}}\leq 1$. Assume $p\geq 1$. By
commutation relations of $V,W$ and integrating by parts,

$\displaystyle\int\eta(u)\cdot(V^{p}Wh)\circ\phi_{u}^{W}(m)du$
$\displaystyle=\int\eta(u)\cdot
V^{p-1}W(Vh)\circ\phi_{u}^{W}(m)du+\int\eta(u)\cdot
V^{p-1}Zh\circ\phi_{u}^{W}(m)du$ $\displaystyle=\int\eta(u)\cdot
W(V^{p}h)\circ\phi_{u}^{W}(m)du+p\int\eta(u)\cdot
V^{p-1}Zh\circ\phi_{u}^{W}(m)du$ $\displaystyle=-\int\eta^{\prime}(u)\cdot
V^{p}h\circ\phi_{u}^{W}(m)du+p\int\eta(u)\cdot
V^{p-1}Zh\circ\phi_{u}^{W}(m)du.$

Therefore,

$\displaystyle\left|\int\eta(u)\cdot(V^{p}Wh)\circ\phi_{u}^{W}(m)du\right|$
$\displaystyle\leq\left\|V^{p}h\right\|_{0,q}+p\left\|V^{p-1}(Zh)\right\|_{0,q+1}$
$\displaystyle\leq(2\pi iNp+1)\left\|h\right\|_{p,q+1}.$

∎

######  Lemma 2.7.

Suppose that $N\neq 0$, $h\in\mathcal{B}^{p,q}_{N}(M)$ and $Wh=0$. Then $h=0$.

###### Proof.

Recall that the norm is based on linear functions of the form

$\ell_{\eta,m}(h)=\int\eta(u)\cdot h\circ\varphi_{u}^{W}(m)\ du$

and these linear functionals extend to $\mathcal{B}^{p,q}_{N}(M)$. Using the
assumption $Wh=0$ we deduce that $\ell_{\eta,m}(h)=\ell_{\eta_{s},m}(h)$ where
$\eta_{s}(u)=\eta(u-s)$ is the translation of any $\eta$. In other words, the
distribution induced by $h$ on the leaf
$\\{\varphi_{u}^{W}(m):u\in\mathbb{R}\\}$ is translation invariant. This means
that the distribution is equal to a constant multiple of Lebesgue measure
along the leaf.222Let $\varphi\in\mathcal{S}^{\prime}(\mathbb{R})$ such that
$\varphi(f^{\prime})=0$ for any $f\in\mathcal{S}(\mathbb{R})$. Fix $\gamma\in
S(\mathbb{R})$ with $Leb(\gamma)=1$. Set
$\tilde{g}(u)=\int_{-\infty}^{u}g(t)-\gamma(t)Leb(g)dt$. Then
$\tilde{g}\in\mathcal{S}(\mathbb{R})$. Since
$\tilde{g}^{\prime}(u)=g(u)-\gamma(u)Leb(g)$ and
$\varphi(\tilde{g}^{\prime})=0$, $\varphi(g)=\varphi(\gamma)Leb(g)$. Such leaf
are dense in $M$. Moreover, the quantity $\ell_{\eta,m}(h)$ varies
continuously as $m$ moves in the $V$-direction as shown in Lemma 2.1.
Consequently $h$ is equal to a constant multiple of Lebesgue measure on the
entire of $M$. However, since $N\neq 0$, this contradicts the oscillating
behaviour of $h$ in the $Z$-direction. ∎

A key characteristic of the present setting is that Lemma 2.7 doesn’t hold
when $W$ is replaced with $V$. Indeed, as we will see in Section 4, the set
$\\{h\in\mathcal{B}^{p,q}_{N}(M):Vh=0\\}$ is non-empty. This is one of the
consequences of the anisotropic spaces used in the present setting and
essential for this analysis.

##  3\. Transfer operator and norm estimates

This section is devoted to proving several key properties of the transfer
operator in relation to the the anisotropic norms. In particular that the
transfer operator is quasi compact. Again we consider
$M=\raisebox{-1.99997pt}{$\Gamma$}\diagdown\mathbb{H}$, $N\in\mathbb{Z}$ and
$\Phi:M\to M$, as introduced in Section 1, chosen and fixed for the entire
section.

We consider the linear operator
$\mathcal{L}:\mathcal{C}^{\infty}(M)\to\mathcal{C}^{\infty}(M)$, given by

(5) $\mathcal{L}:h\mapsto h\circ\Phi$

and which we call the _transfer operator_.

######  Remark.

In the present affine setting it doesn’t make a difference, apart from a
simple scaling, if we consider the operator corresponding to the measure of
maximal entropy or the one corresponding to the SRB measure. Moreover, since
the system is invertible, we could consider $h\mapsto h\circ\Phi^{-1}$ and
swap $V$ and $W$.

######  Lemma 3.1.

For all $j,k\in\mathbb{N}$, $h\in\mathcal{C}^{\infty}_{N}(M)$,

$V^{j}\mathcal{L}^{k}h=\lambda^{-jk}\mathcal{L}^{k}\left(V^{j}h\right)\quad\text{and}\quad
W^{j}\mathcal{L}^{k}h=\lambda^{jk}\mathcal{L}^{k}\left(W^{j}h\right).$

###### Proof.

We know that (1),

$V\mathcal{L}^{k}h=V(h\circ\Phi^{k})=(Vh\circ\Phi^{k})\lambda^{-k}=\lambda^{-k}\mathcal{L}\left(Vh\right)$

and so $V\mathcal{L}^{k}h=\lambda^{-k}\mathcal{L}^{k}\left(Vh\right)$.
Iterating this leads to the full result. Similarly for $W$. ∎

The above result extends to all spaces on which $V$, $W$ and $\mathcal{L}$
extend continuously, a fact which will soon be useful.

For convenience, for all $j\in\mathbb{N}_{0}$, $q\in\mathbb{N}$, we define the
semi-norm on $\mathcal{C}_{N}^{\infty}(M)$,

(6) $\left|h\right|_{j,q}=\sup\left\\{\left|\ell_{\eta,m}(V^{j}h)\right|:m\in
M,\eta\in\mathcal{C}^{\infty}_{\operatorname{c}}(I_{\delta}),\left\|\eta\right\|_{\mathcal{C}^{q}}\leq
1\right\\}.$

This definition has the consequence that $\left\|h\right\|_{p,q}=\sup_{0\leq
j\leq p}\left|h\right|_{j,q}$.

The norms are defined based on $\delta>0$. The following lemma clarifies the
dependence of the norm on this choice.

######  Lemma 3.2.

There exists $C>0$ such that, for all
$\eta\in\mathcal{C}^{\infty}(\mathbb{R})$ supported in some interval
$A\subset\mathbb{R}$ of length $\left|A\right|$ and for all
$h\in\mathcal{C}^{\infty}_{N}(M)$, $q\in\mathbb{R}$,

$\left|\int\eta(u)\cdot h\circ\varphi_{u}^{W}(m)\ du\right|\leq
C\left|A\right|\left\|\eta\right\|_{\mathcal{C}^{q}}\left|h\right|_{0,q}.$

###### Proof.

If $\left|A\right|$ is less than $2\delta$ then the conclusion is immediate so
we assume the contrary. Using a smooth partition of unity we write
$\eta=\sum_{i=1}^{3\left|A\right|}\tilde{\eta}_{i}$ where each
$\tilde{\eta}_{i}$ is supported on an interval smaller that $2\delta$ and
$\left\|\tilde{\eta}_{i}\right\|_{\mathcal{C}^{q}}\leq C$ for a uniform
constant $C>0$. ∎

######  Lemma 3.3.

For all $q\in\mathbb{N}$ there exists $C>0$ such that, for all
$k\in\mathbb{N}$, $h\in\mathcal{C}^{\infty}_{N}(M)$,

$\left|\smash{\mathcal{L}^{k}h}\right|_{0,q}\leq C\left|h\right|_{0,q}.$

###### Proof.

Let $k\in\mathbb{N}$, $m\in M$, $h\in\mathcal{C}^{\infty}_{N}(M)$ and
$\eta\in\mathcal{C}^{\infty}_{\operatorname{c}}(I_{\delta})$ such that
$\left\|\eta\right\|_{\mathcal{C}^{q}}\leq 1$. We need to estimate

$\ell_{\eta,m}(\mathcal{L}^{k}h)=\int_{-\delta}^{\delta}\eta(u)\cdot\mathcal{L}^{k}h\circ\varphi_{u}^{W}(m)\
du.$

We observe that
$\Phi^{-k}\circ\varphi_{u}^{W}=\varphi_{\lambda^{k}u}^{W}\circ\Phi^{-k}$ and
so

$\mathcal{L}^{k}h\circ\varphi_{u}^{W}=h\circ\Phi^{-k}\circ\varphi_{u}^{W}=h\circ\varphi_{\lambda^{k}u}^{W}\circ\Phi^{-k}.$

Changing variables in the integral ($s=\lambda^{k}u$),

$\ell_{\eta,m}(\mathcal{L}^{k}h)=\lambda^{k}\int_{-\lambda^{k}\delta}^{\lambda^{k}\delta}\eta(\lambda^{-k}s)\cdot
h\circ\varphi_{s}^{W}(m)\ ds.$

We apply Lemma 3.2 and note that $s\mapsto\eta(\lambda^{-k}s)$ is supported on
the interval $(-\lambda^{k}\delta,\lambda^{k}\delta)$. Consequently
$\left|\ell_{\eta,m}(\mathcal{L}^{k}h)\right|\leq C\left|h\right|_{0,q}$. ∎

We observe that $\left|h\right|_{j,q}=\left|V^{j}h\right|_{0,q}$. Hence the
estimate of the above lemma implies also that
$\left|\mathcal{L}^{k}h\right|_{j,q}\leq C\left|h\right|_{j,q}$ for all
$j\in\mathbb{N}$. This suffices to show that the operator $\mathcal{L}$
extends to a continuous operator on $\mathcal{B}_{N}^{p,q}(M)$.

In the following lemma we take advantage of the contraction in the
$V$-direction.

######  Lemma 3.4.

There exists $C>0$ such that, for all $j,q,k\in\mathbb{N}$,
$h\in\mathcal{C}^{\infty}_{N}(M)$,

$\left|\smash{\mathcal{L}^{k}h}\right|_{j,q}\leq
C\lambda^{-jk}\left|h\right|_{j,q}.$

###### Proof.

By definition of the norm and Lemma 3.1,

$\displaystyle\left|\smash{\mathcal{L}^{k}h}\right|_{j,q}$
$\displaystyle=\left|\smash{V^{j}\mathcal{L}^{k}h}\right|_{0,q}$
$\displaystyle=\lambda^{-jk}\left|\smash{\mathcal{L}^{k}(V^{j}h)}\right|_{0,q}.$

Using also the estimate of Lemma 3.3,

$\left|\smash{\mathcal{L}^{k}h}\right|_{j,q}\leq
C\lambda^{-jk}\left|V^{j}h\right|_{0,q}=C\lambda^{-jk}\left|h\right|_{j,q}.\qed$

The above lemma means that $\left\|\mathcal{L}^{k}h\right\|_{p,q}\leq
C\lambda^{-jk}\left\|h\right\|_{p,q}+\sup_{j<p}\left\|\mathcal{L}^{k}h\right\|_{j,q}$.
We now need to get a useful estimate for
$\left\|\mathcal{L}^{k}h\right\|_{j,q}$ when $0\leq j\leq p-1$. We do this in
the following lemma, using the expansion in the $W$-direction.

######  Lemma 3.5.

There exists $C>0$ such that, for all all $j,q,k\in\mathbb{N}$,
$h\in\mathcal{C}^{\infty}_{N}(M)$,

$\left|\smash{\mathcal{L}^{k}h}\right|_{j,q}\leq
C\lambda^{-qk}\left|h\right|_{j,q}+C_{k}\left|h\right|_{j,q+1}.$

###### Proof.

By Lemma 3.1 it suffices to prove the case when $j=0$ because
$\left|\smash{\mathcal{L}^{k}h}\right|_{j,q}=\left|\smash{V^{j}\mathcal{L}^{k}h}\right|_{0,q}=\lambda^{-j}\left|\smash{\mathcal{L}^{k}V^{j}h}\right|_{0,q}$.
Let $k\in\mathbb{N}$, $m\in M$, $h\in\mathcal{C}^{\infty}_{N}(M)$ and
$\eta\in\mathcal{C}^{\infty}_{\operatorname{c}}(I_{\delta})$ such that
$\left\|\eta\right\|_{\mathcal{C}^{q}}\leq 1$. We need to estimate

$\ell_{\eta,m}(\mathcal{L}^{k}h)=\int\eta(u)\cdot\mathcal{L}^{k}h\circ\varphi_{u}^{W}(m)\
du.$

For $\epsilon>0$ we fix $\rho_{\epsilon}$ an $\epsilon$-sized mollifier and
define the smoothed version $\eta_{\epsilon}=\eta\star\rho_{\epsilon}$. This
construction means that

$\left\|\eta-\eta_{\epsilon}\right\|_{\mathcal{C}^{q-1}}\leq
C\epsilon,\quad\left\|\eta_{\epsilon}\right\|_{\mathcal{C}^{q}}\leq
C,\quad\left\|\eta_{\epsilon}\right\|_{\mathcal{C}^{q+1}}\leq C\epsilon^{-1}.$

We choose $\epsilon=\lambda^{-qk}$. We write

$\ell_{\eta,m}(\mathcal{L}^{k}h)=\int\eta_{\epsilon}(u)\cdot\mathcal{L}^{k}h\circ\varphi_{u}^{W}(m)\
du+\int\left(\eta-\eta_{\epsilon}\right)(u)\cdot\mathcal{L}^{k}h\circ\varphi_{u}^{W}(m)\
du.$

For the first integral we have the estimate

$\left|\int\eta_{\epsilon}(u)\cdot\mathcal{L}^{k}h\circ\varphi_{u}^{W}(m)\
du\right|\leq\left\|\eta_{\epsilon}\right\|_{\mathcal{C}^{q+1}}\left|\smash{\mathcal{L}^{k}h}\right|_{0,q+1}\leq
C\epsilon^{-1}\left|\smash{\mathcal{L}^{k}h}\right|_{0,q+1}.$

For the second integral, let
$\tilde{\eta}(u)=\left(\eta-\eta_{\epsilon}\right)(\lambda^{-k}u)$ (similar to
the proof of Lemma 3.3).

$\displaystyle\left\|\tilde{\eta}\right\|_{\mathcal{C}^{q}}$
$\displaystyle\leq\left\|\tilde{\eta}\right\|_{\mathcal{C}^{q}}+\left\|\tilde{\eta}\right\|_{\mathcal{C}^{q-1}}$
$\displaystyle\leq\lambda^{-qk}\left\|\eta-\eta_{\epsilon}\right\|_{\mathcal{C}^{q}}+\left\|\eta-\eta_{\epsilon}\right\|_{\mathcal{C}^{q-1}}$
$\displaystyle\leq C\lambda^{-qk}+C\epsilon.$

Again we use the estimate of Lemma 3.2. Summing the above two estimates
completes the proof of the lemma. ∎

######  Lemma 3.6.

Let $p\in\mathbb{N}_{0}$, $q\in\mathbb{N}$. The operator $\mathcal{L}$ extends
to a continuous operator on $\mathcal{B}_{N}^{p,q}(M)$. Moreover there exists
$C>0$ such that, for every $h\in\mathcal{B}_{N}^{p,q}(M)$, $k\in\mathbb{N}$,

$\left\|\smash{\mathcal{L}^{k}h}\right\|_{p,q}\leq C\left\|h\right\|_{p,q}.$

Furthermore, for any $\tilde{\lambda}\in(1,\lambda)$, there exists $C>0$ such
that, for every $h\in\mathcal{B}_{N}^{p,q}(M)$, $k\in\mathbb{N}$,

$\left\|\smash{\mathcal{L}^{k}h}\right\|_{0,q}\leq
C\tilde{\lambda}^{-qk}\left\|h\right\|_{p,q},$

and, when $p\geq 1$,

$\left\|\smash{\mathcal{L}^{k}h}\right\|_{p,q}\leq
C\tilde{\lambda}^{-\min(p,q)k}\left\|h\right\|_{p,q}+C\left\|h\right\|_{p-1,q+1}.$

###### Proof.

By density, it suffices to prove the inequality for
$h\in\mathcal{C}^{\infty}_{N}(M)$. Let $p,q\in\mathbb{N}$. Combining the
estimates of Lemma 3.4 and Lemma 3.5 gives the following estimate.

$\displaystyle\left\|\smash{\mathcal{L}^{k}h}\right\|_{p,q}$
$\displaystyle\leq\left|\smash{\mathcal{L}^{k}h}\right|_{p,q}+\sup_{0\leq
j\leq p-1}\left|\smash{\mathcal{L}^{k}h}\right|_{j,q}$ $\displaystyle\leq
C\lambda^{-pk}\left|h\right|_{p,q}+C\lambda^{-qk}\left\|h\right\|_{p-1,q}+C_{k}\left\|h\right\|_{p-1,q+1}$
$\displaystyle\leq
C\lambda^{-\min(p,q)k}\left\|h\right\|_{p,q}+C_{k}\left\|h\right\|_{p-1,q+1}.$

The uniform in $k$ estimate of Lemma 3.3, together with iterating the above
estimate allows the removal of the $k$ dependence in the second term. ∎

######  Lemma 3.7.

Let $N\in\mathbb{Z}$, $p,q\in\mathbb{N}$. The operator
$\mathcal{L}:\mathcal{B}_{N}^{p,q}(M)\to\mathcal{B}_{N}^{p,q}(M)$ has spectral
radius not greater than $1$ and essential spectral radius not greater than
$\lambda^{-\min{\\{p,q\\}}}$. Moreover the spectral radius of
$\mathcal{L}:\mathcal{B}_{0}^{p,q}(M)\to\mathcal{B}_{0}^{p,q}(M)$ is equal to
$1$.

###### Proof.

The estimates on the spectral and essential spectral radius follow from Lemma
3.6 and Lemma 2.4 by Hennion’s argument [18]. The observation that constant
functions are contained within $\mathcal{C}^{\infty}_{0}(M)$ and are invariant
under $\mathcal{L}$ implies that the spectral radius of
$\mathcal{L}:\mathcal{B}_{0}^{p,q}(M)\to\mathcal{B}_{0}^{p,q}(M)$ is equal to
$1$. ∎

##  4\. Peripheral spectrum

This section is devoted to determining the spectrum of the transfer operator
when restricted to the kernel of $V$. The motive for this will become clear in
Section 5, particularly Lemma 5.1. We will see that the kernel consists of a
finite number of eigenvalues of absolute value $\lambda^{\frac{1}{2}}$. As
before we consider $\Phi:M\to M$, as introduced in Section 1 and
$N\in\mathbb{Z}\setminus\\{0\\}$, chosen and fixed for the entire section. For
convenience let,

$\mathcal{I}_{N}=\\{h\in\mathcal{B}_{N}^{p,q}(M):Vh=0\\}.$

Subsequently we will see that $\mathcal{I}_{N}$ does not depend on $p,q$, so
justifying this choice of notation. Recall that, according to Lemma 2.2, the
anisotropic spaces are continuously embedded in the space of distributions. To
proceed it is convenient to show that the space $\mathcal{B}_{N}^{p,q}(M)$ is
sufficiently large that we see the “invariant distributions” which were
identified by Flaminio & Forni [10]. Recall, as introduced in Section 1,
$\mathcal{D}_{N}^{1}(M)$ denotes the space of distributions on
$\mathcal{C}^{1}(M)$, supported on $\mathcal{C}^{1}_{N}(M)$.

######  Lemma 4.1.

Suppose that $\theta\in\mathcal{D}_{N}^{1}(M)$ and that $\theta(Vh)=0$ for all
$h\in\mathcal{C}^{\infty}(M)$. Then, for every $p\in\mathbb{N}_{0}$,
$q\in\mathbb{N}$ there exists $g\in\mathcal{B}_{N}^{p,q}(M)$ such that
$\theta=\iota g$.

###### Proof.

Since, for all $q\in\mathbb{N}$,
$\mathcal{B}_{N}^{p,q}(M)\subset\mathcal{B}_{N}^{p,1}(M)$ (Lemma 2.3), it
suffices to prove the case $q=1$. Moreover it suffices to work locally in
$\mathbb{R}^{2}$. Consequently, without loss of generality, we work with the
seminorm,

$\left|h\right|_{p,1}=\sup_{\eta,x}\left|\int_{-\delta}^{\delta}\tfrac{\partial^{p}h}{\partial
x_{1}^{p}}(x_{1},x_{2}+u)\cdot\eta(u)\ du\right|$

where the supremum is over $\eta\in\mathcal{C}^{1}_{c}(\mathbb{R})$,
$\left\|\eta\right\|_{\mathcal{C}^{1}}\leq 1$ and
$x=(x_{1},x_{2})\in\mathbb{R}^{2}$. Moreover we assume that
$\theta(\frac{\partial h}{\partial x_{1}})=0$ for all
$h\in\mathcal{C}^{\infty}(\mathbb{R}^{2})$. Let
$\rho\in\mathcal{C}^{\infty}(\mathbb{R}^{2},\mathbb{R}_{+})$ be the standard
mollifier (in particular $\rho$ is supported on $\\{\left|x\right|<1\\}$ and
$\int\rho(x)\ dx=1$). For $\epsilon>0$, let
$\rho_{\epsilon}(x)=\epsilon^{-2}\rho(\epsilon^{-1}x)$. The smoothed version
of a function is defined as $h_{\epsilon}(x)=\int\rho_{\epsilon}(x-y)\cdot
h(y)\ dy$. (See, e.g., [6, §4] for standard details related to the mollifier
and smoothing which we will use in the remainder of this proof.) Additionally,
for $x\in\mathbb{R}^{2}$, let $\rho_{\epsilon,x}(y)=\rho_{\epsilon}(y-x)$.
Observe that $\rho_{\epsilon,x}$ is an $\epsilon$-sized “bump” centred at $x$.
Let $g_{\epsilon}\in\mathcal{C}^{\infty}(\mathbb{R}^{2})$ be defined as

$g_{\epsilon}(x)=\theta(\rho_{\epsilon,x}).$

We will show that $g_{\epsilon}$ converges to $g\in\mathcal{B}_{N}^{p,1}(M)$
and satisfies the required properties. That $\theta(\frac{\partial h}{\partial
x_{1}})=0$ implies that $\tfrac{\partial g_{\epsilon}}{\partial x_{1}}=0$.
Furthermore, observe that, for any $h\in\mathcal{C}^{\infty}(\mathbb{R}^{2})$,

$\displaystyle\langle g_{\epsilon},h\rangle$
$\displaystyle=\int\theta(\rho_{\epsilon,x})\cdot h(x)\
dx=\int\theta(\rho_{\epsilon,x}\cdot h(x))\ dx$
$\displaystyle=\theta\left(\int\rho_{\epsilon,x}\cdot h(x)\
dx\right)=\theta\left(h_{\epsilon}\right)$

since $\left(\int\rho_{\epsilon,x}\cdot h(x)\
dx\right)(y)=\int\rho_{\epsilon}(y-x)\cdot h(x)\ dx=h_{\epsilon}(y)$.
Consequently $\langle g_{\epsilon},h\rangle-\theta(h)=\theta(h_{\epsilon}-h)$.
Since $h_{\epsilon}\to h$ this means that $\langle
g_{\epsilon},\cdot\rangle\to\theta(\cdot)$ in the space of distributions of
order $1$. Using the invariance of $g_{\epsilon}$ in the first coordinate we
need only consider the case $p=0$ and,

$\int_{-\delta}^{\delta}g_{\epsilon}(x_{1},x_{2}+u)\cdot\eta(u)\
du=\iint_{\mathbb{R}^{2}}g_{\epsilon}(y_{1},y_{2})\cdot\rho(y_{1})\cdot\eta(y_{2}-x_{2})\
dy_{2}dy_{1}.$

The “test function” in this integral,
$(y_{1},y_{2})\mapsto\rho(y_{1})\cdot\eta(y_{2}-x_{2})$ has uniformly bounded
$\mathcal{C}^{1}$ norm. Consequently the convergence in the space of
distributions proves that the norm is uniformly bounded and so $g_{\epsilon}$
converges in the anisotropic norm $\left|\cdot\right|_{p,1}$. ∎

######  Lemma 4.2.

$\mathcal{I}_{N}$ has dimension $K\left|N\right|$.

###### Proof.

Let $\mathcal{J}_{N}$ denote the space of $V$-invariant distributions, i.e.,
the set of $D\in\mathcal{C}^{\infty}(M)^{\prime}$ such that $D(Vh)=0$ for all
$h\in\mathcal{C}^{\infty}(M)$ and supported on $\mathcal{C}^{\infty}_{N}(M)$.
It is known [10, Proposition 4.4] that $\mathcal{J}_{N}$ has dimension
$K\left|N\right|$ and comprises of distributions of Sobolev order
$\frac{1}{2}$. We know (Lemma 2.2) that $\iota\mathcal{I}_{N}$ is a subset of
$\mathcal{C}^{\infty}(M)^{\prime}$ and that (Lemma 4.1) $\iota\mathcal{I}_{N}$
contains $\mathcal{J}_{N}$. Consequently the dimension of $\mathcal{I}_{N}$ is
equal to the dimension of $\mathcal{J}_{N}$. ∎

######  Lemma 4.3.

$\lambda^{-1/2}\mathcal{L}$ is an isometry on $\mathcal{I}_{N}$.

###### Proof.

Lemma 4.2 tells that $\mathcal{I}_{N}$ is finite dimensional and every norm is
equivalent. Consequently this result follows from the known result [10,
Proposition 4.8] even though the reference uses a different norm. ∎

######  Lemma 4.4.

There exists a set of $K\left|N\right|$ unit complex numbers $\\{\mu_{j}\\}$
such that the spectrum of $\mathcal{L}$ restricted to $\mathcal{I}_{N}$ is
$\\{\lambda^{-\frac{1}{2}}\mu_{j}\\}_{j=1}^{K\left|N\right|}$, repeated
according to algebraic multiplicity.

###### Proof.

This result follows from the fact that $\mathcal{L}$ is an isometry (Lemma
4.3) on the subspace $\mathcal{I}_{N}$ which has dimension $K\left|N\right|$
(Lemma 4.2). ∎

##  5\. Zooming in

This section is devoted to the part of the argument which allows us to derive
the full spectrum of
$\mathcal{L}:\mathcal{B}_{N}^{p,q}(M)\to\mathcal{B}_{N}^{p,q}(M)$, $N\neq 0$
from the peripheral spectrum proven in Section 4. As before,
$M=\raisebox{-1.99997pt}{$\Gamma$}\diagdown\mathbb{H}$ and
$N\in\mathbb{Z}\setminus\\{0\\}$ are considered chosen and fixed for the
entire section.

We will repeatedly take advantage of the lemma of Baladi & Tsujii [2, Lemma
A.1] which tells us when the point spectrum of an operator considered on two
different Banach spaces will coincide. We repeat here the full statement.

######  Lemma ([2, Lemma A.1]).

Let $B$ be a separated topological linear space and let $B_{1}$ and $B_{2}$ be
Banach spaces that are continuously embedded in $B$. Suppose further that
there is a subspace $B_{0}\subset B_{1}\cap B_{2}$ that is dense both in the
Banach spaces $B_{1}$ and $B_{2}$. Let $L:B\to B$ be a continuous linear map,
which preserves the subspaces $B_{0}$, $B_{1}$ and $B_{2}$. Suppose that the
restriction of $L$ to $B_{1}$ and $B_{2}$ are bounded operators whose
essential spectral radii are both strictly smaller than $\rho>0$. Then the
eigenvalues of $L:B_{1}\to B_{1}$ and $L:B_{2}\to B_{2}$ in
$\\{z\in\mathbb{C}:\left|z\right|>\rho\\}$ coincide. Furthermore the
corresponding generalized eigenspaces coincide and are contained in the
intersection $B_{1}\cap B_{2}$.

In our setting $B_{0}$ is the space of $\mathcal{C}^{\infty}$ functions whilst
$B$ is the space of distributions (Lemma 2.2). The following tells us that the
outer part of the spectrum is given by Lemma 4.4.

######  Lemma 5.1.

Let $p,q\in\mathbb{N}$. The spectrum of
$\mathcal{L}:\mathcal{B}_{N}^{p,q}(M)\to\mathcal{B}_{N}^{p,q}(M)$, $N\neq 0$,
restricted to $\\{\left|z\right|>\lambda^{-1}\\}$ is equal to the spectrum of
${\left.\mathcal{L}\right|}_{\mathcal{I}_{N}}$.

###### Proof.

Suppose that $z$ is an eigenvalue of
$\mathcal{L}:\mathcal{B}_{N}^{p,q}(M)\to\mathcal{B}_{N}^{p,q}(M)$ and that
$\left|z\right|>\lambda^{-1}$. Consequently there exists
$h_{z}\in\mathcal{B}_{N}^{p,q}(M)$ such that $\mathcal{L}h_{z}=zh_{z}$. By
Lemma 3.1,

$\lambda V\mathcal{L}h_{z}=\lambda z(Vh_{z})=\mathcal{L}(Vh_{z}).$

This means that, either $\lambda z$ is an eigenvalue of
$\mathcal{L}:\mathcal{B}_{N}^{p-1,q}(M)\to\mathcal{B}_{N}^{p-1,q}(M)$ or that
$Vh_{z}=0$. However $\left|\lambda z\right|>1$ but the spectral radius of
$\mathcal{L}$ is at most 1 (Lemma 3.7). This means that $Vh_{z}=0$, i.e.,
$h_{z}\in\mathcal{I}_{N}$.

For the other direction, observe that Lemma 4.4 in particular implies that the
spectrum of ${\left.\mathcal{L}\right|}_{\mathcal{I}_{N}}$ is a subset of
$\\{\left|z\right|=\lambda^{-1/2}\\}$. ∎

######  Lemma 5.2.

Let $p,q,k\in\mathbb{N}$, $p,q>k$. The spectrum of
$\mathcal{L}:\mathcal{B}_{N}^{p,q}(M)\to\mathcal{B}_{N}^{p,q}(M)$, $N\neq 0$,
restricted to $\\{z:\lambda^{-(k+1)}<\left|z\right|\leq\lambda^{-k}\\}$ is
contained within
$\\{z:\lambda^{k}z\in\operatorname{Spec}({\left.\mathcal{L}\right|}_{\mathcal{I}_{N}})\\}$.

###### Proof.

We take advantage of the fact that the point spectrum is independent on the
Banach space [2, Lemma A.1]. We will prove, by induction, that for all
$k\in\mathbb{N}_{0}$,

$\text{if $z\in\operatorname{Spec}(\mathcal{L})$ and
$\lambda^{-(k+1)}<\left|z\right|\leq\lambda^{-k}$},\quad\text{then
$\lambda^{k}z\in\operatorname{Spec}({\left.\mathcal{L}\right|}_{\mathcal{I}_{N}})$}.$

The case $k=0$ is given by Lemma 5.1. We now suppose the statement is already
proven for $k$. Suppose that $z$ is an eigenvalue of $\mathcal{L}$ and that
$\lambda^{-(k+2)}<\left|z\right|\leq\lambda^{-(k+1)}$. Let $h_{z}\neq 0$ be
such that $\mathcal{L}h_{z}=zh_{z}$. By Lemma 3.1,

$\lambda V\mathcal{L}h_{z}=\lambda z(Vh_{z})=\mathcal{L}(Vh_{z}).$

This means that, either $\lambda z$ is an eigenvalue of $\mathcal{L}$ or
$Vh_{z}=0$. In the second case, $h_{z}\in\ker(V)$ and so, by Lemma 5.1,
$\left|z\right|>\lambda^{-1}$ but this contradicts the fact that
$\left|z\right|\leq\lambda^{-(k+1)}\leq\lambda^{-1}$. This means that the
first case is the only possibility. Consequently we know that $\lambda
z\in\operatorname{Spec}(\mathcal{L})$ and $\lambda^{-(k+1)}<\left|\lambda
z\right|\leq\lambda^{-k}$. We then apply the inductive hypothesis and
conclude. ∎

Next we take advantage of the operator $W$ in order to upgrade Lemma 5.2 to
equality. The technique we will use has some similarity to the argument used
for pseudo Anosov maps [8]. It is very different to the technique of Flaminio
& Forni [10, A.3] who took advantage of a basis of Hermite functions to write
a formal inverse of $V$ and then further argue for the “iterated invariant
distributions” (see also [13]).

######  Lemma 5.3.

Let $p,q\in\mathbb{N}$. The spectrum of
$\mathcal{L}:\mathcal{B}_{N}^{p,q}(M)\to\mathcal{B}_{N}^{p,q}(M)$, $N\neq 0$,
restricted to $\\{z:\lambda^{-(k+1)}<\left|z\right|\leq\lambda^{-k}\\}$ is
equal to
$\\{z:\lambda^{k}z\in\operatorname{Spec}({\left.\mathcal{L}\right|}_{\mathcal{I}_{N}})\\}$.

###### Proof.

In light of Lemma 5.2, we need only prove that $z$ in the spectrum implies
that $\lambda^{-1}z$ is also in the spectrum. For this we take advantage of
the fact, established in Lemma 2.7, that
$Y:\mathcal{B}_{N}^{p,q}(M)\to\mathcal{B}_{N}^{p,q+1}(M)$ is injective. Let
$h_{z}\neq 0$ be such that $\mathcal{L}h_{z}=zh_{z}$. By Lemma 3.1,

$\lambda^{-1}W\mathcal{L}h_{z}=\lambda^{-1}z(Wh_{z})=\mathcal{L}(Wh_{z}).$

Here again we take advantage of the fact that the point spectrum is
independent on the Banach space [2, Lemma A.1]. By Lemma 2.7 $Wh_{z}\neq 0$
and so $\lambda^{-1}z$ is in the spectrum. ∎

##  Appendix A Partially hyperbolic automorphisms

In this section we discuss the explicit form of partially hyperbolic
automorphisms. For our present purposes we say that an automorphism $\Phi:M\to
M$ is _partially hyperbolic with neutral centre_ if the tangent bundle admits
a splitting $TM=E_{s}\oplus E_{c}\oplus E_{u}$ into one dimensional sub-
bundles such that: $E_{c}$ corresponds to $Z$ and $\Phi$ is an orientation-
preserving isometry on this bundle; $\Phi$ is uniformly contracting on $E_{s}$
and uniformly expanding on $E_{u}$. In this section, to match with the
pertinent references, we work in _polarised_ coordinates, i.e.,
$\mathbb{H}^{\mathrm{pol}}$ is equal to $\mathbb{R}^{3}$ with the group law

$(x,y,z)*(x^{\prime},y^{\prime},z^{\prime})=(x+x^{\prime},y+y^{\prime},z+z^{\prime}+xy^{\prime}).$

We recall the following result which describes the structure of such
automorphisms on Heisenberg nil-manifolds.

######  Lemma A.1 (Shi [20, 21]).

Let $M=\raisebox{-1.99997pt}{$\Gamma$}\diagdown\mathbb{H}^{\mathrm{pol}}$. The
function $\Phi:M\to M$ is an automorphism, partially hyperbolic with neutral
centre, if and only if, it has the form, $\Phi=\exp\circ\,\phi\circ\exp^{-1}$
where333We identify the Lie algebra with $\mathbb{R}^{3}$ using the basis
$\\{X,Y,Z\\}$ where,  $X=\left(\begin{smallmatrix}0&1&0\\\ 0&0&0\\\
0&0&0\end{smallmatrix}\right),\quad Y=\left(\begin{smallmatrix}0&0&0\\\
0&0&1\\\ 0&0&0\end{smallmatrix}\right),\quad
Z=\left(\begin{smallmatrix}0&0&1\\\ 0&0&0\\\ 0&0&0\end{smallmatrix}\right).$

(7) $\phi=\begin{pmatrix}a&b&0\\\ c&d&0\\\
\tfrac{ac}{2}+\ell&\tfrac{bd}{2}+m&1\end{pmatrix}$

for some $a,b,c,d,\ell,m\in\mathbb{Z}$ such that
$\det\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)=1$ and
$\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)$ has eigenvalues
$\lambda,\lambda^{-1}$ for some $\lambda>1$. Equivalently,

(8) $\displaystyle\Phi(x,y,z)$
$\displaystyle=\left(ax+by,cx+dy,z+\tau(x,y)\right),$ $\displaystyle\tau(x,y)$
$\displaystyle=\tfrac{ac}{2}x^{2}+bcxy+\tfrac{bd}{2}y^{2}+\left(\tfrac{ac}{2}+\ell\right)x+\left(\tfrac{bd}{2}+m\right)y.$

###### Sketch of proof.

For full details of these calculations consult the work of Shi [20, 21], here
we present an overview. There is a one-to-one correspondence between the
automorphisms of the Lie algebra and the automorphisms of $\mathbb{H}$ [12,
Proposition 1.21]. The Lie bracket must be preserved by any automorphism of
the Lie algebra which gives several restrictions on the possible form. The
matrix (8) represents the associated automorphism of the Lie algebra. Applying
the exponential function, the formula of the automorphism on $\mathbb{H}$ can
then be deduced. Further observing that the automorphism must preserve the
lattice $\Gamma$ and be an orientation preserving isometry in $Z$ gives the
final restrictions on the form of the automorphism. ∎

An automorphism associated to $\left(\begin{smallmatrix}2&1\\\
1&1\end{smallmatrix}\right)$, the classic Anosov map of the torus, is given by
Lemma A.1 with the coefficients $a=2,b=c=d=1,l=m=0$.

Let $\Phi:M\to M$ be an automorphism, partially hyperbolic with neutral
centre. I.e., $a,b,c,d,\ell,m\in\mathbb{Z}$ are fixed and satisfy the
requirements detailed in Lemma A.1. Let $\left(\begin{smallmatrix}\alpha\\\
\beta\end{smallmatrix}\right)$, $\left(\begin{smallmatrix}\beta\\\
-\alpha\end{smallmatrix}\right)$ be the normalised eigenvectors of
$\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)$. Without loss of
generality, we suppose that $\alpha,\beta>0$ are such that
$\alpha^{2}+\beta^{2}=1$ and

$\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\left(\begin{smallmatrix}\alpha\\\
\beta\end{smallmatrix}\right)=\lambda\left(\begin{smallmatrix}\alpha\\\
\beta\end{smallmatrix}\right),\quad\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\left(\begin{smallmatrix}\beta\\\
-\alpha\end{smallmatrix}\right)=\lambda^{-1}\left(\begin{smallmatrix}\beta\\\
-\alpha\end{smallmatrix}\right).$

Let

$V=\alpha X+\beta Y+\gamma Z,\quad W=-\beta X+\alpha Y+\gamma^{\prime}Z$

where

$\displaystyle\gamma$
$\displaystyle=\tfrac{1}{\lambda-1}\left(\alpha(\tfrac{ac}{2}+\ell)+\beta(\tfrac{bd}{2}+m)\right),$
$\displaystyle\gamma^{\prime}$
$\displaystyle=\tfrac{1}{1-\lambda^{-1}}\left(\beta(\tfrac{ac}{2}+\ell)-\alpha(\tfrac{bd}{2}+m)\right).$

######  Lemma A.2.

Let $V$, $W$ be defined as above. Then $[V,W]=Z$, $[V,Z]=[W,Z]=0$. I.e.,
$\\{V,W,Z\\}$ is a Heisenberg frame. Moreover

$\Phi_{*}V=\lambda V,\quad\Phi_{*}W=\lambda^{-1}W.$

###### Proof.

The commutation relations follow from the commutation relations for $X,Y,Z$,
together with the chosen normalization. The two equalities are verified using
the matrix form of the automorphism of the Lie algebra (7). ∎

## Acknowledgements

We are grateful to Giovanni Forni and Carlangelo Liverani for several
enlightening discussions. O.B. is particularly grateful to Lucia Simonelli
who, several years ago, searched out details, figured out the calculations and
explained much of this material to them.

M.K acknowledges the Center of Excellence “Dynamics, mathematical analysis and
artificial intelligence” at the Nicolaus Copernicus University in Toruń and
University of Rome Tor Vergata for hospitality. This work was partially
supported by PRIN Grant “Regular and stochastic behaviour in dynamical
systems” (PRIN 2017S35EHN) and the MIUR Excellence Department Project
MatMod@TOV awarded to the Department of Mathematics, University of Rome Tor
Vergata.

## Statements and Declarations

The authors have no relevant financial or non-financial interests to disclose.
Data sharing is not applicable to this article as no datasets were generated
or analysed during the current study.

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